research article analysis of synchronization impairments
TRANSCRIPT
Research ArticleAnalysis of Synchronization Impairments for Cooperative BaseStations Using OFDM
Konstantinos Manolakis1 Christian Oberli2 Volker Jungnickel3 and Fernando Rosas4
1Huawei Technologies European Research Center 80892 Munich Germany2Pontificia Universidad Catolica de Chile 7820436 Santiago Chile3Technische Universitat Berlin 10623 Berlin Germany4KU Leuven 3001 Leuven Belgium
Correspondence should be addressed to Konstantinos Manolakis konstantinosmanolakishuaweicom
Received 21 August 2014 Revised 10 February 2015 Accepted 12 February 2015
Academic Editor Feifei Gao
Copyright copy 2015 Konstantinos Manolakis et alThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
Base station cooperation is envisioned as a key technology for future cellular networks as it has the potential to eliminate intercellinterference and to enhance spectral efficiency To date there is still lack of understanding of how imperfect carrier and samplingfrequency synchronization between transmitters and receivers limit the potential gains and what the actual system requirementsare In this paper OFDM signal model is established for multiuser multicellular networks describing the joint effect of multiplecarrier and sampling frequency offsets It is shown that the impact of sampling offsets is much smaller than the impact of carrierfrequency offsets The model is extended to the downlink of base-coordinated networks and closed-form expressions are derivedfor the mean power of usersrsquo self-signal interuser and intercarrier interference whereas it is shown that interuser interference isthe main source of degradation The SIR is inverse to the base stationsrsquo carrier frequency variance and to the square of time sincethe last precoder update whereas it grows with the number of base stations and drops with the number of users Through userselection the derived SIR upper bound can be approached Finally system design recommendations for meeting synchronizationrequirements are provided
1 Introduction
Base station cooperation also known as coordinated multi-point (CoMP) is an ambitious multiple-antenna techniquewhere antennas of multiple distributed base stations andthose of multiple terminals served within those cells areconsidered as a distributed multiple-input multiple-output(MIMO) system [1ndash3] In the downlink also known as jointtransmission (JT) CoMP signal preprocessing at the basestations is applied to eliminate the intercell interference andto enhance the spectral efficiency In the simplest case datasymbols are precoded with the pseudoinverse of the MIMOchannel matrix this method is known as zero-forcing (ZF)precoding [4] Using ZF precoding system performancebecomes close to optimal in the high signal-to-noise ratio(SNR) regime as shown in [5] Deployment concepts forJT CoMP and field trial results have been reported in [6]whereas recent progress can be found in [7 8] The role of
CoMP and integration aspects into next generation cellularsystems are highlighted in [9] Finally an overview oncooperative communications can be found in [10]
The combination of MIMO techniques with orthogonalfrequency division multiplexing (OFDM) has been a suc-cessful concept for broadband cellular networks and hasenabled a significant increase of the spectral efficiency duringthe last years [11 12] However it quickly became clear thatprecise synchronization is vital for realizing the potential ofMIMO-OFDM systems It is known from [13] that the carrierfrequency offset (CFO) causes intercarrier interference (ICI)as well as a phase drift on all OFDM subcarriers knownas common phase error (CPE) The sampling frequencyoffset (SFO) is also a source of ICI and implies a phasedrift that grows linearly with frequency thus affecting eachsubcarrier differently Accurate maximum likelihood (ML)tracking algorithms have been developed and optimized forsingle-user point-to-point MIMO-OFDM in [14] whereas
Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 125653 14 pageshttpdxdoiorg1011552015125653
2 International Journal of Antennas and Propagation
synchronization for multiuser MIMO within one cell hasbeen studied in [15] For the OFDM-based multiuser uplinka signal model and compensation techniques have beendeveloped in [16]
Considering distributed JT CoMP cooperative base sta-tions are located at different sites which implies that theirfrequency up- and downconverters are driven by theirown local oscillators while sampling frequencies also differamong them Signal modeling of JT CoMP with individualoffsets in carrier and sampling frequencies in [17] revealedthat orthogonality between multiple usersrsquo data signals ismisaligned and interuser interference (IUI) arises Firstinsights into the performance degradation were obtained bynumerical evaluation In chapter 8 of [18] the sensitivityof CoMP to the CFO was analyzed for a scenario withtwo cooperating base stations Similar observations havebeen reported in [19 20] whereas methods for estimatingmultiple CFOs based on training signals have been developedin [21] The problem of nonsynchronized cooperating basestations has been also investigated in [22ndash24] where thefocus has been on how to estimate and compensate themultiple CFOs In [25] propagation delay differences werealso included for transmissions fromdistributed base stationswithmultiple CFOs In [26] a scheme for synchronizing basestations has been proposed based on a time-slotted round-trip carrier synchronization protocol The implementationof Global Positioning System- (GPS-) based synchronizationfor distributed base stations in an outdoor testbed has beenreported by the authors in [27] More recently an over-the-air synchronization protocol has been proposed in [28]which is also applicable for networks with a large numberof access points A survey on physical layer synchroniza-tion for distributed wireless networks can be found in[29]
The first objective of the present paper is to investi-gate the synchronization requirements for base-coordinatedmulticellular MIMO networks A major contribution of thiswork is the derivation of an exact signal model capturingthe joint effect of multiple CFOs and SFOs at transmittersand receivers in a MIMO-OFDM system and over the timeBased on this model it is shown that the impact of the SFO isnegligible compared to the one of the CFO Application of themodel to the distributed CoMP downlink with ZF precodingleads to analytical closed-form expressions for the meanpower of the usersrsquo self-signal interuser and intercarrierinterferences It is found that the interuser interference isthe dominant source of signal degradation and that syn-chronization requirements for cooperating base stations arevery high compared to the ones in single-cell transmissionThe mean signal-to-interference ratio (SIR) is analyzed andis approximately found to degrade quadratically with timeand to be inversely proportional to the variance of thebase stationsrsquo CFO The SIR further grows with the numberof base stations and drops with the number of users Inaddition to the SIR analysis for the Rayleigh fading channelan SIR upper bound is derived which can be approachedby appropriate user selection Finally recommendations forpractical synchronization of distributedwireless networks aregiven
The paper is organized as follows In Section 2 a generalsignal model for a MIMO-OFDM communication systemin the presence of multiple CFOs and SFOs is derived InSection 3 the model is applied to the CoMP downlink andexpressions are derived formobile usersrsquo self-signal interuserand intercarrier interferences Analysis in Section 4 leads toclosed-form expressions for the mean power of the abovesignals and the resulting SIR The system performance isevaluated analytically and verified bymeans of simulations inSection 5 Synchronization requirements are established andpractical methods to fulfill them are discussed in Section 6Finally conclusions are summarized in Section 7
2 General Signal Model for DistributedMIMO-OFDM Systems
In the following a distributed MIMO network is consideredwith an arbitrary number of antenna branches at every basestation and at every userThe cellular network usesOFDM forthe air interface with119873
119904subcarriers which are indexed with
119896 in the range minus1198731199042 119873
1199042minus1 An entireOFDMsymbol
is 119873119892samples long equal to 119873
119904samples plus the number
of samples of the cyclic prefix Integer 119899 indexes successiveOFDM symbols and is hence a measure of time
Each base station and each mobile are assumed to havetheir own carrier and sampling frequency within typicalranges The total number of transmit branches is 119873
119905 Each
transmit branch (can be a base station in the downlink or auser in the uplink) denoted by subindex 119894 has its individualsampling period119879
119894 carrier frequency119891
119894and respective initial
phase parameters 120591119894and 120593
119894 In Figure 1 it is shown for a
point-to-point transmission how sampling and carrier offsetsmisalign analog-to-digital and digital-to-analog conversionas well as frequency conversion respectively The corre-sponding receiver parameters are denoted as 119879
119895 119891
119895 120591
119895 and
120593119895 while symbol 120485 is used for radicminus1 The digital modulation
of subcarrier 119896 on transmit branch 119894 is represented by thecomplex-valued symbol 119883
119894(119896) Due to different sampling
timings between transmit branches of different base stations(downlink) or among mobile users (uplink) the intercarrierspacing is transmit-branch-specific and measures 120575
119894=
(119873119904119879119894)minus1 Hertz The ideal carrier frequency is denoted by 119891
119888
and the ideal sampling period with 119879 For any transmitter orreceiver its CFO and SFO are defined as the the deviationfrom the ideal carrier frequency and sampling period respec-tively The complex baseband-equivalent frequency responseof the passband channel between transmitter 119894 and receiver119895 at frequency 119891 is denoted by119867
119895119894(119891) It includes frequency-
flat path loss and shadow fading as well as frequency-selectivesmall-scale fading
By following similar arguments as the ones leading toequation (8) in [14] and by considering the clarificationin [30] that is corrections in magnitude in order to keepsignal energies consistent it is found that the spectrum of theOFDM signal observed at any given receive branch 119895 has theform
119884119895(119891) = 119880
119895(119891 119896) + 119880
119895(119891 119896) + 119873
119895(119891) (1)
International Journal of Antennas and Propagation 3
Baseband processing
Baseband processing
Transmitter Receiver
DA AD
Ti Tjfi fi
Figure 1 Transmitter and receiver have individual sampling periods 119879119894and 119879
119895for digital-to-analog and analog-to-digital conversion and
individual carrier frequencies 119891119894and 119891
119895for upconversion and downconversion
where 119880119895(119891 119896) represents the continuous-frequency spec-
trum of the received multiuser signal at receive antenna 119895for transmitted subcarrier 119896 (on-carrier signal) It is notedthat 119880
119895(119891 119896) is the received spectrum of the multiuser signal
of all other subcarriers ] = 119896 that is the multiuser ICI(it is arbitrary which subcarrier is designated as 119896 but ouranalysis requires to single out one of them) The additivewhite Gaussian noise (AWGN) contributed by the front-endof receive antenna 119895 is denoted by119873
119895(119891)The above terms are
given by
119880119895(119891 119896) = 119879
119895
119873119905
sum
119894=1
119890120485(120593119895minus120593119894)
120573119895119894(119891 119896)
sdot 119883119894(119896)119867
119895119894(119896120575
119894+ 119891
119894minus 119891
119888)
(2)
119880119895(119891 119896) = 119879
119895
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119905
sum
119894=1
119890120485(120593119895minus120593119894)
120573119895119894(119891 ])
sdot 119883119894(])119867
119895119894(]120575
119894+ 119891
119894minus 119891
119888)
(3)
with
120573119895119894(119891 119896) = 119890
minus1204852120587(119891119895minus119891119894)(119899119873119892119879119895+120591119895)
sdot 119890minus1204852120587119896120575
119894(119899119873119892119879119894+120591119894minus119899119873119892119879119895minus120591119895)
sdot 119890minus120485120587(119891+119891
119895minus119891119894minus119896120575119894)119879119895(119873119904minus1)
sdot
sin [120587 (119891 + 119891119895minus 119891
119894minus 119896120575
119894) 119879
119895119873119904]
sin [120587 (119891 + 119891119895minus 119891
119894minus 119896120575
119894) 119879
119895]
(4)
The exponential terms in (4) are phase shifts due to carrierand sampling frequency misalignment while the fractionalterm describes the loss of orthogonality among OFDMsubcarriers causing a leakage of the signal transmitted onsubcarrier 119896 Note that 119867
119895119894(119896120575
119894+ 119891
119894minus 119891
119888) expresses the
channel at the frequency of subcarrier 119896 plus a shift due tothe transmitterrsquos SFO and CFO
Themodel given by (1) through (4) makes no assumptionabout the synchronization among branches and it is gen-eral for any MIMO-OFDM communication also includingcellular networks with base station cooperation It describeshow successive OFDM symbols are degraded under constantand uncompensated CFOs and SFOs as time goes by that
is OFDM symbol index 119899 grows The CFO must be smallerthan half of a subcarrier spacing For typical mobile receiversthis implies that an earlier coarse frequency synchronizationstage has succeeded which we assume henceforth RegardingSFO the expressions are valid well beyond the typical rangespecified for SFO in commercial OFDM systems It is to benoted however that for a given SFO the FFT window ofeach receiver branch does eventually drift away to a point atwhich intersymbol interference (ISI) arises between OFDMsymbols From that point on severe degradation ensues andthe model stops being valid It is also implicit in our modelthat OFDM symbol timing has been acquired in a priorstage The result assumes that the time dispersion of all theMIMOchannel impulse responses also fromdistributed basestations and mobile users is shorter than the OFDM cyclicprefix in use
Returning to the model it can be observed in (2) and(3) that the phase offsets due to 120591
119894 120591
119895 120593
119894 and 120593
119895may be
considered without loss of generality as part of the channelIt follows that we may choose 120591
119894= 0 120591
119895= 0 120593
119894= 0 and
120593119895= 0 We also point out that in a practical implementation
for the 119895th OFDM receiver branch the output of its FFTcorresponds to a sequence of samples of 119884
119895(119891) taken at
frequencies 119891 = 119897119873119904119879119895= 119897120575
119895 where 119897 is the subcarrier
index (minus1198731199042 le 119897 le 119873
1199042 minus 1) and 120575
119895is the receiver-side
intercarrier spacing Note that 119897 points to a slightly differentfrequency at each receive branch due to the different SFO andgenerally also to a different frequency than that pointed atby index 119896 at the various transmit branches 119894 Imposing theabove conditions on (2) and (3) and focusing on an arbitraryreceived subcarrier 119897 = 119896 we obtain the following discrete-frequency signal model
119884119895(119896) = 119880
119895(119896) + 119880
119895(119896) + 119873
119895(119896) (5)
with
119880119895(119896) =
119873119905
sum
119894=1
120573119895119894(119896 119896)119883
119894(119896)119867
119895119894(119896) (6)
119880119895(119896) =
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119905
sum
119894=1
120573119895119894(119896 ]) 119883
119894(])119867
119895119894(]) (7)
4 International Journal of Antennas and Propagation
120573119895119894(119896 ]) = 119890minus1204852120587119899119873119892119879119895(119891119895minus119891119894)
sdot 119890minus1204852120587119899119873
119892]120575119894(119879119894minus119879119895)
sdot 119890minus120485120587((119873
119904minus1)119873
119904)[(119896minus](119879
119895119879119894))+(119891119895minus119891119894)119879119895119873119904]
sdot1
119873119904
sdot
sin [120587 (119896 minus ] (119879119895119879
119894)) + 120587 (119891
119895minus 119891
119894) 119879
119895119873119904]
sin [(120587119873119904) (119896 minus ] (119879
119895119879
119894)) + 120587 (119891
119895minus 119891
119894) 119879
119895]
(8)
and with 119873119895(119896) being the receiver-side AWGN of power
1198730120575119895 Note that in (6) and (7) we have approximated and
defined119867119895119894(119896120575
119894+119891
119894minus119891
119888) asymp 119867
119895119894(119896120575
119894) ≜ 119867
119895119894(119896) because |119891
119894minus119891
119888|
is assumed to bemuch smaller than the coherence bandwidthof the channel
In practical system implementation the carrier andsampling frequency clocks of a transmitter or receiver arederived from the same reference that is from the same localoscillator Thus for the product of an arbitrary 119894th carrierfrequency and sampling period119891
119894sdot119879119894≜ 120581 it holds that 120581 ≫ 1
as the (ideal) carrier frequency 119891119888is two to three orders of
magnitude larger than the (ideal) sampling frequency 1119879The constant 120581 depends only on the system and hardwaredesign and is independent of 119894
Considering this relationship in (8) it is straightforwardto see that the exponent in the expressionrsquos second line whichcaptures the SFO effect on 120573
119895119894(119896 ]) and which is maximized
for ] = 119873119904 still remains 120581 times smaller than the exponent
in the first line capturing the CFO effect Similar findings canbe observed comparing the influence of SFO with the one ofthe CFO onto the terms in the third and the fourth line of(8) Thus it can be safely said that the impact of the SFO on120573119895119894(119896 ]) is significantly weaker than the impact of the CFO
when using the same oscillator reference for both at eachindividual transmitter and receiver
The derivations up to here have been presented includingboth CFO and SFO for the sake of completeness as well asfor further reference Expressions (5) through (8) provide theexact signal model which characterizes the joint effect ofmultiple CFOs and SFOs over consecutive OFDM symbolsin MIMO transmissions and is one important contributionof this work However beyond signal modeling this workfurther aims to identify the major degradation sourcesanalyze the performance degradation and determine fromthere the actual system requirements To this end in thefollowing we focus on the CFO and neglect the SFO byassuming ideal sampling for all transmitters and receiversWith 119879
119894= 119879
119895= 119879 (8) becomes
120573119895119894(119896 ]) = 119890minus1204852120587[(119891119895minus119891119894)119905119899+((119873119904minus1)2119873119904)(119896minus])]
sdot1
119873119904
sin [120587 (119896 minus ]) + 120587 (119891119895minus 119891
119894) 119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119894) 119879]
(9)
The discrete time variable 119905119899≜ (119899119873
119892+ (119873
119904minus 1)2)119879 has been
defined for convenience and is measured in secondsExpressions (5) to (9) provide the discrete-frequency
signal model for a MIMO-OFDM communication system in
the presence of multiple CFOs and SFOs as a function of timeand will be used in the following section
3 Precoded Multicell Downlink Signal Model
Now we specialize the model obtained in Section 2 to theCoMPdownlink including joint precoding of the data signalsHere the required channel state information (CSI) for theprecoder calculation is estimated either at the base stations orat the terminals and fed back to the base stations dependingonwhether timedivision duplex (TDD)or frequency divisionduplex (FDD) is used For distributed base stations coordi-nation by means of a backhaul network is considered whichenables fast exchange of data and CSI
It is to be noted that channel estimation errors and CSIdelays due to the feedback and the backhaul network arenot considered in this work Their effect on JT CoMP isimportant andmight even overwhelm the effects of imperfectsynchronization which we are analyzing here A distinctanalysis by the authors which includes the effect of imperfectchannel knowledge and derives the resulting performancelimitations can be found in [33] We will therefore assumehere that perfect knowledge of the downlink MIMO channelmatrix is available at the base stations and is used for real-timeprecoding of the downlink signals As already mentioned inSection 2 residual phase terms due toCFOs and SFOs are alsoconsidered as part of the (perfectly estimated) channel Weconsider 119873
119906users equipped with single-antenna terminals
which are jointly served by119873119887coordinated antenna branches
among all base stations forming the cooperation clusterThere is no exchange of information between mobile usersand out-of-cluster transmission is not considered in ourmodel
Transmissions are precoded on each OFDM subcarrier 119896with the right-hand pseudoinverse of the119873
119906times 119873
119887downlink
MIMO channel matrixH(119896) for that subcarrier given by
W (119896) = H119867
(119896) [H (119896)H119867
(119896)]minus1
(10)
The inverse is of size 119873119887times 119873
119906and we assume it exists for
119873119887gt 119873
119906 Note that this condition can be met in practice
by appropriate user selection and clustering of base stationswhich is thereby assumed
Next consider S(119896) to be an 119873119906times 1 vector that contains
the complex-valued data symbols 119904119906(119896) to be transmitted to
the119873119906users on subcarrier 119896 Then the119873
119887times 1 vector X(119896) of
precoded symbols119883119887(119896) to be transmitted on subcarrier 119896 by
the119873119887base stations is X(119896) = W(119896)S(119896) where elements are
given by
119883119887(119896) =
119873119906
sum
119906=1
119908119887119906(119896) 119904
119906(119896) (11)
and 119908119887119906(119896) are the elements ofW(119896) Transmitter index 119894 has
been replaced with 119887 to stress that from here on transmittersare base stations Imposing the expression of the precodedsymbol (11) into 119880
119895(119896) given by (6) we obtain the on-carrier
signal on subcarrier 119896 of user 119895 given by119880119895(119896) = 119904
119895(119896) + 119904
119895(119896) (12)
International Journal of Antennas and Propagation 5
with
119904119895(119896) = 119904
119895(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119895(119896) (13)
119904119895(119896) =
119873119906
sum
119906=1
119906 =119895
119904119906(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119906(119896) (14)
An arbitrary user 119895 has been singled out in (13) from theremaining users Thus 119904
119895(119896) represents the self-signal while
119904119895(119896) represents the IUI observed by user 119895 given by (14)This
interference is due to the loss of orthogonality of the precodedtransmission to multiple users caused by synchronizationimpairments Imposing (11) into 119880
119895(119896) in (7) we obtain the
ICI on subcarrier 119896 of user 119895 given by
119880119895(119896) =
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119904119906(])
119873119887
sum
119887=1
120573119895119887(119896 ])119867
119895119887(]) 119908
119887119906(]) (15)
Our complete discrete-frequency CoMP downlink signalmodel is then
119884119895(119896) = 119904
119895(119896) + 119904
119895(119896) + 119880
119895(119896) + 119873
119895(119896) (16)
with 119904119895(119896) 119904
119895(119896) and 119880
119895(119896) given by (13) (14) and (15)
respectively and 119873119895(119896) being the AWGN of power 119873
0120575119895 It
is noted that the model of Section 3 is valid independently iffor 120573
119895119894(119896 ]) the exact expression (8) or its simplification (9) is
used where the SFO is neglected
4 Analysis of Signal and Interference Powers
The following section contains an in-depth analysis of theimpact of synchronization impairments onto the perfor-mance It is organized as follows first we study the powerof a userrsquos self-signal (useful signal) and then the IUI andICI Next we show that ICI is small compared to IUI andprovide analytical expressions for the mean SIR Finally wehighlight the value of user selection and show its impact ontothe performance
Conceptually the rise of IUI and ICI due to imperfectsynchronization can be understood as a dispersion of theenergy allocated on a specific subcarrier of a specific user toother users (IUI) and to other subcarriers (ICI) This impliesa drop of the self-signal power and a rise on that user andthat subcarrier of the power of IUI and ICI from other usersrsquoand other subcarriersrsquo losses In what follows we proceedunder the assumption that data symbols are statisticallyindependent between users and across subcarriers that isE119904
119895(119896)119904
lowast
119906(]) = 119864
119904120575119895119906120575119896] where Esdot denotes the expectation
operator 119864119904the mean energy per data symbol and 120575
119909119910the
Kronecker delta between 119909 and 119910
41 Power of the Userrsquos Self-Signal Since there is statisticalindependence between data symbols channel coefficients
and synchronization parameters themeanpower of the userrsquosself-signal (13) is
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
E 1198671198951198871
1199081198871119895119867lowast
1198951198872
119908lowast
1198872119895
(17)
where subcarrier index 119896 has been omitted for simplicity ofnotation and cross-terms equal to zero have been alreadydisregarded
In a first step we analyze E1205731198951198871
120573lowast
1198951198872
which appears in(17) Using therefore ] = 119896 in (9) and replacing transmitterindex 119894 with base station index 119887 we reach
120573119895119887=
1
119873119904
sdot 119890minus1204852120587(119891
119895minus119891119887)119905119899 sdot
sin [120587 (119891119895minus 119891
119887) 119879119873
119904]
sin [120587 (119891119895minus 119891
119887) 119879]
(18)
which does not depend on the subcarrier index 119896 By notingthat |119891
119895minus 119891
119887| ≪ 1119879 we can safely say that the argument 119909 =
120587(119891119895minus 119891
119887)119879 of the sin(119909) in the denominator of the fraction
on the right-hand side of (18) is very small The same termalso appears in the exponential term and both multiplicativeterms in (18) are close to one But comparing the magnitudedeviations from unity we see that for typical synchroniza-tion parameters |1 minus (1119873
119904)(sin(119873
119904119909) sin(119909))| ≪ |1 minus
119890minus120485(214119899+1)119873
119904119909
| (this results in119873119904≫ 1 and a cyclic prefix equal
to 007119873119904[31]) In absolute terms we can further use the first
order Taylor series expansion (1119873119904)(sin(119873
119904119909) sin(119909)) asymp 1
and reach
1205731198951198871
120573lowast
1198951198872
asymp 119890minus1204852120587(119891
119895minus1198911198871)119905119899 sdot 119890
1204852120587(119891119895minus1198911198872)119905119899 = 119890
1204852120587(1198911198871minus1198911198872)119905119899 (19)
It is interesting to observe that the power of the exponentialterm as approximated in (19) depends on the differencebetween the base stationsrsquo carrier frequenciesThe power losseffect of the symbol transmitted on subcarrier 119896 which is thegenerating factor of ICI depends on both the CFOs of thebase stations and of the mobile usersThe effect that has beenneglected here due to its relatively small role compared tothe one of the exponential term will be however thoroughlyanalyzed in Section 42
For transmission from one base station that is for case1198871= 119887
2 it is immediate that (19) equals one For 119887
1= 119887
2
it is reasonable to assume that different base stationsrsquo carrierfrequencies 119891
1198871
and 1198911198872
are statistically independent whichallows for expressing the expectation of (19) as a product oftwo terms each depending on one base stationrsquos CFO
E 1205731198951198871
120573lowast
1198951198872
= E 1198901204852120587(119891
1198871minus119891119888)119905119899 sdot E 119890
minus1204852120587(1198911198872minus119891119888)119905119899 (20)
By assuming further that theCFOs are identically distributedthe second term of the right-hand side in (20) is the complex-conjugate of the first one hence (20) can be developed as
E 1205731198951198871
120573lowast
1198951198872
=10038161003816100381610038161003816E 119890
minus1204852120587(119891119887minus119891119888)11990511989910038161003816100381610038161003816
2
=
1003816100381610038161003816100381610038161003816int 119901
(119891119887minus119891119888)119890minus1204852120587(119891
119887minus119891119888)119905119899119889 (119891
119887minus 119891
119888)
1003816100381610038161003816100381610038161003816
2
(21)
6 International Journal of Antennas and Propagation
where 119901(119891119887minus119891119888)denotes the probability distribution function
(pdf) of the CFO (119891119887minus 119891
119888) and is independent on index 119887 It
is to be noted that the integral in (21) is a Fourier transformof the CFOrsquos pdf Hence we can write
E 1205731198951198871
120573lowast
1198951198872
=
1 1198871= 119887
2
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
1198871
= 1198872
(22)
where F119901(sdot) denotes the Fourier transform of 119901
(119891119887minus119891119888)and is
here a function of time For being a characteristic functionthat is the Fourier transform of a pdf it is guaranteedthat |F
119901(119905119899)| le 1 (see [32]) which means that the power
of the userrsquos self-signal (17) drops under imperfect carriersynchronization conditions
Result (22) can be used for developing (17) further byseparating the sum over 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= 119864119904sdot (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
) sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
(23)
The double sum in the last line in (23) can be formulated as
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= E
119873119887
sum
1198871=1
1198671198951198871
1199081198871119895sdot
119873119887
sum
1198872=1
119867lowast
1198951198872
119908lowast
1198872119895
(24)
which by considering the ZF condition
119873119887
sum
119887=1
119867119895119887119908119887119906= 120575
119895119906 (25)
is found to be equal to 1 Thus (23) can be written as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904[1 minus (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870U] (26)
where we defined119870U = 1 minussum119873119887
119887=1E|119867
119895119887119908119887119895|2
Note that (25)cannot be applied to the sum terms in the first and third line of(23) because of the power indexThe constant119870U depends onthe precoder and the channel statistical properties As shownin Appendix A for ZF precoding given a channel matrix with119873119887gt 119873
119906and complex Gaussian independent and identically
distributed (iid) entries (Rayleigh fading channel) withzero mean and variance 1205902
ℎ119870U can be approximated as
119870U asymp 1 minus1
119873119887minus 119873
119906
(27)
If we further consider the case in which the CFOs of thebase stations are Gaussian iid with zero mean and variance1205902
119891 the Fourier transform in (22) becomes
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
= 119890minus412058721205902
1198911199052
119899 asymp 1 minus 41205872
1205902
1198911199052
119899 (28)
where the first order Taylor series expansion 119890minus119909 asymp 1 minus 119909 hasbeen used It is to be noted that this approximation is accuratefor typical values of 120590
119891and 119905
119899 Using approximations (27) and
(28) we can formulate themeanpower of the userrsquos self-signal(26) as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904[1 minus 4120587
2
1205902
1198911199052
119899sdot (1 minus
1
119873119887minus 119873
119906
)] (29)
from where we see that it decreases linearly with the basestationsrsquo CFO variance and quadratically with time Result(29) also shows that the mean power of the self-signal growswith the number of base stations and drops with the numberof users It is noteworthy that for 119873
119906= 119873
119887minus 1 the mean
power of the self-signal remains constant However thisshould not be used as a system design rule as for determiningthe system performance the degradation due to IUI and ICImust be considered as well
42 Power of the Interuser Interference The derivation of themean power of the IUI (14) follows similar steps as the onesin Section 41 Concretely
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119906
sum
119906=1
119906 =119895
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
sdot E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(30)
and noting that in this case always 119895 = 119906 in (25) we find that
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904(1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI (31)
where119870IUI = sum119873119906
119906=1119906 =119895sum119873119887
119887=1E|119867
119895119887119908119887119906|2
If all users undergoidentical channel statistics119870IUI simplifies to119870IUI = (119873119906
minus1) sdot
sum119873119887
119887=1E|119867
119895119887119908119887119906|2
whereas using the results of Appendix Afor the Rayleigh fading channel and119873
119887gt 119873
119906 we find
119870IUI asymp119873119906minus 1
119873119887minus 119873
119906
(32)
Using (32) and (28) in (31) we obtain
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904sdot 4120587
2
1205902
1198911199052
119899sdot119873119906minus 1
119873119887minus 119873
119906
(33)
Result (33) reveals that the IUI power grows linearly withthe base stationsrsquo CFO variance and quadratically with time
International Journal of Antennas and Propagation 7
Using more base stations decreases the IUI power whileadding more users results in increasing it Due to theapproximation used in (19) it can be said that the impactof the mobile usersrsquo CFO onto the IUI power level is lesssignificant than the impact of the base stationsrsquo CFOs
43 Power of the Intercarrier Interference Following similarsteps as before now for (15) the mean power of the ICI is
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
sdot E 1198671198951198871
(]) 1199081198871119906(])119867lowast
1198951198872
(]) 119908lowast1198872119906(])
(34)
Assuming identical channel statistics for all subcarriers thelast term in (34) does not depend on subcarrier index ] andbecomes E119867
1198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906 already seen in Sections 41
and 42 For convenience we define1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) ≜
1198611 119887
1= 119887
2
1198612 119887
1= 1198872
(35)
which is analyzed inAppendix B In (34) we separate the sumover 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871as in (23) and obtain
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot 119861
1sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
= 119864119904sdot (119861
1minus 119861
2) sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(36)
We use the expressions for 1198611and 119861
2in Appendix B which
provide accurate approximations for Gaussian distributedCFOs for the base stations and the mobile user 119895 with zeromean and variances 1205902
119891and 1205902
119891119895 respectively and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot 119870ICI + 120590
2
119891119895) (37)
Here 119870ICI = sum119873119906
119906=1sum119873119887
119887=1E|119867
119895119887119908119887119906|2
= 119870IUI minus 119870U + 1 is aconstant For aRayleigh fadingMIMOchannel we use in (37)the expression provided by Appendix A for119870ICI and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot
119873119906
119873119887minus 119873
119906
+ 1205902
119891119895) (38)
Expression (38) shows that the mean power of the userrsquos ICIcan be split into two additive parts the first part dependingon the base stationsrsquo CFO variance which is multiplied witha weighting factor 119870ICI according to the cluster size and thesecond part depending on the userrsquos own CFO Furthermore(38) reveals that the power of the ICI does not vary with timeand that it is inverse to the square of the OFDM subcarrierspacing
44 ICI-to-IUI Ratio and SIR The relative magnitudes of thepower terms derived so far in this section are analyzed nextA simple expression for the mean ICI-to-IUI power ratio ofa user 119895 (an interference-to-interference ratio IIR) can beobtained from (33) and (38)
IIR =
E 10038161003816100381610038161003816119880119895
10038161003816100381610038161003816
2
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp1
1212057521199052119899
sdot (119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (39)
with 119873119887gt 119873
119906ge 2 and 120585 = 120590
119891119895120590
119891defined as the ratio
between the standard deviations of the userrsquos and the basestationsrsquo CFOs By replacing 120575 = (119873
119904119879)
minus1 and discrete time119905119899= (107119899 + 05)119873
119904119879 (this results in 119873
119904≫ 1 and a cyclic
prefix equal to 007119873119904[33]) relation (39) becomes
IIR asymp1
12 (107119899 + 05)2sdot (
119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (40)
It is interesting to observe that in the approximation (40)the IIR does not depend on the OFDM system parametersbut only on the OFDM symbols index 119899
Regarding the ratio 120585 it has been shown in [14 21]that mobile terminals attain a carrier frequency accuracywhich is at least one order of magnitude below the accuracyof typical base station oscillators used in Third GenerationPartnership Project (3GPP) Long Term Evolution (LTE) [31]Assuming for instance 1205852 ≫ 1 the first additive term in (40)becomes much smaller than the second term indicating thatthe terminalsrsquo CFO is the main source of ICI Using 120585 = 0one can evaluate the IIR including only the ICI part due to thebase stationsrsquo CFOs in this case the ratio does not depend onthe base stationsrsquo CFOs at all Using a more practical value offor example 120585 = 10 the largest possible IIR after 119899 = 14
OFDM symbols (119905119899asymp 1ms) which occurs with 119873
119906= 2
users given 119873119887= 7 equals minus73 dB The IIR drops quickly
down already to minus273 dB for 119899 = 140 (119905119899asymp 10ms) These
quantitive results show that for typical JT CoMP scenariosthe IUI dominates over the ICI
Wewill nowneglect the ICI and define themean SIR (self-user signal-to-IUI ratio) by the ratio between (26) and (31)
SIR =
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
(1 minus10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI
minus1
119873119906minus 1
(41)
For a MIMO channel with iid Rayleigh fading entries (119870IUIgiven by (32)) and for iid Gaussian CFOs (|F
119901(119905119899)|2 given
by (28)) (41) can be simplified as
SIR asymp1
412058721205902
1198911199052119899
sdot119873119887minus 119873
119906
119873119906minus 1
minus119873119887minus 119873
119906+ 1
119873119906minus 1
(42)
8 International Journal of Antennas and Propagation
Expressions (41) and (42) are valid for 119873119887gt 119873
119906ge 2
Expression (42) reveals that for ZF precoding the mean SIRis in an approximately inverse relation to the base stationsrsquoCFO variance and to the square of time Furthermore it isshown that the SIR grows with the number of base stationsand drops with the number of jointly served users
45 TheValue of User Selection in JTCoMPand an SIRBoundThe SNR gains in JT CoMP are increased if a schedulerselects the appropriate users to be jointly served on thesame time and frequency resources As shown in [34] theselection criteria are based on the rule that all users gainfrom the cooperation in the cluster compared to the case ofnoncoordinated transmission Evaluated on a field scenariothe resulting singular value statistics after such user selectionwas found to be comparable to the one of a Rayleigh fadingchannel a fact that also relates to scenarios where users arelocated close to the cell edge that is in which gains throughcoordination are particularly large
Considering now an idealized scenario from the precod-ing point of view with orthogonal channel vectors of equalpower among the users scenario which has been analyzed in[5 11] an upper bound can be derived for the mean SIR usingZF precoding in JT CoMP Using the expression for 119870IUI asprovided in Appendix A for 119873
119887gt 119873
119906ge 2 we obtain the
following expression
SIRmax asymp1
412058721205902
1198911199052119899
sdot119873119887
119873119906minus 1
minus119873119887+ 1
119873119906minus 1
(43)
The distance between (42) and (43) reveals the potential forSIR enhancement if the usersrsquo selection process considers theorthogonality among their channel vectors
It should be clarified at this point that (42) and (43) do notconsider any gains from resource allocation the evaluationof which not in the scope of this work In an orthogonalfrequency division multiple access (OFDMA) system eachuser can be assigned a part of the spectrum in a way that hisperformance and the network performance can be optimizedas known from [35] and also shown in [36] for the multiuserdownlink
The signal model given in Section 3 for the impairedOFDM-based JT CoMP is valid for any OFDMA schemeUsing OFDMA in the downlink does not affect the signalstructure of a userrsquos received self-signal and the IUI TheICI also has the same structure and statistical propertiesas typically data are transmitted on all subcarriers which isrecommended for keeping frequency-flat distribution of theICI power [13] The degradation mechanisms due to multipleCFOs and SFOs as described in our model will be the samefor any system using OFDM
What indeed changes in OFDMA is the channel statis-tics for each user after resource allocation Therefore ifintended to evaluate the overall performance of a coordinatedmultipoint system using OFDMA the general mean powerexpressions (26) (31) and (37) would need to be evaluated forthe actual usersrsquo channel statistics This procedure practicallyimplies a calculation of constants119870U119870IUI and119870ICI
In this section the received power of a user was studiedin relation to the interference from other users and othersubcarriers in amulticellular multiuser downlink with coop-erative base stations Closed-form expressions and accurateapproximations for all relevant termswere derivedMoreoverit was demonstrated that interuser interference has the mostrelevant effect Finally the impact of the radio channel wasinvestigated and analytical SIR expressions were derived forzero-forcing precoding
5 Evaluation and Numerical Validation
In this section analytical results of Section 4 are evaluatedand verified by simulations Based on those results adequaterequirements are determined for the base stationsrsquo oscillatorsin CoMP systems
A JT CoMP scenario is considered where a cooperationcluster of 119873
119887= 7 base stations transmits jointly by using
zero-forcing precoding to 119873119906terminals on the same time
and frequency resource with 119873119887gt 119873
119906ge 2 Out-of-cluster
interference is not consideredAt time instant 119905 = 0 the base stations receive an update
of the downlink channel matrix and compute a precodingmatrix This will be used during the following 10ms timeat which a new channel update will be obtained and a newprecoder will be calculated This routine agrees with thecurrent 3GPP LTE channel and precoder updating cycle [31]As already mentioned in Section 3 it is assumed that theradio channel is perfectly estimated and available at the basestations at 119905 = 0 whereas channel aging effects are notconsidered either Between updating instants the phases ofthe local base stationsrsquo oscillators slowly drift away fromeach other because of the individual frequency offsets withrespect to the ideal carrier frequency 119891
119888 As a consequence
self-signal drops IUI grows and overall the SIR drops withtime In a practical system each user can use downlink pilotsymbols to estimate and equalize its own CFO and avoid ICIenhancement However mobiles cannot stop the IUI fromrising because the degradation is caused jointly by all themisaligned base station oscillators
The oscillator accuracy Osc typically specified in partsper million (ppm) or parts per billion (ppb) relates to thestandard deviation of the resulting carrier frequency as 120590
119891=
Osc sdot 119891119888 The CFO variation over time is assumed to be slow
enough to be considered static with respect to the signalingand precoder updating timescale Here Gaussian iid CFOsare randomly assigned to base stations and mobile users allwith zero mean and standard deviations given by 120590
119891and 120590
119891119895
respectivelyTypical 3GPP LTE parameters are used specified in [31]
The broadband OFDM signal has 119873119904= 2048 subcarriers
separated by 120575 = 15 kHzThe length of the cyclic prefix is 144samples resulting in OFDM symbols with a length of 119873
119892=
2192 samples The ideal carrier and sampling frequencies are119891119888= 265GHz and 1119879 = 3072MHz respectively The
energy per data symbol is set to 119864119904= 1
Figure 2 depicts the mean power of the IUI and ICI overtime passed from the last precoder update (in logarithmicscale) for 3 and 6 mobile users a base station oscillator
International Journal of Antennas and Propagation 9
0
Time (ms)
Mea
n po
wer
(dB)
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
10minus1 100 101
IUI Nu = 3 Osc = 1ppbICI Nu = 3 Osc = 1ppb 120585 = 0
ICI Nu = 3 Osc = 1ppb 120585 = 10
IUI Nu = 6 Osc = 1ppbICI Nu = 6 Osc = 1ppb 120585 = 0
ICI Nu = 6 Osc = 1ppb 120585 = 0
Figure 2 Mean power of interuser and intercarrier interferences ina Rayleigh fading channel Here 7 base stations serve 3 and 6 usersrespectively Analytical results are shown by lines while markersshow numerical evaluations
accuracy of 1 ppb and Rayleigh fading channel conditionsThe curves illustrate analytical results given by (33) and(38) whereas solid lines depict results for 3 users anddashed lines for 6 users respectivelyMarkers show respectivenumerical evaluations of themeanpower of (14) and (15) over10
3 independent realizations of the MIMO channel matrixentries with 120590
2
ℎ= 1 and of the basesrsquo CFOs Regarding
the ICI two scenarios are considered a scenario with 120585 =
0 that is perfectly synchronized mobile users in order toevaluate solely the ICI due to the base stationsrsquo CFOs (red)and a second scenario with nonsynchronized mobiles with aCFO accuracy of an order of magnitude lower than the basestationsrsquo one that is 120585 = 10 (black) where the total ICI isevaluated
First of all Figure 2 verifies the analytical expres-sions (includingmathematical approximations) by numericalmeans It is further observed that the IUI grows approxi-mately with the square of time and that it increases with thenumber of users Comparing the mean power levels of IUI(blue curves) with the ICI both caused by the base stationsrsquoCFOs (red curves for 120585 = 0) we see that independently of thenumber of users and for typical feedback delays between 2msand 10ms the IUI lies between 40 dB and 50 dB above thepower of ICIThis clearly indicates that the IUI is significantlystronger than the ICI caused by the base stationsrsquo CFOsConsidering now the second scenario with nonperfectlysynchronized mobile users (120585 = 10) it can be observed thatthe part of the ICI caused by the usersrsquo CFO overwhelms theICI caused by the base stationsrsquo CFOs According to (38) the(dominant) ICI due to a mobile userrsquos CFO does not depend
SIR
(dB)
0
10
20
30
40
50
60
70
80
90
Time (ms)10minus1 100 101
20dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 3 Mean SIR over time for Rayleigh fading channel and SIRupper bound Here 7 base stations using oscillators of accuracygiven by Osc serve jointly 6 users Analytical results are shown bylines simulations by markers
on the number of served users which is reflected in the factthat the black curves in Figure 2 evaluating the total ICI for120585 = 10 almost overlap for the cases of 3 and 6 users It isalso interesting to observe that the IUI power level due tothe cooperating base stationsrsquo CFOs is still around 20 dB to30 dB (for 3 users) and 30 dB to 40 dB (for 6 users) higherthan the total ICI power considering typical feedback delaysbetween 2ms and 10ms These results clearly show that inthe a cooperative network even with typically nonperfectlysynchronized mobile users the main interference source liesin the base stationsrsquo CFOs
Figure 3 shows the SIR over the time as defined in (41)for119873
119887= 7 base stations serving jointly119873
119906= 6mobile users
which is also the highest possible number for the Rayleighfading channel The influence of the base stationsrsquo oscillatoraccuracy is evaluated and the corresponding ICI is neglectedas it is significantly smaller than the IUI Disregarding the ICIalso means that the synchronicity level of the mobile users isnot relevant as it does not contribute the IUI
For the Rayleigh fading channel the analytical expression(42) is compared with the ratio between the numericallyevaluated mean power of (13) and (14) over 103 independentchannel and CFO realizations which validates our analysisincluding the accuracy of the mathematical approximationsSimilarly for the SIR upper bound expression the analyticalexpression (43) is compared with results from numericalevaluation From Figure 3 it is seen that a degradation of thebase station oscillatorsrsquo accuracy by one order of magnitudeincreases the IUI and thus decreases the SIR by around 20 dBFurthermore the SIR drops approximately quadratically withtime that is 20 dB per time decade In terms of accuracyrequirements it is found that for reaching in average anSNR of 20 dB 10ms after the precoder update (free-running)oscillators of 1 ppb are needed
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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DistributedSensor Networks
International Journal of
2 International Journal of Antennas and Propagation
synchronization for multiuser MIMO within one cell hasbeen studied in [15] For the OFDM-based multiuser uplinka signal model and compensation techniques have beendeveloped in [16]
Considering distributed JT CoMP cooperative base sta-tions are located at different sites which implies that theirfrequency up- and downconverters are driven by theirown local oscillators while sampling frequencies also differamong them Signal modeling of JT CoMP with individualoffsets in carrier and sampling frequencies in [17] revealedthat orthogonality between multiple usersrsquo data signals ismisaligned and interuser interference (IUI) arises Firstinsights into the performance degradation were obtained bynumerical evaluation In chapter 8 of [18] the sensitivityof CoMP to the CFO was analyzed for a scenario withtwo cooperating base stations Similar observations havebeen reported in [19 20] whereas methods for estimatingmultiple CFOs based on training signals have been developedin [21] The problem of nonsynchronized cooperating basestations has been also investigated in [22ndash24] where thefocus has been on how to estimate and compensate themultiple CFOs In [25] propagation delay differences werealso included for transmissions fromdistributed base stationswithmultiple CFOs In [26] a scheme for synchronizing basestations has been proposed based on a time-slotted round-trip carrier synchronization protocol The implementationof Global Positioning System- (GPS-) based synchronizationfor distributed base stations in an outdoor testbed has beenreported by the authors in [27] More recently an over-the-air synchronization protocol has been proposed in [28]which is also applicable for networks with a large numberof access points A survey on physical layer synchroniza-tion for distributed wireless networks can be found in[29]
The first objective of the present paper is to investi-gate the synchronization requirements for base-coordinatedmulticellular MIMO networks A major contribution of thiswork is the derivation of an exact signal model capturingthe joint effect of multiple CFOs and SFOs at transmittersand receivers in a MIMO-OFDM system and over the timeBased on this model it is shown that the impact of the SFO isnegligible compared to the one of the CFO Application of themodel to the distributed CoMP downlink with ZF precodingleads to analytical closed-form expressions for the meanpower of the usersrsquo self-signal interuser and intercarrierinterferences It is found that the interuser interference isthe dominant source of signal degradation and that syn-chronization requirements for cooperating base stations arevery high compared to the ones in single-cell transmissionThe mean signal-to-interference ratio (SIR) is analyzed andis approximately found to degrade quadratically with timeand to be inversely proportional to the variance of thebase stationsrsquo CFO The SIR further grows with the numberof base stations and drops with the number of users Inaddition to the SIR analysis for the Rayleigh fading channelan SIR upper bound is derived which can be approachedby appropriate user selection Finally recommendations forpractical synchronization of distributedwireless networks aregiven
The paper is organized as follows In Section 2 a generalsignal model for a MIMO-OFDM communication systemin the presence of multiple CFOs and SFOs is derived InSection 3 the model is applied to the CoMP downlink andexpressions are derived formobile usersrsquo self-signal interuserand intercarrier interferences Analysis in Section 4 leads toclosed-form expressions for the mean power of the abovesignals and the resulting SIR The system performance isevaluated analytically and verified bymeans of simulations inSection 5 Synchronization requirements are established andpractical methods to fulfill them are discussed in Section 6Finally conclusions are summarized in Section 7
2 General Signal Model for DistributedMIMO-OFDM Systems
In the following a distributed MIMO network is consideredwith an arbitrary number of antenna branches at every basestation and at every userThe cellular network usesOFDM forthe air interface with119873
119904subcarriers which are indexed with
119896 in the range minus1198731199042 119873
1199042minus1 An entireOFDMsymbol
is 119873119892samples long equal to 119873
119904samples plus the number
of samples of the cyclic prefix Integer 119899 indexes successiveOFDM symbols and is hence a measure of time
Each base station and each mobile are assumed to havetheir own carrier and sampling frequency within typicalranges The total number of transmit branches is 119873
119905 Each
transmit branch (can be a base station in the downlink or auser in the uplink) denoted by subindex 119894 has its individualsampling period119879
119894 carrier frequency119891
119894and respective initial
phase parameters 120591119894and 120593
119894 In Figure 1 it is shown for a
point-to-point transmission how sampling and carrier offsetsmisalign analog-to-digital and digital-to-analog conversionas well as frequency conversion respectively The corre-sponding receiver parameters are denoted as 119879
119895 119891
119895 120591
119895 and
120593119895 while symbol 120485 is used for radicminus1 The digital modulation
of subcarrier 119896 on transmit branch 119894 is represented by thecomplex-valued symbol 119883
119894(119896) Due to different sampling
timings between transmit branches of different base stations(downlink) or among mobile users (uplink) the intercarrierspacing is transmit-branch-specific and measures 120575
119894=
(119873119904119879119894)minus1 Hertz The ideal carrier frequency is denoted by 119891
119888
and the ideal sampling period with 119879 For any transmitter orreceiver its CFO and SFO are defined as the the deviationfrom the ideal carrier frequency and sampling period respec-tively The complex baseband-equivalent frequency responseof the passband channel between transmitter 119894 and receiver119895 at frequency 119891 is denoted by119867
119895119894(119891) It includes frequency-
flat path loss and shadow fading as well as frequency-selectivesmall-scale fading
By following similar arguments as the ones leading toequation (8) in [14] and by considering the clarificationin [30] that is corrections in magnitude in order to keepsignal energies consistent it is found that the spectrum of theOFDM signal observed at any given receive branch 119895 has theform
119884119895(119891) = 119880
119895(119891 119896) + 119880
119895(119891 119896) + 119873
119895(119891) (1)
International Journal of Antennas and Propagation 3
Baseband processing
Baseband processing
Transmitter Receiver
DA AD
Ti Tjfi fi
Figure 1 Transmitter and receiver have individual sampling periods 119879119894and 119879
119895for digital-to-analog and analog-to-digital conversion and
individual carrier frequencies 119891119894and 119891
119895for upconversion and downconversion
where 119880119895(119891 119896) represents the continuous-frequency spec-
trum of the received multiuser signal at receive antenna 119895for transmitted subcarrier 119896 (on-carrier signal) It is notedthat 119880
119895(119891 119896) is the received spectrum of the multiuser signal
of all other subcarriers ] = 119896 that is the multiuser ICI(it is arbitrary which subcarrier is designated as 119896 but ouranalysis requires to single out one of them) The additivewhite Gaussian noise (AWGN) contributed by the front-endof receive antenna 119895 is denoted by119873
119895(119891)The above terms are
given by
119880119895(119891 119896) = 119879
119895
119873119905
sum
119894=1
119890120485(120593119895minus120593119894)
120573119895119894(119891 119896)
sdot 119883119894(119896)119867
119895119894(119896120575
119894+ 119891
119894minus 119891
119888)
(2)
119880119895(119891 119896) = 119879
119895
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119905
sum
119894=1
119890120485(120593119895minus120593119894)
120573119895119894(119891 ])
sdot 119883119894(])119867
119895119894(]120575
119894+ 119891
119894minus 119891
119888)
(3)
with
120573119895119894(119891 119896) = 119890
minus1204852120587(119891119895minus119891119894)(119899119873119892119879119895+120591119895)
sdot 119890minus1204852120587119896120575
119894(119899119873119892119879119894+120591119894minus119899119873119892119879119895minus120591119895)
sdot 119890minus120485120587(119891+119891
119895minus119891119894minus119896120575119894)119879119895(119873119904minus1)
sdot
sin [120587 (119891 + 119891119895minus 119891
119894minus 119896120575
119894) 119879
119895119873119904]
sin [120587 (119891 + 119891119895minus 119891
119894minus 119896120575
119894) 119879
119895]
(4)
The exponential terms in (4) are phase shifts due to carrierand sampling frequency misalignment while the fractionalterm describes the loss of orthogonality among OFDMsubcarriers causing a leakage of the signal transmitted onsubcarrier 119896 Note that 119867
119895119894(119896120575
119894+ 119891
119894minus 119891
119888) expresses the
channel at the frequency of subcarrier 119896 plus a shift due tothe transmitterrsquos SFO and CFO
Themodel given by (1) through (4) makes no assumptionabout the synchronization among branches and it is gen-eral for any MIMO-OFDM communication also includingcellular networks with base station cooperation It describeshow successive OFDM symbols are degraded under constantand uncompensated CFOs and SFOs as time goes by that
is OFDM symbol index 119899 grows The CFO must be smallerthan half of a subcarrier spacing For typical mobile receiversthis implies that an earlier coarse frequency synchronizationstage has succeeded which we assume henceforth RegardingSFO the expressions are valid well beyond the typical rangespecified for SFO in commercial OFDM systems It is to benoted however that for a given SFO the FFT window ofeach receiver branch does eventually drift away to a point atwhich intersymbol interference (ISI) arises between OFDMsymbols From that point on severe degradation ensues andthe model stops being valid It is also implicit in our modelthat OFDM symbol timing has been acquired in a priorstage The result assumes that the time dispersion of all theMIMOchannel impulse responses also fromdistributed basestations and mobile users is shorter than the OFDM cyclicprefix in use
Returning to the model it can be observed in (2) and(3) that the phase offsets due to 120591
119894 120591
119895 120593
119894 and 120593
119895may be
considered without loss of generality as part of the channelIt follows that we may choose 120591
119894= 0 120591
119895= 0 120593
119894= 0 and
120593119895= 0 We also point out that in a practical implementation
for the 119895th OFDM receiver branch the output of its FFTcorresponds to a sequence of samples of 119884
119895(119891) taken at
frequencies 119891 = 119897119873119904119879119895= 119897120575
119895 where 119897 is the subcarrier
index (minus1198731199042 le 119897 le 119873
1199042 minus 1) and 120575
119895is the receiver-side
intercarrier spacing Note that 119897 points to a slightly differentfrequency at each receive branch due to the different SFO andgenerally also to a different frequency than that pointed atby index 119896 at the various transmit branches 119894 Imposing theabove conditions on (2) and (3) and focusing on an arbitraryreceived subcarrier 119897 = 119896 we obtain the following discrete-frequency signal model
119884119895(119896) = 119880
119895(119896) + 119880
119895(119896) + 119873
119895(119896) (5)
with
119880119895(119896) =
119873119905
sum
119894=1
120573119895119894(119896 119896)119883
119894(119896)119867
119895119894(119896) (6)
119880119895(119896) =
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119905
sum
119894=1
120573119895119894(119896 ]) 119883
119894(])119867
119895119894(]) (7)
4 International Journal of Antennas and Propagation
120573119895119894(119896 ]) = 119890minus1204852120587119899119873119892119879119895(119891119895minus119891119894)
sdot 119890minus1204852120587119899119873
119892]120575119894(119879119894minus119879119895)
sdot 119890minus120485120587((119873
119904minus1)119873
119904)[(119896minus](119879
119895119879119894))+(119891119895minus119891119894)119879119895119873119904]
sdot1
119873119904
sdot
sin [120587 (119896 minus ] (119879119895119879
119894)) + 120587 (119891
119895minus 119891
119894) 119879
119895119873119904]
sin [(120587119873119904) (119896 minus ] (119879
119895119879
119894)) + 120587 (119891
119895minus 119891
119894) 119879
119895]
(8)
and with 119873119895(119896) being the receiver-side AWGN of power
1198730120575119895 Note that in (6) and (7) we have approximated and
defined119867119895119894(119896120575
119894+119891
119894minus119891
119888) asymp 119867
119895119894(119896120575
119894) ≜ 119867
119895119894(119896) because |119891
119894minus119891
119888|
is assumed to bemuch smaller than the coherence bandwidthof the channel
In practical system implementation the carrier andsampling frequency clocks of a transmitter or receiver arederived from the same reference that is from the same localoscillator Thus for the product of an arbitrary 119894th carrierfrequency and sampling period119891
119894sdot119879119894≜ 120581 it holds that 120581 ≫ 1
as the (ideal) carrier frequency 119891119888is two to three orders of
magnitude larger than the (ideal) sampling frequency 1119879The constant 120581 depends only on the system and hardwaredesign and is independent of 119894
Considering this relationship in (8) it is straightforwardto see that the exponent in the expressionrsquos second line whichcaptures the SFO effect on 120573
119895119894(119896 ]) and which is maximized
for ] = 119873119904 still remains 120581 times smaller than the exponent
in the first line capturing the CFO effect Similar findings canbe observed comparing the influence of SFO with the one ofthe CFO onto the terms in the third and the fourth line of(8) Thus it can be safely said that the impact of the SFO on120573119895119894(119896 ]) is significantly weaker than the impact of the CFO
when using the same oscillator reference for both at eachindividual transmitter and receiver
The derivations up to here have been presented includingboth CFO and SFO for the sake of completeness as well asfor further reference Expressions (5) through (8) provide theexact signal model which characterizes the joint effect ofmultiple CFOs and SFOs over consecutive OFDM symbolsin MIMO transmissions and is one important contributionof this work However beyond signal modeling this workfurther aims to identify the major degradation sourcesanalyze the performance degradation and determine fromthere the actual system requirements To this end in thefollowing we focus on the CFO and neglect the SFO byassuming ideal sampling for all transmitters and receiversWith 119879
119894= 119879
119895= 119879 (8) becomes
120573119895119894(119896 ]) = 119890minus1204852120587[(119891119895minus119891119894)119905119899+((119873119904minus1)2119873119904)(119896minus])]
sdot1
119873119904
sin [120587 (119896 minus ]) + 120587 (119891119895minus 119891
119894) 119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119894) 119879]
(9)
The discrete time variable 119905119899≜ (119899119873
119892+ (119873
119904minus 1)2)119879 has been
defined for convenience and is measured in secondsExpressions (5) to (9) provide the discrete-frequency
signal model for a MIMO-OFDM communication system in
the presence of multiple CFOs and SFOs as a function of timeand will be used in the following section
3 Precoded Multicell Downlink Signal Model
Now we specialize the model obtained in Section 2 to theCoMPdownlink including joint precoding of the data signalsHere the required channel state information (CSI) for theprecoder calculation is estimated either at the base stations orat the terminals and fed back to the base stations dependingonwhether timedivision duplex (TDD)or frequency divisionduplex (FDD) is used For distributed base stations coordi-nation by means of a backhaul network is considered whichenables fast exchange of data and CSI
It is to be noted that channel estimation errors and CSIdelays due to the feedback and the backhaul network arenot considered in this work Their effect on JT CoMP isimportant andmight even overwhelm the effects of imperfectsynchronization which we are analyzing here A distinctanalysis by the authors which includes the effect of imperfectchannel knowledge and derives the resulting performancelimitations can be found in [33] We will therefore assumehere that perfect knowledge of the downlink MIMO channelmatrix is available at the base stations and is used for real-timeprecoding of the downlink signals As already mentioned inSection 2 residual phase terms due toCFOs and SFOs are alsoconsidered as part of the (perfectly estimated) channel Weconsider 119873
119906users equipped with single-antenna terminals
which are jointly served by119873119887coordinated antenna branches
among all base stations forming the cooperation clusterThere is no exchange of information between mobile usersand out-of-cluster transmission is not considered in ourmodel
Transmissions are precoded on each OFDM subcarrier 119896with the right-hand pseudoinverse of the119873
119906times 119873
119887downlink
MIMO channel matrixH(119896) for that subcarrier given by
W (119896) = H119867
(119896) [H (119896)H119867
(119896)]minus1
(10)
The inverse is of size 119873119887times 119873
119906and we assume it exists for
119873119887gt 119873
119906 Note that this condition can be met in practice
by appropriate user selection and clustering of base stationswhich is thereby assumed
Next consider S(119896) to be an 119873119906times 1 vector that contains
the complex-valued data symbols 119904119906(119896) to be transmitted to
the119873119906users on subcarrier 119896 Then the119873
119887times 1 vector X(119896) of
precoded symbols119883119887(119896) to be transmitted on subcarrier 119896 by
the119873119887base stations is X(119896) = W(119896)S(119896) where elements are
given by
119883119887(119896) =
119873119906
sum
119906=1
119908119887119906(119896) 119904
119906(119896) (11)
and 119908119887119906(119896) are the elements ofW(119896) Transmitter index 119894 has
been replaced with 119887 to stress that from here on transmittersare base stations Imposing the expression of the precodedsymbol (11) into 119880
119895(119896) given by (6) we obtain the on-carrier
signal on subcarrier 119896 of user 119895 given by119880119895(119896) = 119904
119895(119896) + 119904
119895(119896) (12)
International Journal of Antennas and Propagation 5
with
119904119895(119896) = 119904
119895(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119895(119896) (13)
119904119895(119896) =
119873119906
sum
119906=1
119906 =119895
119904119906(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119906(119896) (14)
An arbitrary user 119895 has been singled out in (13) from theremaining users Thus 119904
119895(119896) represents the self-signal while
119904119895(119896) represents the IUI observed by user 119895 given by (14)This
interference is due to the loss of orthogonality of the precodedtransmission to multiple users caused by synchronizationimpairments Imposing (11) into 119880
119895(119896) in (7) we obtain the
ICI on subcarrier 119896 of user 119895 given by
119880119895(119896) =
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119904119906(])
119873119887
sum
119887=1
120573119895119887(119896 ])119867
119895119887(]) 119908
119887119906(]) (15)
Our complete discrete-frequency CoMP downlink signalmodel is then
119884119895(119896) = 119904
119895(119896) + 119904
119895(119896) + 119880
119895(119896) + 119873
119895(119896) (16)
with 119904119895(119896) 119904
119895(119896) and 119880
119895(119896) given by (13) (14) and (15)
respectively and 119873119895(119896) being the AWGN of power 119873
0120575119895 It
is noted that the model of Section 3 is valid independently iffor 120573
119895119894(119896 ]) the exact expression (8) or its simplification (9) is
used where the SFO is neglected
4 Analysis of Signal and Interference Powers
The following section contains an in-depth analysis of theimpact of synchronization impairments onto the perfor-mance It is organized as follows first we study the powerof a userrsquos self-signal (useful signal) and then the IUI andICI Next we show that ICI is small compared to IUI andprovide analytical expressions for the mean SIR Finally wehighlight the value of user selection and show its impact ontothe performance
Conceptually the rise of IUI and ICI due to imperfectsynchronization can be understood as a dispersion of theenergy allocated on a specific subcarrier of a specific user toother users (IUI) and to other subcarriers (ICI) This impliesa drop of the self-signal power and a rise on that user andthat subcarrier of the power of IUI and ICI from other usersrsquoand other subcarriersrsquo losses In what follows we proceedunder the assumption that data symbols are statisticallyindependent between users and across subcarriers that isE119904
119895(119896)119904
lowast
119906(]) = 119864
119904120575119895119906120575119896] where Esdot denotes the expectation
operator 119864119904the mean energy per data symbol and 120575
119909119910the
Kronecker delta between 119909 and 119910
41 Power of the Userrsquos Self-Signal Since there is statisticalindependence between data symbols channel coefficients
and synchronization parameters themeanpower of the userrsquosself-signal (13) is
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
E 1198671198951198871
1199081198871119895119867lowast
1198951198872
119908lowast
1198872119895
(17)
where subcarrier index 119896 has been omitted for simplicity ofnotation and cross-terms equal to zero have been alreadydisregarded
In a first step we analyze E1205731198951198871
120573lowast
1198951198872
which appears in(17) Using therefore ] = 119896 in (9) and replacing transmitterindex 119894 with base station index 119887 we reach
120573119895119887=
1
119873119904
sdot 119890minus1204852120587(119891
119895minus119891119887)119905119899 sdot
sin [120587 (119891119895minus 119891
119887) 119879119873
119904]
sin [120587 (119891119895minus 119891
119887) 119879]
(18)
which does not depend on the subcarrier index 119896 By notingthat |119891
119895minus 119891
119887| ≪ 1119879 we can safely say that the argument 119909 =
120587(119891119895minus 119891
119887)119879 of the sin(119909) in the denominator of the fraction
on the right-hand side of (18) is very small The same termalso appears in the exponential term and both multiplicativeterms in (18) are close to one But comparing the magnitudedeviations from unity we see that for typical synchroniza-tion parameters |1 minus (1119873
119904)(sin(119873
119904119909) sin(119909))| ≪ |1 minus
119890minus120485(214119899+1)119873
119904119909
| (this results in119873119904≫ 1 and a cyclic prefix equal
to 007119873119904[31]) In absolute terms we can further use the first
order Taylor series expansion (1119873119904)(sin(119873
119904119909) sin(119909)) asymp 1
and reach
1205731198951198871
120573lowast
1198951198872
asymp 119890minus1204852120587(119891
119895minus1198911198871)119905119899 sdot 119890
1204852120587(119891119895minus1198911198872)119905119899 = 119890
1204852120587(1198911198871minus1198911198872)119905119899 (19)
It is interesting to observe that the power of the exponentialterm as approximated in (19) depends on the differencebetween the base stationsrsquo carrier frequenciesThe power losseffect of the symbol transmitted on subcarrier 119896 which is thegenerating factor of ICI depends on both the CFOs of thebase stations and of the mobile usersThe effect that has beenneglected here due to its relatively small role compared tothe one of the exponential term will be however thoroughlyanalyzed in Section 42
For transmission from one base station that is for case1198871= 119887
2 it is immediate that (19) equals one For 119887
1= 119887
2
it is reasonable to assume that different base stationsrsquo carrierfrequencies 119891
1198871
and 1198911198872
are statistically independent whichallows for expressing the expectation of (19) as a product oftwo terms each depending on one base stationrsquos CFO
E 1205731198951198871
120573lowast
1198951198872
= E 1198901204852120587(119891
1198871minus119891119888)119905119899 sdot E 119890
minus1204852120587(1198911198872minus119891119888)119905119899 (20)
By assuming further that theCFOs are identically distributedthe second term of the right-hand side in (20) is the complex-conjugate of the first one hence (20) can be developed as
E 1205731198951198871
120573lowast
1198951198872
=10038161003816100381610038161003816E 119890
minus1204852120587(119891119887minus119891119888)11990511989910038161003816100381610038161003816
2
=
1003816100381610038161003816100381610038161003816int 119901
(119891119887minus119891119888)119890minus1204852120587(119891
119887minus119891119888)119905119899119889 (119891
119887minus 119891
119888)
1003816100381610038161003816100381610038161003816
2
(21)
6 International Journal of Antennas and Propagation
where 119901(119891119887minus119891119888)denotes the probability distribution function
(pdf) of the CFO (119891119887minus 119891
119888) and is independent on index 119887 It
is to be noted that the integral in (21) is a Fourier transformof the CFOrsquos pdf Hence we can write
E 1205731198951198871
120573lowast
1198951198872
=
1 1198871= 119887
2
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
1198871
= 1198872
(22)
where F119901(sdot) denotes the Fourier transform of 119901
(119891119887minus119891119888)and is
here a function of time For being a characteristic functionthat is the Fourier transform of a pdf it is guaranteedthat |F
119901(119905119899)| le 1 (see [32]) which means that the power
of the userrsquos self-signal (17) drops under imperfect carriersynchronization conditions
Result (22) can be used for developing (17) further byseparating the sum over 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= 119864119904sdot (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
) sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
(23)
The double sum in the last line in (23) can be formulated as
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= E
119873119887
sum
1198871=1
1198671198951198871
1199081198871119895sdot
119873119887
sum
1198872=1
119867lowast
1198951198872
119908lowast
1198872119895
(24)
which by considering the ZF condition
119873119887
sum
119887=1
119867119895119887119908119887119906= 120575
119895119906 (25)
is found to be equal to 1 Thus (23) can be written as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904[1 minus (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870U] (26)
where we defined119870U = 1 minussum119873119887
119887=1E|119867
119895119887119908119887119895|2
Note that (25)cannot be applied to the sum terms in the first and third line of(23) because of the power indexThe constant119870U depends onthe precoder and the channel statistical properties As shownin Appendix A for ZF precoding given a channel matrix with119873119887gt 119873
119906and complex Gaussian independent and identically
distributed (iid) entries (Rayleigh fading channel) withzero mean and variance 1205902
ℎ119870U can be approximated as
119870U asymp 1 minus1
119873119887minus 119873
119906
(27)
If we further consider the case in which the CFOs of thebase stations are Gaussian iid with zero mean and variance1205902
119891 the Fourier transform in (22) becomes
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
= 119890minus412058721205902
1198911199052
119899 asymp 1 minus 41205872
1205902
1198911199052
119899 (28)
where the first order Taylor series expansion 119890minus119909 asymp 1 minus 119909 hasbeen used It is to be noted that this approximation is accuratefor typical values of 120590
119891and 119905
119899 Using approximations (27) and
(28) we can formulate themeanpower of the userrsquos self-signal(26) as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904[1 minus 4120587
2
1205902
1198911199052
119899sdot (1 minus
1
119873119887minus 119873
119906
)] (29)
from where we see that it decreases linearly with the basestationsrsquo CFO variance and quadratically with time Result(29) also shows that the mean power of the self-signal growswith the number of base stations and drops with the numberof users It is noteworthy that for 119873
119906= 119873
119887minus 1 the mean
power of the self-signal remains constant However thisshould not be used as a system design rule as for determiningthe system performance the degradation due to IUI and ICImust be considered as well
42 Power of the Interuser Interference The derivation of themean power of the IUI (14) follows similar steps as the onesin Section 41 Concretely
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119906
sum
119906=1
119906 =119895
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
sdot E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(30)
and noting that in this case always 119895 = 119906 in (25) we find that
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904(1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI (31)
where119870IUI = sum119873119906
119906=1119906 =119895sum119873119887
119887=1E|119867
119895119887119908119887119906|2
If all users undergoidentical channel statistics119870IUI simplifies to119870IUI = (119873119906
minus1) sdot
sum119873119887
119887=1E|119867
119895119887119908119887119906|2
whereas using the results of Appendix Afor the Rayleigh fading channel and119873
119887gt 119873
119906 we find
119870IUI asymp119873119906minus 1
119873119887minus 119873
119906
(32)
Using (32) and (28) in (31) we obtain
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904sdot 4120587
2
1205902
1198911199052
119899sdot119873119906minus 1
119873119887minus 119873
119906
(33)
Result (33) reveals that the IUI power grows linearly withthe base stationsrsquo CFO variance and quadratically with time
International Journal of Antennas and Propagation 7
Using more base stations decreases the IUI power whileadding more users results in increasing it Due to theapproximation used in (19) it can be said that the impactof the mobile usersrsquo CFO onto the IUI power level is lesssignificant than the impact of the base stationsrsquo CFOs
43 Power of the Intercarrier Interference Following similarsteps as before now for (15) the mean power of the ICI is
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
sdot E 1198671198951198871
(]) 1199081198871119906(])119867lowast
1198951198872
(]) 119908lowast1198872119906(])
(34)
Assuming identical channel statistics for all subcarriers thelast term in (34) does not depend on subcarrier index ] andbecomes E119867
1198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906 already seen in Sections 41
and 42 For convenience we define1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) ≜
1198611 119887
1= 119887
2
1198612 119887
1= 1198872
(35)
which is analyzed inAppendix B In (34) we separate the sumover 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871as in (23) and obtain
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot 119861
1sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
= 119864119904sdot (119861
1minus 119861
2) sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(36)
We use the expressions for 1198611and 119861
2in Appendix B which
provide accurate approximations for Gaussian distributedCFOs for the base stations and the mobile user 119895 with zeromean and variances 1205902
119891and 1205902
119891119895 respectively and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot 119870ICI + 120590
2
119891119895) (37)
Here 119870ICI = sum119873119906
119906=1sum119873119887
119887=1E|119867
119895119887119908119887119906|2
= 119870IUI minus 119870U + 1 is aconstant For aRayleigh fadingMIMOchannel we use in (37)the expression provided by Appendix A for119870ICI and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot
119873119906
119873119887minus 119873
119906
+ 1205902
119891119895) (38)
Expression (38) shows that the mean power of the userrsquos ICIcan be split into two additive parts the first part dependingon the base stationsrsquo CFO variance which is multiplied witha weighting factor 119870ICI according to the cluster size and thesecond part depending on the userrsquos own CFO Furthermore(38) reveals that the power of the ICI does not vary with timeand that it is inverse to the square of the OFDM subcarrierspacing
44 ICI-to-IUI Ratio and SIR The relative magnitudes of thepower terms derived so far in this section are analyzed nextA simple expression for the mean ICI-to-IUI power ratio ofa user 119895 (an interference-to-interference ratio IIR) can beobtained from (33) and (38)
IIR =
E 10038161003816100381610038161003816119880119895
10038161003816100381610038161003816
2
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp1
1212057521199052119899
sdot (119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (39)
with 119873119887gt 119873
119906ge 2 and 120585 = 120590
119891119895120590
119891defined as the ratio
between the standard deviations of the userrsquos and the basestationsrsquo CFOs By replacing 120575 = (119873
119904119879)
minus1 and discrete time119905119899= (107119899 + 05)119873
119904119879 (this results in 119873
119904≫ 1 and a cyclic
prefix equal to 007119873119904[33]) relation (39) becomes
IIR asymp1
12 (107119899 + 05)2sdot (
119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (40)
It is interesting to observe that in the approximation (40)the IIR does not depend on the OFDM system parametersbut only on the OFDM symbols index 119899
Regarding the ratio 120585 it has been shown in [14 21]that mobile terminals attain a carrier frequency accuracywhich is at least one order of magnitude below the accuracyof typical base station oscillators used in Third GenerationPartnership Project (3GPP) Long Term Evolution (LTE) [31]Assuming for instance 1205852 ≫ 1 the first additive term in (40)becomes much smaller than the second term indicating thatthe terminalsrsquo CFO is the main source of ICI Using 120585 = 0one can evaluate the IIR including only the ICI part due to thebase stationsrsquo CFOs in this case the ratio does not depend onthe base stationsrsquo CFOs at all Using a more practical value offor example 120585 = 10 the largest possible IIR after 119899 = 14
OFDM symbols (119905119899asymp 1ms) which occurs with 119873
119906= 2
users given 119873119887= 7 equals minus73 dB The IIR drops quickly
down already to minus273 dB for 119899 = 140 (119905119899asymp 10ms) These
quantitive results show that for typical JT CoMP scenariosthe IUI dominates over the ICI
Wewill nowneglect the ICI and define themean SIR (self-user signal-to-IUI ratio) by the ratio between (26) and (31)
SIR =
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
(1 minus10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI
minus1
119873119906minus 1
(41)
For a MIMO channel with iid Rayleigh fading entries (119870IUIgiven by (32)) and for iid Gaussian CFOs (|F
119901(119905119899)|2 given
by (28)) (41) can be simplified as
SIR asymp1
412058721205902
1198911199052119899
sdot119873119887minus 119873
119906
119873119906minus 1
minus119873119887minus 119873
119906+ 1
119873119906minus 1
(42)
8 International Journal of Antennas and Propagation
Expressions (41) and (42) are valid for 119873119887gt 119873
119906ge 2
Expression (42) reveals that for ZF precoding the mean SIRis in an approximately inverse relation to the base stationsrsquoCFO variance and to the square of time Furthermore it isshown that the SIR grows with the number of base stationsand drops with the number of jointly served users
45 TheValue of User Selection in JTCoMPand an SIRBoundThe SNR gains in JT CoMP are increased if a schedulerselects the appropriate users to be jointly served on thesame time and frequency resources As shown in [34] theselection criteria are based on the rule that all users gainfrom the cooperation in the cluster compared to the case ofnoncoordinated transmission Evaluated on a field scenariothe resulting singular value statistics after such user selectionwas found to be comparable to the one of a Rayleigh fadingchannel a fact that also relates to scenarios where users arelocated close to the cell edge that is in which gains throughcoordination are particularly large
Considering now an idealized scenario from the precod-ing point of view with orthogonal channel vectors of equalpower among the users scenario which has been analyzed in[5 11] an upper bound can be derived for the mean SIR usingZF precoding in JT CoMP Using the expression for 119870IUI asprovided in Appendix A for 119873
119887gt 119873
119906ge 2 we obtain the
following expression
SIRmax asymp1
412058721205902
1198911199052119899
sdot119873119887
119873119906minus 1
minus119873119887+ 1
119873119906minus 1
(43)
The distance between (42) and (43) reveals the potential forSIR enhancement if the usersrsquo selection process considers theorthogonality among their channel vectors
It should be clarified at this point that (42) and (43) do notconsider any gains from resource allocation the evaluationof which not in the scope of this work In an orthogonalfrequency division multiple access (OFDMA) system eachuser can be assigned a part of the spectrum in a way that hisperformance and the network performance can be optimizedas known from [35] and also shown in [36] for the multiuserdownlink
The signal model given in Section 3 for the impairedOFDM-based JT CoMP is valid for any OFDMA schemeUsing OFDMA in the downlink does not affect the signalstructure of a userrsquos received self-signal and the IUI TheICI also has the same structure and statistical propertiesas typically data are transmitted on all subcarriers which isrecommended for keeping frequency-flat distribution of theICI power [13] The degradation mechanisms due to multipleCFOs and SFOs as described in our model will be the samefor any system using OFDM
What indeed changes in OFDMA is the channel statis-tics for each user after resource allocation Therefore ifintended to evaluate the overall performance of a coordinatedmultipoint system using OFDMA the general mean powerexpressions (26) (31) and (37) would need to be evaluated forthe actual usersrsquo channel statistics This procedure practicallyimplies a calculation of constants119870U119870IUI and119870ICI
In this section the received power of a user was studiedin relation to the interference from other users and othersubcarriers in amulticellular multiuser downlink with coop-erative base stations Closed-form expressions and accurateapproximations for all relevant termswere derivedMoreoverit was demonstrated that interuser interference has the mostrelevant effect Finally the impact of the radio channel wasinvestigated and analytical SIR expressions were derived forzero-forcing precoding
5 Evaluation and Numerical Validation
In this section analytical results of Section 4 are evaluatedand verified by simulations Based on those results adequaterequirements are determined for the base stationsrsquo oscillatorsin CoMP systems
A JT CoMP scenario is considered where a cooperationcluster of 119873
119887= 7 base stations transmits jointly by using
zero-forcing precoding to 119873119906terminals on the same time
and frequency resource with 119873119887gt 119873
119906ge 2 Out-of-cluster
interference is not consideredAt time instant 119905 = 0 the base stations receive an update
of the downlink channel matrix and compute a precodingmatrix This will be used during the following 10ms timeat which a new channel update will be obtained and a newprecoder will be calculated This routine agrees with thecurrent 3GPP LTE channel and precoder updating cycle [31]As already mentioned in Section 3 it is assumed that theradio channel is perfectly estimated and available at the basestations at 119905 = 0 whereas channel aging effects are notconsidered either Between updating instants the phases ofthe local base stationsrsquo oscillators slowly drift away fromeach other because of the individual frequency offsets withrespect to the ideal carrier frequency 119891
119888 As a consequence
self-signal drops IUI grows and overall the SIR drops withtime In a practical system each user can use downlink pilotsymbols to estimate and equalize its own CFO and avoid ICIenhancement However mobiles cannot stop the IUI fromrising because the degradation is caused jointly by all themisaligned base station oscillators
The oscillator accuracy Osc typically specified in partsper million (ppm) or parts per billion (ppb) relates to thestandard deviation of the resulting carrier frequency as 120590
119891=
Osc sdot 119891119888 The CFO variation over time is assumed to be slow
enough to be considered static with respect to the signalingand precoder updating timescale Here Gaussian iid CFOsare randomly assigned to base stations and mobile users allwith zero mean and standard deviations given by 120590
119891and 120590
119891119895
respectivelyTypical 3GPP LTE parameters are used specified in [31]
The broadband OFDM signal has 119873119904= 2048 subcarriers
separated by 120575 = 15 kHzThe length of the cyclic prefix is 144samples resulting in OFDM symbols with a length of 119873
119892=
2192 samples The ideal carrier and sampling frequencies are119891119888= 265GHz and 1119879 = 3072MHz respectively The
energy per data symbol is set to 119864119904= 1
Figure 2 depicts the mean power of the IUI and ICI overtime passed from the last precoder update (in logarithmicscale) for 3 and 6 mobile users a base station oscillator
International Journal of Antennas and Propagation 9
0
Time (ms)
Mea
n po
wer
(dB)
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
10minus1 100 101
IUI Nu = 3 Osc = 1ppbICI Nu = 3 Osc = 1ppb 120585 = 0
ICI Nu = 3 Osc = 1ppb 120585 = 10
IUI Nu = 6 Osc = 1ppbICI Nu = 6 Osc = 1ppb 120585 = 0
ICI Nu = 6 Osc = 1ppb 120585 = 0
Figure 2 Mean power of interuser and intercarrier interferences ina Rayleigh fading channel Here 7 base stations serve 3 and 6 usersrespectively Analytical results are shown by lines while markersshow numerical evaluations
accuracy of 1 ppb and Rayleigh fading channel conditionsThe curves illustrate analytical results given by (33) and(38) whereas solid lines depict results for 3 users anddashed lines for 6 users respectivelyMarkers show respectivenumerical evaluations of themeanpower of (14) and (15) over10
3 independent realizations of the MIMO channel matrixentries with 120590
2
ℎ= 1 and of the basesrsquo CFOs Regarding
the ICI two scenarios are considered a scenario with 120585 =
0 that is perfectly synchronized mobile users in order toevaluate solely the ICI due to the base stationsrsquo CFOs (red)and a second scenario with nonsynchronized mobiles with aCFO accuracy of an order of magnitude lower than the basestationsrsquo one that is 120585 = 10 (black) where the total ICI isevaluated
First of all Figure 2 verifies the analytical expres-sions (includingmathematical approximations) by numericalmeans It is further observed that the IUI grows approxi-mately with the square of time and that it increases with thenumber of users Comparing the mean power levels of IUI(blue curves) with the ICI both caused by the base stationsrsquoCFOs (red curves for 120585 = 0) we see that independently of thenumber of users and for typical feedback delays between 2msand 10ms the IUI lies between 40 dB and 50 dB above thepower of ICIThis clearly indicates that the IUI is significantlystronger than the ICI caused by the base stationsrsquo CFOsConsidering now the second scenario with nonperfectlysynchronized mobile users (120585 = 10) it can be observed thatthe part of the ICI caused by the usersrsquo CFO overwhelms theICI caused by the base stationsrsquo CFOs According to (38) the(dominant) ICI due to a mobile userrsquos CFO does not depend
SIR
(dB)
0
10
20
30
40
50
60
70
80
90
Time (ms)10minus1 100 101
20dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 3 Mean SIR over time for Rayleigh fading channel and SIRupper bound Here 7 base stations using oscillators of accuracygiven by Osc serve jointly 6 users Analytical results are shown bylines simulations by markers
on the number of served users which is reflected in the factthat the black curves in Figure 2 evaluating the total ICI for120585 = 10 almost overlap for the cases of 3 and 6 users It isalso interesting to observe that the IUI power level due tothe cooperating base stationsrsquo CFOs is still around 20 dB to30 dB (for 3 users) and 30 dB to 40 dB (for 6 users) higherthan the total ICI power considering typical feedback delaysbetween 2ms and 10ms These results clearly show that inthe a cooperative network even with typically nonperfectlysynchronized mobile users the main interference source liesin the base stationsrsquo CFOs
Figure 3 shows the SIR over the time as defined in (41)for119873
119887= 7 base stations serving jointly119873
119906= 6mobile users
which is also the highest possible number for the Rayleighfading channel The influence of the base stationsrsquo oscillatoraccuracy is evaluated and the corresponding ICI is neglectedas it is significantly smaller than the IUI Disregarding the ICIalso means that the synchronicity level of the mobile users isnot relevant as it does not contribute the IUI
For the Rayleigh fading channel the analytical expression(42) is compared with the ratio between the numericallyevaluated mean power of (13) and (14) over 103 independentchannel and CFO realizations which validates our analysisincluding the accuracy of the mathematical approximationsSimilarly for the SIR upper bound expression the analyticalexpression (43) is compared with results from numericalevaluation From Figure 3 it is seen that a degradation of thebase station oscillatorsrsquo accuracy by one order of magnitudeincreases the IUI and thus decreases the SIR by around 20 dBFurthermore the SIR drops approximately quadratically withtime that is 20 dB per time decade In terms of accuracyrequirements it is found that for reaching in average anSNR of 20 dB 10ms after the precoder update (free-running)oscillators of 1 ppb are needed
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
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DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 3
Baseband processing
Baseband processing
Transmitter Receiver
DA AD
Ti Tjfi fi
Figure 1 Transmitter and receiver have individual sampling periods 119879119894and 119879
119895for digital-to-analog and analog-to-digital conversion and
individual carrier frequencies 119891119894and 119891
119895for upconversion and downconversion
where 119880119895(119891 119896) represents the continuous-frequency spec-
trum of the received multiuser signal at receive antenna 119895for transmitted subcarrier 119896 (on-carrier signal) It is notedthat 119880
119895(119891 119896) is the received spectrum of the multiuser signal
of all other subcarriers ] = 119896 that is the multiuser ICI(it is arbitrary which subcarrier is designated as 119896 but ouranalysis requires to single out one of them) The additivewhite Gaussian noise (AWGN) contributed by the front-endof receive antenna 119895 is denoted by119873
119895(119891)The above terms are
given by
119880119895(119891 119896) = 119879
119895
119873119905
sum
119894=1
119890120485(120593119895minus120593119894)
120573119895119894(119891 119896)
sdot 119883119894(119896)119867
119895119894(119896120575
119894+ 119891
119894minus 119891
119888)
(2)
119880119895(119891 119896) = 119879
119895
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119905
sum
119894=1
119890120485(120593119895minus120593119894)
120573119895119894(119891 ])
sdot 119883119894(])119867
119895119894(]120575
119894+ 119891
119894minus 119891
119888)
(3)
with
120573119895119894(119891 119896) = 119890
minus1204852120587(119891119895minus119891119894)(119899119873119892119879119895+120591119895)
sdot 119890minus1204852120587119896120575
119894(119899119873119892119879119894+120591119894minus119899119873119892119879119895minus120591119895)
sdot 119890minus120485120587(119891+119891
119895minus119891119894minus119896120575119894)119879119895(119873119904minus1)
sdot
sin [120587 (119891 + 119891119895minus 119891
119894minus 119896120575
119894) 119879
119895119873119904]
sin [120587 (119891 + 119891119895minus 119891
119894minus 119896120575
119894) 119879
119895]
(4)
The exponential terms in (4) are phase shifts due to carrierand sampling frequency misalignment while the fractionalterm describes the loss of orthogonality among OFDMsubcarriers causing a leakage of the signal transmitted onsubcarrier 119896 Note that 119867
119895119894(119896120575
119894+ 119891
119894minus 119891
119888) expresses the
channel at the frequency of subcarrier 119896 plus a shift due tothe transmitterrsquos SFO and CFO
Themodel given by (1) through (4) makes no assumptionabout the synchronization among branches and it is gen-eral for any MIMO-OFDM communication also includingcellular networks with base station cooperation It describeshow successive OFDM symbols are degraded under constantand uncompensated CFOs and SFOs as time goes by that
is OFDM symbol index 119899 grows The CFO must be smallerthan half of a subcarrier spacing For typical mobile receiversthis implies that an earlier coarse frequency synchronizationstage has succeeded which we assume henceforth RegardingSFO the expressions are valid well beyond the typical rangespecified for SFO in commercial OFDM systems It is to benoted however that for a given SFO the FFT window ofeach receiver branch does eventually drift away to a point atwhich intersymbol interference (ISI) arises between OFDMsymbols From that point on severe degradation ensues andthe model stops being valid It is also implicit in our modelthat OFDM symbol timing has been acquired in a priorstage The result assumes that the time dispersion of all theMIMOchannel impulse responses also fromdistributed basestations and mobile users is shorter than the OFDM cyclicprefix in use
Returning to the model it can be observed in (2) and(3) that the phase offsets due to 120591
119894 120591
119895 120593
119894 and 120593
119895may be
considered without loss of generality as part of the channelIt follows that we may choose 120591
119894= 0 120591
119895= 0 120593
119894= 0 and
120593119895= 0 We also point out that in a practical implementation
for the 119895th OFDM receiver branch the output of its FFTcorresponds to a sequence of samples of 119884
119895(119891) taken at
frequencies 119891 = 119897119873119904119879119895= 119897120575
119895 where 119897 is the subcarrier
index (minus1198731199042 le 119897 le 119873
1199042 minus 1) and 120575
119895is the receiver-side
intercarrier spacing Note that 119897 points to a slightly differentfrequency at each receive branch due to the different SFO andgenerally also to a different frequency than that pointed atby index 119896 at the various transmit branches 119894 Imposing theabove conditions on (2) and (3) and focusing on an arbitraryreceived subcarrier 119897 = 119896 we obtain the following discrete-frequency signal model
119884119895(119896) = 119880
119895(119896) + 119880
119895(119896) + 119873
119895(119896) (5)
with
119880119895(119896) =
119873119905
sum
119894=1
120573119895119894(119896 119896)119883
119894(119896)119867
119895119894(119896) (6)
119880119895(119896) =
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119905
sum
119894=1
120573119895119894(119896 ]) 119883
119894(])119867
119895119894(]) (7)
4 International Journal of Antennas and Propagation
120573119895119894(119896 ]) = 119890minus1204852120587119899119873119892119879119895(119891119895minus119891119894)
sdot 119890minus1204852120587119899119873
119892]120575119894(119879119894minus119879119895)
sdot 119890minus120485120587((119873
119904minus1)119873
119904)[(119896minus](119879
119895119879119894))+(119891119895minus119891119894)119879119895119873119904]
sdot1
119873119904
sdot
sin [120587 (119896 minus ] (119879119895119879
119894)) + 120587 (119891
119895minus 119891
119894) 119879
119895119873119904]
sin [(120587119873119904) (119896 minus ] (119879
119895119879
119894)) + 120587 (119891
119895minus 119891
119894) 119879
119895]
(8)
and with 119873119895(119896) being the receiver-side AWGN of power
1198730120575119895 Note that in (6) and (7) we have approximated and
defined119867119895119894(119896120575
119894+119891
119894minus119891
119888) asymp 119867
119895119894(119896120575
119894) ≜ 119867
119895119894(119896) because |119891
119894minus119891
119888|
is assumed to bemuch smaller than the coherence bandwidthof the channel
In practical system implementation the carrier andsampling frequency clocks of a transmitter or receiver arederived from the same reference that is from the same localoscillator Thus for the product of an arbitrary 119894th carrierfrequency and sampling period119891
119894sdot119879119894≜ 120581 it holds that 120581 ≫ 1
as the (ideal) carrier frequency 119891119888is two to three orders of
magnitude larger than the (ideal) sampling frequency 1119879The constant 120581 depends only on the system and hardwaredesign and is independent of 119894
Considering this relationship in (8) it is straightforwardto see that the exponent in the expressionrsquos second line whichcaptures the SFO effect on 120573
119895119894(119896 ]) and which is maximized
for ] = 119873119904 still remains 120581 times smaller than the exponent
in the first line capturing the CFO effect Similar findings canbe observed comparing the influence of SFO with the one ofthe CFO onto the terms in the third and the fourth line of(8) Thus it can be safely said that the impact of the SFO on120573119895119894(119896 ]) is significantly weaker than the impact of the CFO
when using the same oscillator reference for both at eachindividual transmitter and receiver
The derivations up to here have been presented includingboth CFO and SFO for the sake of completeness as well asfor further reference Expressions (5) through (8) provide theexact signal model which characterizes the joint effect ofmultiple CFOs and SFOs over consecutive OFDM symbolsin MIMO transmissions and is one important contributionof this work However beyond signal modeling this workfurther aims to identify the major degradation sourcesanalyze the performance degradation and determine fromthere the actual system requirements To this end in thefollowing we focus on the CFO and neglect the SFO byassuming ideal sampling for all transmitters and receiversWith 119879
119894= 119879
119895= 119879 (8) becomes
120573119895119894(119896 ]) = 119890minus1204852120587[(119891119895minus119891119894)119905119899+((119873119904minus1)2119873119904)(119896minus])]
sdot1
119873119904
sin [120587 (119896 minus ]) + 120587 (119891119895minus 119891
119894) 119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119894) 119879]
(9)
The discrete time variable 119905119899≜ (119899119873
119892+ (119873
119904minus 1)2)119879 has been
defined for convenience and is measured in secondsExpressions (5) to (9) provide the discrete-frequency
signal model for a MIMO-OFDM communication system in
the presence of multiple CFOs and SFOs as a function of timeand will be used in the following section
3 Precoded Multicell Downlink Signal Model
Now we specialize the model obtained in Section 2 to theCoMPdownlink including joint precoding of the data signalsHere the required channel state information (CSI) for theprecoder calculation is estimated either at the base stations orat the terminals and fed back to the base stations dependingonwhether timedivision duplex (TDD)or frequency divisionduplex (FDD) is used For distributed base stations coordi-nation by means of a backhaul network is considered whichenables fast exchange of data and CSI
It is to be noted that channel estimation errors and CSIdelays due to the feedback and the backhaul network arenot considered in this work Their effect on JT CoMP isimportant andmight even overwhelm the effects of imperfectsynchronization which we are analyzing here A distinctanalysis by the authors which includes the effect of imperfectchannel knowledge and derives the resulting performancelimitations can be found in [33] We will therefore assumehere that perfect knowledge of the downlink MIMO channelmatrix is available at the base stations and is used for real-timeprecoding of the downlink signals As already mentioned inSection 2 residual phase terms due toCFOs and SFOs are alsoconsidered as part of the (perfectly estimated) channel Weconsider 119873
119906users equipped with single-antenna terminals
which are jointly served by119873119887coordinated antenna branches
among all base stations forming the cooperation clusterThere is no exchange of information between mobile usersand out-of-cluster transmission is not considered in ourmodel
Transmissions are precoded on each OFDM subcarrier 119896with the right-hand pseudoinverse of the119873
119906times 119873
119887downlink
MIMO channel matrixH(119896) for that subcarrier given by
W (119896) = H119867
(119896) [H (119896)H119867
(119896)]minus1
(10)
The inverse is of size 119873119887times 119873
119906and we assume it exists for
119873119887gt 119873
119906 Note that this condition can be met in practice
by appropriate user selection and clustering of base stationswhich is thereby assumed
Next consider S(119896) to be an 119873119906times 1 vector that contains
the complex-valued data symbols 119904119906(119896) to be transmitted to
the119873119906users on subcarrier 119896 Then the119873
119887times 1 vector X(119896) of
precoded symbols119883119887(119896) to be transmitted on subcarrier 119896 by
the119873119887base stations is X(119896) = W(119896)S(119896) where elements are
given by
119883119887(119896) =
119873119906
sum
119906=1
119908119887119906(119896) 119904
119906(119896) (11)
and 119908119887119906(119896) are the elements ofW(119896) Transmitter index 119894 has
been replaced with 119887 to stress that from here on transmittersare base stations Imposing the expression of the precodedsymbol (11) into 119880
119895(119896) given by (6) we obtain the on-carrier
signal on subcarrier 119896 of user 119895 given by119880119895(119896) = 119904
119895(119896) + 119904
119895(119896) (12)
International Journal of Antennas and Propagation 5
with
119904119895(119896) = 119904
119895(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119895(119896) (13)
119904119895(119896) =
119873119906
sum
119906=1
119906 =119895
119904119906(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119906(119896) (14)
An arbitrary user 119895 has been singled out in (13) from theremaining users Thus 119904
119895(119896) represents the self-signal while
119904119895(119896) represents the IUI observed by user 119895 given by (14)This
interference is due to the loss of orthogonality of the precodedtransmission to multiple users caused by synchronizationimpairments Imposing (11) into 119880
119895(119896) in (7) we obtain the
ICI on subcarrier 119896 of user 119895 given by
119880119895(119896) =
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119904119906(])
119873119887
sum
119887=1
120573119895119887(119896 ])119867
119895119887(]) 119908
119887119906(]) (15)
Our complete discrete-frequency CoMP downlink signalmodel is then
119884119895(119896) = 119904
119895(119896) + 119904
119895(119896) + 119880
119895(119896) + 119873
119895(119896) (16)
with 119904119895(119896) 119904
119895(119896) and 119880
119895(119896) given by (13) (14) and (15)
respectively and 119873119895(119896) being the AWGN of power 119873
0120575119895 It
is noted that the model of Section 3 is valid independently iffor 120573
119895119894(119896 ]) the exact expression (8) or its simplification (9) is
used where the SFO is neglected
4 Analysis of Signal and Interference Powers
The following section contains an in-depth analysis of theimpact of synchronization impairments onto the perfor-mance It is organized as follows first we study the powerof a userrsquos self-signal (useful signal) and then the IUI andICI Next we show that ICI is small compared to IUI andprovide analytical expressions for the mean SIR Finally wehighlight the value of user selection and show its impact ontothe performance
Conceptually the rise of IUI and ICI due to imperfectsynchronization can be understood as a dispersion of theenergy allocated on a specific subcarrier of a specific user toother users (IUI) and to other subcarriers (ICI) This impliesa drop of the self-signal power and a rise on that user andthat subcarrier of the power of IUI and ICI from other usersrsquoand other subcarriersrsquo losses In what follows we proceedunder the assumption that data symbols are statisticallyindependent between users and across subcarriers that isE119904
119895(119896)119904
lowast
119906(]) = 119864
119904120575119895119906120575119896] where Esdot denotes the expectation
operator 119864119904the mean energy per data symbol and 120575
119909119910the
Kronecker delta between 119909 and 119910
41 Power of the Userrsquos Self-Signal Since there is statisticalindependence between data symbols channel coefficients
and synchronization parameters themeanpower of the userrsquosself-signal (13) is
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
E 1198671198951198871
1199081198871119895119867lowast
1198951198872
119908lowast
1198872119895
(17)
where subcarrier index 119896 has been omitted for simplicity ofnotation and cross-terms equal to zero have been alreadydisregarded
In a first step we analyze E1205731198951198871
120573lowast
1198951198872
which appears in(17) Using therefore ] = 119896 in (9) and replacing transmitterindex 119894 with base station index 119887 we reach
120573119895119887=
1
119873119904
sdot 119890minus1204852120587(119891
119895minus119891119887)119905119899 sdot
sin [120587 (119891119895minus 119891
119887) 119879119873
119904]
sin [120587 (119891119895minus 119891
119887) 119879]
(18)
which does not depend on the subcarrier index 119896 By notingthat |119891
119895minus 119891
119887| ≪ 1119879 we can safely say that the argument 119909 =
120587(119891119895minus 119891
119887)119879 of the sin(119909) in the denominator of the fraction
on the right-hand side of (18) is very small The same termalso appears in the exponential term and both multiplicativeterms in (18) are close to one But comparing the magnitudedeviations from unity we see that for typical synchroniza-tion parameters |1 minus (1119873
119904)(sin(119873
119904119909) sin(119909))| ≪ |1 minus
119890minus120485(214119899+1)119873
119904119909
| (this results in119873119904≫ 1 and a cyclic prefix equal
to 007119873119904[31]) In absolute terms we can further use the first
order Taylor series expansion (1119873119904)(sin(119873
119904119909) sin(119909)) asymp 1
and reach
1205731198951198871
120573lowast
1198951198872
asymp 119890minus1204852120587(119891
119895minus1198911198871)119905119899 sdot 119890
1204852120587(119891119895minus1198911198872)119905119899 = 119890
1204852120587(1198911198871minus1198911198872)119905119899 (19)
It is interesting to observe that the power of the exponentialterm as approximated in (19) depends on the differencebetween the base stationsrsquo carrier frequenciesThe power losseffect of the symbol transmitted on subcarrier 119896 which is thegenerating factor of ICI depends on both the CFOs of thebase stations and of the mobile usersThe effect that has beenneglected here due to its relatively small role compared tothe one of the exponential term will be however thoroughlyanalyzed in Section 42
For transmission from one base station that is for case1198871= 119887
2 it is immediate that (19) equals one For 119887
1= 119887
2
it is reasonable to assume that different base stationsrsquo carrierfrequencies 119891
1198871
and 1198911198872
are statistically independent whichallows for expressing the expectation of (19) as a product oftwo terms each depending on one base stationrsquos CFO
E 1205731198951198871
120573lowast
1198951198872
= E 1198901204852120587(119891
1198871minus119891119888)119905119899 sdot E 119890
minus1204852120587(1198911198872minus119891119888)119905119899 (20)
By assuming further that theCFOs are identically distributedthe second term of the right-hand side in (20) is the complex-conjugate of the first one hence (20) can be developed as
E 1205731198951198871
120573lowast
1198951198872
=10038161003816100381610038161003816E 119890
minus1204852120587(119891119887minus119891119888)11990511989910038161003816100381610038161003816
2
=
1003816100381610038161003816100381610038161003816int 119901
(119891119887minus119891119888)119890minus1204852120587(119891
119887minus119891119888)119905119899119889 (119891
119887minus 119891
119888)
1003816100381610038161003816100381610038161003816
2
(21)
6 International Journal of Antennas and Propagation
where 119901(119891119887minus119891119888)denotes the probability distribution function
(pdf) of the CFO (119891119887minus 119891
119888) and is independent on index 119887 It
is to be noted that the integral in (21) is a Fourier transformof the CFOrsquos pdf Hence we can write
E 1205731198951198871
120573lowast
1198951198872
=
1 1198871= 119887
2
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
1198871
= 1198872
(22)
where F119901(sdot) denotes the Fourier transform of 119901
(119891119887minus119891119888)and is
here a function of time For being a characteristic functionthat is the Fourier transform of a pdf it is guaranteedthat |F
119901(119905119899)| le 1 (see [32]) which means that the power
of the userrsquos self-signal (17) drops under imperfect carriersynchronization conditions
Result (22) can be used for developing (17) further byseparating the sum over 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= 119864119904sdot (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
) sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
(23)
The double sum in the last line in (23) can be formulated as
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= E
119873119887
sum
1198871=1
1198671198951198871
1199081198871119895sdot
119873119887
sum
1198872=1
119867lowast
1198951198872
119908lowast
1198872119895
(24)
which by considering the ZF condition
119873119887
sum
119887=1
119867119895119887119908119887119906= 120575
119895119906 (25)
is found to be equal to 1 Thus (23) can be written as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904[1 minus (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870U] (26)
where we defined119870U = 1 minussum119873119887
119887=1E|119867
119895119887119908119887119895|2
Note that (25)cannot be applied to the sum terms in the first and third line of(23) because of the power indexThe constant119870U depends onthe precoder and the channel statistical properties As shownin Appendix A for ZF precoding given a channel matrix with119873119887gt 119873
119906and complex Gaussian independent and identically
distributed (iid) entries (Rayleigh fading channel) withzero mean and variance 1205902
ℎ119870U can be approximated as
119870U asymp 1 minus1
119873119887minus 119873
119906
(27)
If we further consider the case in which the CFOs of thebase stations are Gaussian iid with zero mean and variance1205902
119891 the Fourier transform in (22) becomes
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
= 119890minus412058721205902
1198911199052
119899 asymp 1 minus 41205872
1205902
1198911199052
119899 (28)
where the first order Taylor series expansion 119890minus119909 asymp 1 minus 119909 hasbeen used It is to be noted that this approximation is accuratefor typical values of 120590
119891and 119905
119899 Using approximations (27) and
(28) we can formulate themeanpower of the userrsquos self-signal(26) as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904[1 minus 4120587
2
1205902
1198911199052
119899sdot (1 minus
1
119873119887minus 119873
119906
)] (29)
from where we see that it decreases linearly with the basestationsrsquo CFO variance and quadratically with time Result(29) also shows that the mean power of the self-signal growswith the number of base stations and drops with the numberof users It is noteworthy that for 119873
119906= 119873
119887minus 1 the mean
power of the self-signal remains constant However thisshould not be used as a system design rule as for determiningthe system performance the degradation due to IUI and ICImust be considered as well
42 Power of the Interuser Interference The derivation of themean power of the IUI (14) follows similar steps as the onesin Section 41 Concretely
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119906
sum
119906=1
119906 =119895
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
sdot E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(30)
and noting that in this case always 119895 = 119906 in (25) we find that
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904(1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI (31)
where119870IUI = sum119873119906
119906=1119906 =119895sum119873119887
119887=1E|119867
119895119887119908119887119906|2
If all users undergoidentical channel statistics119870IUI simplifies to119870IUI = (119873119906
minus1) sdot
sum119873119887
119887=1E|119867
119895119887119908119887119906|2
whereas using the results of Appendix Afor the Rayleigh fading channel and119873
119887gt 119873
119906 we find
119870IUI asymp119873119906minus 1
119873119887minus 119873
119906
(32)
Using (32) and (28) in (31) we obtain
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904sdot 4120587
2
1205902
1198911199052
119899sdot119873119906minus 1
119873119887minus 119873
119906
(33)
Result (33) reveals that the IUI power grows linearly withthe base stationsrsquo CFO variance and quadratically with time
International Journal of Antennas and Propagation 7
Using more base stations decreases the IUI power whileadding more users results in increasing it Due to theapproximation used in (19) it can be said that the impactof the mobile usersrsquo CFO onto the IUI power level is lesssignificant than the impact of the base stationsrsquo CFOs
43 Power of the Intercarrier Interference Following similarsteps as before now for (15) the mean power of the ICI is
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
sdot E 1198671198951198871
(]) 1199081198871119906(])119867lowast
1198951198872
(]) 119908lowast1198872119906(])
(34)
Assuming identical channel statistics for all subcarriers thelast term in (34) does not depend on subcarrier index ] andbecomes E119867
1198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906 already seen in Sections 41
and 42 For convenience we define1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) ≜
1198611 119887
1= 119887
2
1198612 119887
1= 1198872
(35)
which is analyzed inAppendix B In (34) we separate the sumover 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871as in (23) and obtain
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot 119861
1sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
= 119864119904sdot (119861
1minus 119861
2) sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(36)
We use the expressions for 1198611and 119861
2in Appendix B which
provide accurate approximations for Gaussian distributedCFOs for the base stations and the mobile user 119895 with zeromean and variances 1205902
119891and 1205902
119891119895 respectively and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot 119870ICI + 120590
2
119891119895) (37)
Here 119870ICI = sum119873119906
119906=1sum119873119887
119887=1E|119867
119895119887119908119887119906|2
= 119870IUI minus 119870U + 1 is aconstant For aRayleigh fadingMIMOchannel we use in (37)the expression provided by Appendix A for119870ICI and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot
119873119906
119873119887minus 119873
119906
+ 1205902
119891119895) (38)
Expression (38) shows that the mean power of the userrsquos ICIcan be split into two additive parts the first part dependingon the base stationsrsquo CFO variance which is multiplied witha weighting factor 119870ICI according to the cluster size and thesecond part depending on the userrsquos own CFO Furthermore(38) reveals that the power of the ICI does not vary with timeand that it is inverse to the square of the OFDM subcarrierspacing
44 ICI-to-IUI Ratio and SIR The relative magnitudes of thepower terms derived so far in this section are analyzed nextA simple expression for the mean ICI-to-IUI power ratio ofa user 119895 (an interference-to-interference ratio IIR) can beobtained from (33) and (38)
IIR =
E 10038161003816100381610038161003816119880119895
10038161003816100381610038161003816
2
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp1
1212057521199052119899
sdot (119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (39)
with 119873119887gt 119873
119906ge 2 and 120585 = 120590
119891119895120590
119891defined as the ratio
between the standard deviations of the userrsquos and the basestationsrsquo CFOs By replacing 120575 = (119873
119904119879)
minus1 and discrete time119905119899= (107119899 + 05)119873
119904119879 (this results in 119873
119904≫ 1 and a cyclic
prefix equal to 007119873119904[33]) relation (39) becomes
IIR asymp1
12 (107119899 + 05)2sdot (
119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (40)
It is interesting to observe that in the approximation (40)the IIR does not depend on the OFDM system parametersbut only on the OFDM symbols index 119899
Regarding the ratio 120585 it has been shown in [14 21]that mobile terminals attain a carrier frequency accuracywhich is at least one order of magnitude below the accuracyof typical base station oscillators used in Third GenerationPartnership Project (3GPP) Long Term Evolution (LTE) [31]Assuming for instance 1205852 ≫ 1 the first additive term in (40)becomes much smaller than the second term indicating thatthe terminalsrsquo CFO is the main source of ICI Using 120585 = 0one can evaluate the IIR including only the ICI part due to thebase stationsrsquo CFOs in this case the ratio does not depend onthe base stationsrsquo CFOs at all Using a more practical value offor example 120585 = 10 the largest possible IIR after 119899 = 14
OFDM symbols (119905119899asymp 1ms) which occurs with 119873
119906= 2
users given 119873119887= 7 equals minus73 dB The IIR drops quickly
down already to minus273 dB for 119899 = 140 (119905119899asymp 10ms) These
quantitive results show that for typical JT CoMP scenariosthe IUI dominates over the ICI
Wewill nowneglect the ICI and define themean SIR (self-user signal-to-IUI ratio) by the ratio between (26) and (31)
SIR =
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
(1 minus10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI
minus1
119873119906minus 1
(41)
For a MIMO channel with iid Rayleigh fading entries (119870IUIgiven by (32)) and for iid Gaussian CFOs (|F
119901(119905119899)|2 given
by (28)) (41) can be simplified as
SIR asymp1
412058721205902
1198911199052119899
sdot119873119887minus 119873
119906
119873119906minus 1
minus119873119887minus 119873
119906+ 1
119873119906minus 1
(42)
8 International Journal of Antennas and Propagation
Expressions (41) and (42) are valid for 119873119887gt 119873
119906ge 2
Expression (42) reveals that for ZF precoding the mean SIRis in an approximately inverse relation to the base stationsrsquoCFO variance and to the square of time Furthermore it isshown that the SIR grows with the number of base stationsand drops with the number of jointly served users
45 TheValue of User Selection in JTCoMPand an SIRBoundThe SNR gains in JT CoMP are increased if a schedulerselects the appropriate users to be jointly served on thesame time and frequency resources As shown in [34] theselection criteria are based on the rule that all users gainfrom the cooperation in the cluster compared to the case ofnoncoordinated transmission Evaluated on a field scenariothe resulting singular value statistics after such user selectionwas found to be comparable to the one of a Rayleigh fadingchannel a fact that also relates to scenarios where users arelocated close to the cell edge that is in which gains throughcoordination are particularly large
Considering now an idealized scenario from the precod-ing point of view with orthogonal channel vectors of equalpower among the users scenario which has been analyzed in[5 11] an upper bound can be derived for the mean SIR usingZF precoding in JT CoMP Using the expression for 119870IUI asprovided in Appendix A for 119873
119887gt 119873
119906ge 2 we obtain the
following expression
SIRmax asymp1
412058721205902
1198911199052119899
sdot119873119887
119873119906minus 1
minus119873119887+ 1
119873119906minus 1
(43)
The distance between (42) and (43) reveals the potential forSIR enhancement if the usersrsquo selection process considers theorthogonality among their channel vectors
It should be clarified at this point that (42) and (43) do notconsider any gains from resource allocation the evaluationof which not in the scope of this work In an orthogonalfrequency division multiple access (OFDMA) system eachuser can be assigned a part of the spectrum in a way that hisperformance and the network performance can be optimizedas known from [35] and also shown in [36] for the multiuserdownlink
The signal model given in Section 3 for the impairedOFDM-based JT CoMP is valid for any OFDMA schemeUsing OFDMA in the downlink does not affect the signalstructure of a userrsquos received self-signal and the IUI TheICI also has the same structure and statistical propertiesas typically data are transmitted on all subcarriers which isrecommended for keeping frequency-flat distribution of theICI power [13] The degradation mechanisms due to multipleCFOs and SFOs as described in our model will be the samefor any system using OFDM
What indeed changes in OFDMA is the channel statis-tics for each user after resource allocation Therefore ifintended to evaluate the overall performance of a coordinatedmultipoint system using OFDMA the general mean powerexpressions (26) (31) and (37) would need to be evaluated forthe actual usersrsquo channel statistics This procedure practicallyimplies a calculation of constants119870U119870IUI and119870ICI
In this section the received power of a user was studiedin relation to the interference from other users and othersubcarriers in amulticellular multiuser downlink with coop-erative base stations Closed-form expressions and accurateapproximations for all relevant termswere derivedMoreoverit was demonstrated that interuser interference has the mostrelevant effect Finally the impact of the radio channel wasinvestigated and analytical SIR expressions were derived forzero-forcing precoding
5 Evaluation and Numerical Validation
In this section analytical results of Section 4 are evaluatedand verified by simulations Based on those results adequaterequirements are determined for the base stationsrsquo oscillatorsin CoMP systems
A JT CoMP scenario is considered where a cooperationcluster of 119873
119887= 7 base stations transmits jointly by using
zero-forcing precoding to 119873119906terminals on the same time
and frequency resource with 119873119887gt 119873
119906ge 2 Out-of-cluster
interference is not consideredAt time instant 119905 = 0 the base stations receive an update
of the downlink channel matrix and compute a precodingmatrix This will be used during the following 10ms timeat which a new channel update will be obtained and a newprecoder will be calculated This routine agrees with thecurrent 3GPP LTE channel and precoder updating cycle [31]As already mentioned in Section 3 it is assumed that theradio channel is perfectly estimated and available at the basestations at 119905 = 0 whereas channel aging effects are notconsidered either Between updating instants the phases ofthe local base stationsrsquo oscillators slowly drift away fromeach other because of the individual frequency offsets withrespect to the ideal carrier frequency 119891
119888 As a consequence
self-signal drops IUI grows and overall the SIR drops withtime In a practical system each user can use downlink pilotsymbols to estimate and equalize its own CFO and avoid ICIenhancement However mobiles cannot stop the IUI fromrising because the degradation is caused jointly by all themisaligned base station oscillators
The oscillator accuracy Osc typically specified in partsper million (ppm) or parts per billion (ppb) relates to thestandard deviation of the resulting carrier frequency as 120590
119891=
Osc sdot 119891119888 The CFO variation over time is assumed to be slow
enough to be considered static with respect to the signalingand precoder updating timescale Here Gaussian iid CFOsare randomly assigned to base stations and mobile users allwith zero mean and standard deviations given by 120590
119891and 120590
119891119895
respectivelyTypical 3GPP LTE parameters are used specified in [31]
The broadband OFDM signal has 119873119904= 2048 subcarriers
separated by 120575 = 15 kHzThe length of the cyclic prefix is 144samples resulting in OFDM symbols with a length of 119873
119892=
2192 samples The ideal carrier and sampling frequencies are119891119888= 265GHz and 1119879 = 3072MHz respectively The
energy per data symbol is set to 119864119904= 1
Figure 2 depicts the mean power of the IUI and ICI overtime passed from the last precoder update (in logarithmicscale) for 3 and 6 mobile users a base station oscillator
International Journal of Antennas and Propagation 9
0
Time (ms)
Mea
n po
wer
(dB)
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
10minus1 100 101
IUI Nu = 3 Osc = 1ppbICI Nu = 3 Osc = 1ppb 120585 = 0
ICI Nu = 3 Osc = 1ppb 120585 = 10
IUI Nu = 6 Osc = 1ppbICI Nu = 6 Osc = 1ppb 120585 = 0
ICI Nu = 6 Osc = 1ppb 120585 = 0
Figure 2 Mean power of interuser and intercarrier interferences ina Rayleigh fading channel Here 7 base stations serve 3 and 6 usersrespectively Analytical results are shown by lines while markersshow numerical evaluations
accuracy of 1 ppb and Rayleigh fading channel conditionsThe curves illustrate analytical results given by (33) and(38) whereas solid lines depict results for 3 users anddashed lines for 6 users respectivelyMarkers show respectivenumerical evaluations of themeanpower of (14) and (15) over10
3 independent realizations of the MIMO channel matrixentries with 120590
2
ℎ= 1 and of the basesrsquo CFOs Regarding
the ICI two scenarios are considered a scenario with 120585 =
0 that is perfectly synchronized mobile users in order toevaluate solely the ICI due to the base stationsrsquo CFOs (red)and a second scenario with nonsynchronized mobiles with aCFO accuracy of an order of magnitude lower than the basestationsrsquo one that is 120585 = 10 (black) where the total ICI isevaluated
First of all Figure 2 verifies the analytical expres-sions (includingmathematical approximations) by numericalmeans It is further observed that the IUI grows approxi-mately with the square of time and that it increases with thenumber of users Comparing the mean power levels of IUI(blue curves) with the ICI both caused by the base stationsrsquoCFOs (red curves for 120585 = 0) we see that independently of thenumber of users and for typical feedback delays between 2msand 10ms the IUI lies between 40 dB and 50 dB above thepower of ICIThis clearly indicates that the IUI is significantlystronger than the ICI caused by the base stationsrsquo CFOsConsidering now the second scenario with nonperfectlysynchronized mobile users (120585 = 10) it can be observed thatthe part of the ICI caused by the usersrsquo CFO overwhelms theICI caused by the base stationsrsquo CFOs According to (38) the(dominant) ICI due to a mobile userrsquos CFO does not depend
SIR
(dB)
0
10
20
30
40
50
60
70
80
90
Time (ms)10minus1 100 101
20dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 3 Mean SIR over time for Rayleigh fading channel and SIRupper bound Here 7 base stations using oscillators of accuracygiven by Osc serve jointly 6 users Analytical results are shown bylines simulations by markers
on the number of served users which is reflected in the factthat the black curves in Figure 2 evaluating the total ICI for120585 = 10 almost overlap for the cases of 3 and 6 users It isalso interesting to observe that the IUI power level due tothe cooperating base stationsrsquo CFOs is still around 20 dB to30 dB (for 3 users) and 30 dB to 40 dB (for 6 users) higherthan the total ICI power considering typical feedback delaysbetween 2ms and 10ms These results clearly show that inthe a cooperative network even with typically nonperfectlysynchronized mobile users the main interference source liesin the base stationsrsquo CFOs
Figure 3 shows the SIR over the time as defined in (41)for119873
119887= 7 base stations serving jointly119873
119906= 6mobile users
which is also the highest possible number for the Rayleighfading channel The influence of the base stationsrsquo oscillatoraccuracy is evaluated and the corresponding ICI is neglectedas it is significantly smaller than the IUI Disregarding the ICIalso means that the synchronicity level of the mobile users isnot relevant as it does not contribute the IUI
For the Rayleigh fading channel the analytical expression(42) is compared with the ratio between the numericallyevaluated mean power of (13) and (14) over 103 independentchannel and CFO realizations which validates our analysisincluding the accuracy of the mathematical approximationsSimilarly for the SIR upper bound expression the analyticalexpression (43) is compared with results from numericalevaluation From Figure 3 it is seen that a degradation of thebase station oscillatorsrsquo accuracy by one order of magnitudeincreases the IUI and thus decreases the SIR by around 20 dBFurthermore the SIR drops approximately quadratically withtime that is 20 dB per time decade In terms of accuracyrequirements it is found that for reaching in average anSNR of 20 dB 10ms after the precoder update (free-running)oscillators of 1 ppb are needed
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
4 International Journal of Antennas and Propagation
120573119895119894(119896 ]) = 119890minus1204852120587119899119873119892119879119895(119891119895minus119891119894)
sdot 119890minus1204852120587119899119873
119892]120575119894(119879119894minus119879119895)
sdot 119890minus120485120587((119873
119904minus1)119873
119904)[(119896minus](119879
119895119879119894))+(119891119895minus119891119894)119879119895119873119904]
sdot1
119873119904
sdot
sin [120587 (119896 minus ] (119879119895119879
119894)) + 120587 (119891
119895minus 119891
119894) 119879
119895119873119904]
sin [(120587119873119904) (119896 minus ] (119879
119895119879
119894)) + 120587 (119891
119895minus 119891
119894) 119879
119895]
(8)
and with 119873119895(119896) being the receiver-side AWGN of power
1198730120575119895 Note that in (6) and (7) we have approximated and
defined119867119895119894(119896120575
119894+119891
119894minus119891
119888) asymp 119867
119895119894(119896120575
119894) ≜ 119867
119895119894(119896) because |119891
119894minus119891
119888|
is assumed to bemuch smaller than the coherence bandwidthof the channel
In practical system implementation the carrier andsampling frequency clocks of a transmitter or receiver arederived from the same reference that is from the same localoscillator Thus for the product of an arbitrary 119894th carrierfrequency and sampling period119891
119894sdot119879119894≜ 120581 it holds that 120581 ≫ 1
as the (ideal) carrier frequency 119891119888is two to three orders of
magnitude larger than the (ideal) sampling frequency 1119879The constant 120581 depends only on the system and hardwaredesign and is independent of 119894
Considering this relationship in (8) it is straightforwardto see that the exponent in the expressionrsquos second line whichcaptures the SFO effect on 120573
119895119894(119896 ]) and which is maximized
for ] = 119873119904 still remains 120581 times smaller than the exponent
in the first line capturing the CFO effect Similar findings canbe observed comparing the influence of SFO with the one ofthe CFO onto the terms in the third and the fourth line of(8) Thus it can be safely said that the impact of the SFO on120573119895119894(119896 ]) is significantly weaker than the impact of the CFO
when using the same oscillator reference for both at eachindividual transmitter and receiver
The derivations up to here have been presented includingboth CFO and SFO for the sake of completeness as well asfor further reference Expressions (5) through (8) provide theexact signal model which characterizes the joint effect ofmultiple CFOs and SFOs over consecutive OFDM symbolsin MIMO transmissions and is one important contributionof this work However beyond signal modeling this workfurther aims to identify the major degradation sourcesanalyze the performance degradation and determine fromthere the actual system requirements To this end in thefollowing we focus on the CFO and neglect the SFO byassuming ideal sampling for all transmitters and receiversWith 119879
119894= 119879
119895= 119879 (8) becomes
120573119895119894(119896 ]) = 119890minus1204852120587[(119891119895minus119891119894)119905119899+((119873119904minus1)2119873119904)(119896minus])]
sdot1
119873119904
sin [120587 (119896 minus ]) + 120587 (119891119895minus 119891
119894) 119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119894) 119879]
(9)
The discrete time variable 119905119899≜ (119899119873
119892+ (119873
119904minus 1)2)119879 has been
defined for convenience and is measured in secondsExpressions (5) to (9) provide the discrete-frequency
signal model for a MIMO-OFDM communication system in
the presence of multiple CFOs and SFOs as a function of timeand will be used in the following section
3 Precoded Multicell Downlink Signal Model
Now we specialize the model obtained in Section 2 to theCoMPdownlink including joint precoding of the data signalsHere the required channel state information (CSI) for theprecoder calculation is estimated either at the base stations orat the terminals and fed back to the base stations dependingonwhether timedivision duplex (TDD)or frequency divisionduplex (FDD) is used For distributed base stations coordi-nation by means of a backhaul network is considered whichenables fast exchange of data and CSI
It is to be noted that channel estimation errors and CSIdelays due to the feedback and the backhaul network arenot considered in this work Their effect on JT CoMP isimportant andmight even overwhelm the effects of imperfectsynchronization which we are analyzing here A distinctanalysis by the authors which includes the effect of imperfectchannel knowledge and derives the resulting performancelimitations can be found in [33] We will therefore assumehere that perfect knowledge of the downlink MIMO channelmatrix is available at the base stations and is used for real-timeprecoding of the downlink signals As already mentioned inSection 2 residual phase terms due toCFOs and SFOs are alsoconsidered as part of the (perfectly estimated) channel Weconsider 119873
119906users equipped with single-antenna terminals
which are jointly served by119873119887coordinated antenna branches
among all base stations forming the cooperation clusterThere is no exchange of information between mobile usersand out-of-cluster transmission is not considered in ourmodel
Transmissions are precoded on each OFDM subcarrier 119896with the right-hand pseudoinverse of the119873
119906times 119873
119887downlink
MIMO channel matrixH(119896) for that subcarrier given by
W (119896) = H119867
(119896) [H (119896)H119867
(119896)]minus1
(10)
The inverse is of size 119873119887times 119873
119906and we assume it exists for
119873119887gt 119873
119906 Note that this condition can be met in practice
by appropriate user selection and clustering of base stationswhich is thereby assumed
Next consider S(119896) to be an 119873119906times 1 vector that contains
the complex-valued data symbols 119904119906(119896) to be transmitted to
the119873119906users on subcarrier 119896 Then the119873
119887times 1 vector X(119896) of
precoded symbols119883119887(119896) to be transmitted on subcarrier 119896 by
the119873119887base stations is X(119896) = W(119896)S(119896) where elements are
given by
119883119887(119896) =
119873119906
sum
119906=1
119908119887119906(119896) 119904
119906(119896) (11)
and 119908119887119906(119896) are the elements ofW(119896) Transmitter index 119894 has
been replaced with 119887 to stress that from here on transmittersare base stations Imposing the expression of the precodedsymbol (11) into 119880
119895(119896) given by (6) we obtain the on-carrier
signal on subcarrier 119896 of user 119895 given by119880119895(119896) = 119904
119895(119896) + 119904
119895(119896) (12)
International Journal of Antennas and Propagation 5
with
119904119895(119896) = 119904
119895(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119895(119896) (13)
119904119895(119896) =
119873119906
sum
119906=1
119906 =119895
119904119906(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119906(119896) (14)
An arbitrary user 119895 has been singled out in (13) from theremaining users Thus 119904
119895(119896) represents the self-signal while
119904119895(119896) represents the IUI observed by user 119895 given by (14)This
interference is due to the loss of orthogonality of the precodedtransmission to multiple users caused by synchronizationimpairments Imposing (11) into 119880
119895(119896) in (7) we obtain the
ICI on subcarrier 119896 of user 119895 given by
119880119895(119896) =
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119904119906(])
119873119887
sum
119887=1
120573119895119887(119896 ])119867
119895119887(]) 119908
119887119906(]) (15)
Our complete discrete-frequency CoMP downlink signalmodel is then
119884119895(119896) = 119904
119895(119896) + 119904
119895(119896) + 119880
119895(119896) + 119873
119895(119896) (16)
with 119904119895(119896) 119904
119895(119896) and 119880
119895(119896) given by (13) (14) and (15)
respectively and 119873119895(119896) being the AWGN of power 119873
0120575119895 It
is noted that the model of Section 3 is valid independently iffor 120573
119895119894(119896 ]) the exact expression (8) or its simplification (9) is
used where the SFO is neglected
4 Analysis of Signal and Interference Powers
The following section contains an in-depth analysis of theimpact of synchronization impairments onto the perfor-mance It is organized as follows first we study the powerof a userrsquos self-signal (useful signal) and then the IUI andICI Next we show that ICI is small compared to IUI andprovide analytical expressions for the mean SIR Finally wehighlight the value of user selection and show its impact ontothe performance
Conceptually the rise of IUI and ICI due to imperfectsynchronization can be understood as a dispersion of theenergy allocated on a specific subcarrier of a specific user toother users (IUI) and to other subcarriers (ICI) This impliesa drop of the self-signal power and a rise on that user andthat subcarrier of the power of IUI and ICI from other usersrsquoand other subcarriersrsquo losses In what follows we proceedunder the assumption that data symbols are statisticallyindependent between users and across subcarriers that isE119904
119895(119896)119904
lowast
119906(]) = 119864
119904120575119895119906120575119896] where Esdot denotes the expectation
operator 119864119904the mean energy per data symbol and 120575
119909119910the
Kronecker delta between 119909 and 119910
41 Power of the Userrsquos Self-Signal Since there is statisticalindependence between data symbols channel coefficients
and synchronization parameters themeanpower of the userrsquosself-signal (13) is
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
E 1198671198951198871
1199081198871119895119867lowast
1198951198872
119908lowast
1198872119895
(17)
where subcarrier index 119896 has been omitted for simplicity ofnotation and cross-terms equal to zero have been alreadydisregarded
In a first step we analyze E1205731198951198871
120573lowast
1198951198872
which appears in(17) Using therefore ] = 119896 in (9) and replacing transmitterindex 119894 with base station index 119887 we reach
120573119895119887=
1
119873119904
sdot 119890minus1204852120587(119891
119895minus119891119887)119905119899 sdot
sin [120587 (119891119895minus 119891
119887) 119879119873
119904]
sin [120587 (119891119895minus 119891
119887) 119879]
(18)
which does not depend on the subcarrier index 119896 By notingthat |119891
119895minus 119891
119887| ≪ 1119879 we can safely say that the argument 119909 =
120587(119891119895minus 119891
119887)119879 of the sin(119909) in the denominator of the fraction
on the right-hand side of (18) is very small The same termalso appears in the exponential term and both multiplicativeterms in (18) are close to one But comparing the magnitudedeviations from unity we see that for typical synchroniza-tion parameters |1 minus (1119873
119904)(sin(119873
119904119909) sin(119909))| ≪ |1 minus
119890minus120485(214119899+1)119873
119904119909
| (this results in119873119904≫ 1 and a cyclic prefix equal
to 007119873119904[31]) In absolute terms we can further use the first
order Taylor series expansion (1119873119904)(sin(119873
119904119909) sin(119909)) asymp 1
and reach
1205731198951198871
120573lowast
1198951198872
asymp 119890minus1204852120587(119891
119895minus1198911198871)119905119899 sdot 119890
1204852120587(119891119895minus1198911198872)119905119899 = 119890
1204852120587(1198911198871minus1198911198872)119905119899 (19)
It is interesting to observe that the power of the exponentialterm as approximated in (19) depends on the differencebetween the base stationsrsquo carrier frequenciesThe power losseffect of the symbol transmitted on subcarrier 119896 which is thegenerating factor of ICI depends on both the CFOs of thebase stations and of the mobile usersThe effect that has beenneglected here due to its relatively small role compared tothe one of the exponential term will be however thoroughlyanalyzed in Section 42
For transmission from one base station that is for case1198871= 119887
2 it is immediate that (19) equals one For 119887
1= 119887
2
it is reasonable to assume that different base stationsrsquo carrierfrequencies 119891
1198871
and 1198911198872
are statistically independent whichallows for expressing the expectation of (19) as a product oftwo terms each depending on one base stationrsquos CFO
E 1205731198951198871
120573lowast
1198951198872
= E 1198901204852120587(119891
1198871minus119891119888)119905119899 sdot E 119890
minus1204852120587(1198911198872minus119891119888)119905119899 (20)
By assuming further that theCFOs are identically distributedthe second term of the right-hand side in (20) is the complex-conjugate of the first one hence (20) can be developed as
E 1205731198951198871
120573lowast
1198951198872
=10038161003816100381610038161003816E 119890
minus1204852120587(119891119887minus119891119888)11990511989910038161003816100381610038161003816
2
=
1003816100381610038161003816100381610038161003816int 119901
(119891119887minus119891119888)119890minus1204852120587(119891
119887minus119891119888)119905119899119889 (119891
119887minus 119891
119888)
1003816100381610038161003816100381610038161003816
2
(21)
6 International Journal of Antennas and Propagation
where 119901(119891119887minus119891119888)denotes the probability distribution function
(pdf) of the CFO (119891119887minus 119891
119888) and is independent on index 119887 It
is to be noted that the integral in (21) is a Fourier transformof the CFOrsquos pdf Hence we can write
E 1205731198951198871
120573lowast
1198951198872
=
1 1198871= 119887
2
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
1198871
= 1198872
(22)
where F119901(sdot) denotes the Fourier transform of 119901
(119891119887minus119891119888)and is
here a function of time For being a characteristic functionthat is the Fourier transform of a pdf it is guaranteedthat |F
119901(119905119899)| le 1 (see [32]) which means that the power
of the userrsquos self-signal (17) drops under imperfect carriersynchronization conditions
Result (22) can be used for developing (17) further byseparating the sum over 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= 119864119904sdot (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
) sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
(23)
The double sum in the last line in (23) can be formulated as
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= E
119873119887
sum
1198871=1
1198671198951198871
1199081198871119895sdot
119873119887
sum
1198872=1
119867lowast
1198951198872
119908lowast
1198872119895
(24)
which by considering the ZF condition
119873119887
sum
119887=1
119867119895119887119908119887119906= 120575
119895119906 (25)
is found to be equal to 1 Thus (23) can be written as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904[1 minus (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870U] (26)
where we defined119870U = 1 minussum119873119887
119887=1E|119867
119895119887119908119887119895|2
Note that (25)cannot be applied to the sum terms in the first and third line of(23) because of the power indexThe constant119870U depends onthe precoder and the channel statistical properties As shownin Appendix A for ZF precoding given a channel matrix with119873119887gt 119873
119906and complex Gaussian independent and identically
distributed (iid) entries (Rayleigh fading channel) withzero mean and variance 1205902
ℎ119870U can be approximated as
119870U asymp 1 minus1
119873119887minus 119873
119906
(27)
If we further consider the case in which the CFOs of thebase stations are Gaussian iid with zero mean and variance1205902
119891 the Fourier transform in (22) becomes
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
= 119890minus412058721205902
1198911199052
119899 asymp 1 minus 41205872
1205902
1198911199052
119899 (28)
where the first order Taylor series expansion 119890minus119909 asymp 1 minus 119909 hasbeen used It is to be noted that this approximation is accuratefor typical values of 120590
119891and 119905
119899 Using approximations (27) and
(28) we can formulate themeanpower of the userrsquos self-signal(26) as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904[1 minus 4120587
2
1205902
1198911199052
119899sdot (1 minus
1
119873119887minus 119873
119906
)] (29)
from where we see that it decreases linearly with the basestationsrsquo CFO variance and quadratically with time Result(29) also shows that the mean power of the self-signal growswith the number of base stations and drops with the numberof users It is noteworthy that for 119873
119906= 119873
119887minus 1 the mean
power of the self-signal remains constant However thisshould not be used as a system design rule as for determiningthe system performance the degradation due to IUI and ICImust be considered as well
42 Power of the Interuser Interference The derivation of themean power of the IUI (14) follows similar steps as the onesin Section 41 Concretely
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119906
sum
119906=1
119906 =119895
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
sdot E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(30)
and noting that in this case always 119895 = 119906 in (25) we find that
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904(1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI (31)
where119870IUI = sum119873119906
119906=1119906 =119895sum119873119887
119887=1E|119867
119895119887119908119887119906|2
If all users undergoidentical channel statistics119870IUI simplifies to119870IUI = (119873119906
minus1) sdot
sum119873119887
119887=1E|119867
119895119887119908119887119906|2
whereas using the results of Appendix Afor the Rayleigh fading channel and119873
119887gt 119873
119906 we find
119870IUI asymp119873119906minus 1
119873119887minus 119873
119906
(32)
Using (32) and (28) in (31) we obtain
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904sdot 4120587
2
1205902
1198911199052
119899sdot119873119906minus 1
119873119887minus 119873
119906
(33)
Result (33) reveals that the IUI power grows linearly withthe base stationsrsquo CFO variance and quadratically with time
International Journal of Antennas and Propagation 7
Using more base stations decreases the IUI power whileadding more users results in increasing it Due to theapproximation used in (19) it can be said that the impactof the mobile usersrsquo CFO onto the IUI power level is lesssignificant than the impact of the base stationsrsquo CFOs
43 Power of the Intercarrier Interference Following similarsteps as before now for (15) the mean power of the ICI is
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
sdot E 1198671198951198871
(]) 1199081198871119906(])119867lowast
1198951198872
(]) 119908lowast1198872119906(])
(34)
Assuming identical channel statistics for all subcarriers thelast term in (34) does not depend on subcarrier index ] andbecomes E119867
1198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906 already seen in Sections 41
and 42 For convenience we define1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) ≜
1198611 119887
1= 119887
2
1198612 119887
1= 1198872
(35)
which is analyzed inAppendix B In (34) we separate the sumover 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871as in (23) and obtain
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot 119861
1sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
= 119864119904sdot (119861
1minus 119861
2) sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(36)
We use the expressions for 1198611and 119861
2in Appendix B which
provide accurate approximations for Gaussian distributedCFOs for the base stations and the mobile user 119895 with zeromean and variances 1205902
119891and 1205902
119891119895 respectively and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot 119870ICI + 120590
2
119891119895) (37)
Here 119870ICI = sum119873119906
119906=1sum119873119887
119887=1E|119867
119895119887119908119887119906|2
= 119870IUI minus 119870U + 1 is aconstant For aRayleigh fadingMIMOchannel we use in (37)the expression provided by Appendix A for119870ICI and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot
119873119906
119873119887minus 119873
119906
+ 1205902
119891119895) (38)
Expression (38) shows that the mean power of the userrsquos ICIcan be split into two additive parts the first part dependingon the base stationsrsquo CFO variance which is multiplied witha weighting factor 119870ICI according to the cluster size and thesecond part depending on the userrsquos own CFO Furthermore(38) reveals that the power of the ICI does not vary with timeand that it is inverse to the square of the OFDM subcarrierspacing
44 ICI-to-IUI Ratio and SIR The relative magnitudes of thepower terms derived so far in this section are analyzed nextA simple expression for the mean ICI-to-IUI power ratio ofa user 119895 (an interference-to-interference ratio IIR) can beobtained from (33) and (38)
IIR =
E 10038161003816100381610038161003816119880119895
10038161003816100381610038161003816
2
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp1
1212057521199052119899
sdot (119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (39)
with 119873119887gt 119873
119906ge 2 and 120585 = 120590
119891119895120590
119891defined as the ratio
between the standard deviations of the userrsquos and the basestationsrsquo CFOs By replacing 120575 = (119873
119904119879)
minus1 and discrete time119905119899= (107119899 + 05)119873
119904119879 (this results in 119873
119904≫ 1 and a cyclic
prefix equal to 007119873119904[33]) relation (39) becomes
IIR asymp1
12 (107119899 + 05)2sdot (
119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (40)
It is interesting to observe that in the approximation (40)the IIR does not depend on the OFDM system parametersbut only on the OFDM symbols index 119899
Regarding the ratio 120585 it has been shown in [14 21]that mobile terminals attain a carrier frequency accuracywhich is at least one order of magnitude below the accuracyof typical base station oscillators used in Third GenerationPartnership Project (3GPP) Long Term Evolution (LTE) [31]Assuming for instance 1205852 ≫ 1 the first additive term in (40)becomes much smaller than the second term indicating thatthe terminalsrsquo CFO is the main source of ICI Using 120585 = 0one can evaluate the IIR including only the ICI part due to thebase stationsrsquo CFOs in this case the ratio does not depend onthe base stationsrsquo CFOs at all Using a more practical value offor example 120585 = 10 the largest possible IIR after 119899 = 14
OFDM symbols (119905119899asymp 1ms) which occurs with 119873
119906= 2
users given 119873119887= 7 equals minus73 dB The IIR drops quickly
down already to minus273 dB for 119899 = 140 (119905119899asymp 10ms) These
quantitive results show that for typical JT CoMP scenariosthe IUI dominates over the ICI
Wewill nowneglect the ICI and define themean SIR (self-user signal-to-IUI ratio) by the ratio between (26) and (31)
SIR =
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
(1 minus10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI
minus1
119873119906minus 1
(41)
For a MIMO channel with iid Rayleigh fading entries (119870IUIgiven by (32)) and for iid Gaussian CFOs (|F
119901(119905119899)|2 given
by (28)) (41) can be simplified as
SIR asymp1
412058721205902
1198911199052119899
sdot119873119887minus 119873
119906
119873119906minus 1
minus119873119887minus 119873
119906+ 1
119873119906minus 1
(42)
8 International Journal of Antennas and Propagation
Expressions (41) and (42) are valid for 119873119887gt 119873
119906ge 2
Expression (42) reveals that for ZF precoding the mean SIRis in an approximately inverse relation to the base stationsrsquoCFO variance and to the square of time Furthermore it isshown that the SIR grows with the number of base stationsand drops with the number of jointly served users
45 TheValue of User Selection in JTCoMPand an SIRBoundThe SNR gains in JT CoMP are increased if a schedulerselects the appropriate users to be jointly served on thesame time and frequency resources As shown in [34] theselection criteria are based on the rule that all users gainfrom the cooperation in the cluster compared to the case ofnoncoordinated transmission Evaluated on a field scenariothe resulting singular value statistics after such user selectionwas found to be comparable to the one of a Rayleigh fadingchannel a fact that also relates to scenarios where users arelocated close to the cell edge that is in which gains throughcoordination are particularly large
Considering now an idealized scenario from the precod-ing point of view with orthogonal channel vectors of equalpower among the users scenario which has been analyzed in[5 11] an upper bound can be derived for the mean SIR usingZF precoding in JT CoMP Using the expression for 119870IUI asprovided in Appendix A for 119873
119887gt 119873
119906ge 2 we obtain the
following expression
SIRmax asymp1
412058721205902
1198911199052119899
sdot119873119887
119873119906minus 1
minus119873119887+ 1
119873119906minus 1
(43)
The distance between (42) and (43) reveals the potential forSIR enhancement if the usersrsquo selection process considers theorthogonality among their channel vectors
It should be clarified at this point that (42) and (43) do notconsider any gains from resource allocation the evaluationof which not in the scope of this work In an orthogonalfrequency division multiple access (OFDMA) system eachuser can be assigned a part of the spectrum in a way that hisperformance and the network performance can be optimizedas known from [35] and also shown in [36] for the multiuserdownlink
The signal model given in Section 3 for the impairedOFDM-based JT CoMP is valid for any OFDMA schemeUsing OFDMA in the downlink does not affect the signalstructure of a userrsquos received self-signal and the IUI TheICI also has the same structure and statistical propertiesas typically data are transmitted on all subcarriers which isrecommended for keeping frequency-flat distribution of theICI power [13] The degradation mechanisms due to multipleCFOs and SFOs as described in our model will be the samefor any system using OFDM
What indeed changes in OFDMA is the channel statis-tics for each user after resource allocation Therefore ifintended to evaluate the overall performance of a coordinatedmultipoint system using OFDMA the general mean powerexpressions (26) (31) and (37) would need to be evaluated forthe actual usersrsquo channel statistics This procedure practicallyimplies a calculation of constants119870U119870IUI and119870ICI
In this section the received power of a user was studiedin relation to the interference from other users and othersubcarriers in amulticellular multiuser downlink with coop-erative base stations Closed-form expressions and accurateapproximations for all relevant termswere derivedMoreoverit was demonstrated that interuser interference has the mostrelevant effect Finally the impact of the radio channel wasinvestigated and analytical SIR expressions were derived forzero-forcing precoding
5 Evaluation and Numerical Validation
In this section analytical results of Section 4 are evaluatedand verified by simulations Based on those results adequaterequirements are determined for the base stationsrsquo oscillatorsin CoMP systems
A JT CoMP scenario is considered where a cooperationcluster of 119873
119887= 7 base stations transmits jointly by using
zero-forcing precoding to 119873119906terminals on the same time
and frequency resource with 119873119887gt 119873
119906ge 2 Out-of-cluster
interference is not consideredAt time instant 119905 = 0 the base stations receive an update
of the downlink channel matrix and compute a precodingmatrix This will be used during the following 10ms timeat which a new channel update will be obtained and a newprecoder will be calculated This routine agrees with thecurrent 3GPP LTE channel and precoder updating cycle [31]As already mentioned in Section 3 it is assumed that theradio channel is perfectly estimated and available at the basestations at 119905 = 0 whereas channel aging effects are notconsidered either Between updating instants the phases ofthe local base stationsrsquo oscillators slowly drift away fromeach other because of the individual frequency offsets withrespect to the ideal carrier frequency 119891
119888 As a consequence
self-signal drops IUI grows and overall the SIR drops withtime In a practical system each user can use downlink pilotsymbols to estimate and equalize its own CFO and avoid ICIenhancement However mobiles cannot stop the IUI fromrising because the degradation is caused jointly by all themisaligned base station oscillators
The oscillator accuracy Osc typically specified in partsper million (ppm) or parts per billion (ppb) relates to thestandard deviation of the resulting carrier frequency as 120590
119891=
Osc sdot 119891119888 The CFO variation over time is assumed to be slow
enough to be considered static with respect to the signalingand precoder updating timescale Here Gaussian iid CFOsare randomly assigned to base stations and mobile users allwith zero mean and standard deviations given by 120590
119891and 120590
119891119895
respectivelyTypical 3GPP LTE parameters are used specified in [31]
The broadband OFDM signal has 119873119904= 2048 subcarriers
separated by 120575 = 15 kHzThe length of the cyclic prefix is 144samples resulting in OFDM symbols with a length of 119873
119892=
2192 samples The ideal carrier and sampling frequencies are119891119888= 265GHz and 1119879 = 3072MHz respectively The
energy per data symbol is set to 119864119904= 1
Figure 2 depicts the mean power of the IUI and ICI overtime passed from the last precoder update (in logarithmicscale) for 3 and 6 mobile users a base station oscillator
International Journal of Antennas and Propagation 9
0
Time (ms)
Mea
n po
wer
(dB)
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
10minus1 100 101
IUI Nu = 3 Osc = 1ppbICI Nu = 3 Osc = 1ppb 120585 = 0
ICI Nu = 3 Osc = 1ppb 120585 = 10
IUI Nu = 6 Osc = 1ppbICI Nu = 6 Osc = 1ppb 120585 = 0
ICI Nu = 6 Osc = 1ppb 120585 = 0
Figure 2 Mean power of interuser and intercarrier interferences ina Rayleigh fading channel Here 7 base stations serve 3 and 6 usersrespectively Analytical results are shown by lines while markersshow numerical evaluations
accuracy of 1 ppb and Rayleigh fading channel conditionsThe curves illustrate analytical results given by (33) and(38) whereas solid lines depict results for 3 users anddashed lines for 6 users respectivelyMarkers show respectivenumerical evaluations of themeanpower of (14) and (15) over10
3 independent realizations of the MIMO channel matrixentries with 120590
2
ℎ= 1 and of the basesrsquo CFOs Regarding
the ICI two scenarios are considered a scenario with 120585 =
0 that is perfectly synchronized mobile users in order toevaluate solely the ICI due to the base stationsrsquo CFOs (red)and a second scenario with nonsynchronized mobiles with aCFO accuracy of an order of magnitude lower than the basestationsrsquo one that is 120585 = 10 (black) where the total ICI isevaluated
First of all Figure 2 verifies the analytical expres-sions (includingmathematical approximations) by numericalmeans It is further observed that the IUI grows approxi-mately with the square of time and that it increases with thenumber of users Comparing the mean power levels of IUI(blue curves) with the ICI both caused by the base stationsrsquoCFOs (red curves for 120585 = 0) we see that independently of thenumber of users and for typical feedback delays between 2msand 10ms the IUI lies between 40 dB and 50 dB above thepower of ICIThis clearly indicates that the IUI is significantlystronger than the ICI caused by the base stationsrsquo CFOsConsidering now the second scenario with nonperfectlysynchronized mobile users (120585 = 10) it can be observed thatthe part of the ICI caused by the usersrsquo CFO overwhelms theICI caused by the base stationsrsquo CFOs According to (38) the(dominant) ICI due to a mobile userrsquos CFO does not depend
SIR
(dB)
0
10
20
30
40
50
60
70
80
90
Time (ms)10minus1 100 101
20dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 3 Mean SIR over time for Rayleigh fading channel and SIRupper bound Here 7 base stations using oscillators of accuracygiven by Osc serve jointly 6 users Analytical results are shown bylines simulations by markers
on the number of served users which is reflected in the factthat the black curves in Figure 2 evaluating the total ICI for120585 = 10 almost overlap for the cases of 3 and 6 users It isalso interesting to observe that the IUI power level due tothe cooperating base stationsrsquo CFOs is still around 20 dB to30 dB (for 3 users) and 30 dB to 40 dB (for 6 users) higherthan the total ICI power considering typical feedback delaysbetween 2ms and 10ms These results clearly show that inthe a cooperative network even with typically nonperfectlysynchronized mobile users the main interference source liesin the base stationsrsquo CFOs
Figure 3 shows the SIR over the time as defined in (41)for119873
119887= 7 base stations serving jointly119873
119906= 6mobile users
which is also the highest possible number for the Rayleighfading channel The influence of the base stationsrsquo oscillatoraccuracy is evaluated and the corresponding ICI is neglectedas it is significantly smaller than the IUI Disregarding the ICIalso means that the synchronicity level of the mobile users isnot relevant as it does not contribute the IUI
For the Rayleigh fading channel the analytical expression(42) is compared with the ratio between the numericallyevaluated mean power of (13) and (14) over 103 independentchannel and CFO realizations which validates our analysisincluding the accuracy of the mathematical approximationsSimilarly for the SIR upper bound expression the analyticalexpression (43) is compared with results from numericalevaluation From Figure 3 it is seen that a degradation of thebase station oscillatorsrsquo accuracy by one order of magnitudeincreases the IUI and thus decreases the SIR by around 20 dBFurthermore the SIR drops approximately quadratically withtime that is 20 dB per time decade In terms of accuracyrequirements it is found that for reaching in average anSNR of 20 dB 10ms after the precoder update (free-running)oscillators of 1 ppb are needed
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
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DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 5
with
119904119895(119896) = 119904
119895(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119895(119896) (13)
119904119895(119896) =
119873119906
sum
119906=1
119906 =119895
119904119906(119896)
119873119887
sum
119887=1
120573119895119887(119896 119896)119867
119895119887(119896) 119908
119887119906(119896) (14)
An arbitrary user 119895 has been singled out in (13) from theremaining users Thus 119904
119895(119896) represents the self-signal while
119904119895(119896) represents the IUI observed by user 119895 given by (14)This
interference is due to the loss of orthogonality of the precodedtransmission to multiple users caused by synchronizationimpairments Imposing (11) into 119880
119895(119896) in (7) we obtain the
ICI on subcarrier 119896 of user 119895 given by
119880119895(119896) =
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119904119906(])
119873119887
sum
119887=1
120573119895119887(119896 ])119867
119895119887(]) 119908
119887119906(]) (15)
Our complete discrete-frequency CoMP downlink signalmodel is then
119884119895(119896) = 119904
119895(119896) + 119904
119895(119896) + 119880
119895(119896) + 119873
119895(119896) (16)
with 119904119895(119896) 119904
119895(119896) and 119880
119895(119896) given by (13) (14) and (15)
respectively and 119873119895(119896) being the AWGN of power 119873
0120575119895 It
is noted that the model of Section 3 is valid independently iffor 120573
119895119894(119896 ]) the exact expression (8) or its simplification (9) is
used where the SFO is neglected
4 Analysis of Signal and Interference Powers
The following section contains an in-depth analysis of theimpact of synchronization impairments onto the perfor-mance It is organized as follows first we study the powerof a userrsquos self-signal (useful signal) and then the IUI andICI Next we show that ICI is small compared to IUI andprovide analytical expressions for the mean SIR Finally wehighlight the value of user selection and show its impact ontothe performance
Conceptually the rise of IUI and ICI due to imperfectsynchronization can be understood as a dispersion of theenergy allocated on a specific subcarrier of a specific user toother users (IUI) and to other subcarriers (ICI) This impliesa drop of the self-signal power and a rise on that user andthat subcarrier of the power of IUI and ICI from other usersrsquoand other subcarriersrsquo losses In what follows we proceedunder the assumption that data symbols are statisticallyindependent between users and across subcarriers that isE119904
119895(119896)119904
lowast
119906(]) = 119864
119904120575119895119906120575119896] where Esdot denotes the expectation
operator 119864119904the mean energy per data symbol and 120575
119909119910the
Kronecker delta between 119909 and 119910
41 Power of the Userrsquos Self-Signal Since there is statisticalindependence between data symbols channel coefficients
and synchronization parameters themeanpower of the userrsquosself-signal (13) is
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
E 1198671198951198871
1199081198871119895119867lowast
1198951198872
119908lowast
1198872119895
(17)
where subcarrier index 119896 has been omitted for simplicity ofnotation and cross-terms equal to zero have been alreadydisregarded
In a first step we analyze E1205731198951198871
120573lowast
1198951198872
which appears in(17) Using therefore ] = 119896 in (9) and replacing transmitterindex 119894 with base station index 119887 we reach
120573119895119887=
1
119873119904
sdot 119890minus1204852120587(119891
119895minus119891119887)119905119899 sdot
sin [120587 (119891119895minus 119891
119887) 119879119873
119904]
sin [120587 (119891119895minus 119891
119887) 119879]
(18)
which does not depend on the subcarrier index 119896 By notingthat |119891
119895minus 119891
119887| ≪ 1119879 we can safely say that the argument 119909 =
120587(119891119895minus 119891
119887)119879 of the sin(119909) in the denominator of the fraction
on the right-hand side of (18) is very small The same termalso appears in the exponential term and both multiplicativeterms in (18) are close to one But comparing the magnitudedeviations from unity we see that for typical synchroniza-tion parameters |1 minus (1119873
119904)(sin(119873
119904119909) sin(119909))| ≪ |1 minus
119890minus120485(214119899+1)119873
119904119909
| (this results in119873119904≫ 1 and a cyclic prefix equal
to 007119873119904[31]) In absolute terms we can further use the first
order Taylor series expansion (1119873119904)(sin(119873
119904119909) sin(119909)) asymp 1
and reach
1205731198951198871
120573lowast
1198951198872
asymp 119890minus1204852120587(119891
119895minus1198911198871)119905119899 sdot 119890
1204852120587(119891119895minus1198911198872)119905119899 = 119890
1204852120587(1198911198871minus1198911198872)119905119899 (19)
It is interesting to observe that the power of the exponentialterm as approximated in (19) depends on the differencebetween the base stationsrsquo carrier frequenciesThe power losseffect of the symbol transmitted on subcarrier 119896 which is thegenerating factor of ICI depends on both the CFOs of thebase stations and of the mobile usersThe effect that has beenneglected here due to its relatively small role compared tothe one of the exponential term will be however thoroughlyanalyzed in Section 42
For transmission from one base station that is for case1198871= 119887
2 it is immediate that (19) equals one For 119887
1= 119887
2
it is reasonable to assume that different base stationsrsquo carrierfrequencies 119891
1198871
and 1198911198872
are statistically independent whichallows for expressing the expectation of (19) as a product oftwo terms each depending on one base stationrsquos CFO
E 1205731198951198871
120573lowast
1198951198872
= E 1198901204852120587(119891
1198871minus119891119888)119905119899 sdot E 119890
minus1204852120587(1198911198872minus119891119888)119905119899 (20)
By assuming further that theCFOs are identically distributedthe second term of the right-hand side in (20) is the complex-conjugate of the first one hence (20) can be developed as
E 1205731198951198871
120573lowast
1198951198872
=10038161003816100381610038161003816E 119890
minus1204852120587(119891119887minus119891119888)11990511989910038161003816100381610038161003816
2
=
1003816100381610038161003816100381610038161003816int 119901
(119891119887minus119891119888)119890minus1204852120587(119891
119887minus119891119888)119905119899119889 (119891
119887minus 119891
119888)
1003816100381610038161003816100381610038161003816
2
(21)
6 International Journal of Antennas and Propagation
where 119901(119891119887minus119891119888)denotes the probability distribution function
(pdf) of the CFO (119891119887minus 119891
119888) and is independent on index 119887 It
is to be noted that the integral in (21) is a Fourier transformof the CFOrsquos pdf Hence we can write
E 1205731198951198871
120573lowast
1198951198872
=
1 1198871= 119887
2
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
1198871
= 1198872
(22)
where F119901(sdot) denotes the Fourier transform of 119901
(119891119887minus119891119888)and is
here a function of time For being a characteristic functionthat is the Fourier transform of a pdf it is guaranteedthat |F
119901(119905119899)| le 1 (see [32]) which means that the power
of the userrsquos self-signal (17) drops under imperfect carriersynchronization conditions
Result (22) can be used for developing (17) further byseparating the sum over 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= 119864119904sdot (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
) sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
(23)
The double sum in the last line in (23) can be formulated as
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= E
119873119887
sum
1198871=1
1198671198951198871
1199081198871119895sdot
119873119887
sum
1198872=1
119867lowast
1198951198872
119908lowast
1198872119895
(24)
which by considering the ZF condition
119873119887
sum
119887=1
119867119895119887119908119887119906= 120575
119895119906 (25)
is found to be equal to 1 Thus (23) can be written as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904[1 minus (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870U] (26)
where we defined119870U = 1 minussum119873119887
119887=1E|119867
119895119887119908119887119895|2
Note that (25)cannot be applied to the sum terms in the first and third line of(23) because of the power indexThe constant119870U depends onthe precoder and the channel statistical properties As shownin Appendix A for ZF precoding given a channel matrix with119873119887gt 119873
119906and complex Gaussian independent and identically
distributed (iid) entries (Rayleigh fading channel) withzero mean and variance 1205902
ℎ119870U can be approximated as
119870U asymp 1 minus1
119873119887minus 119873
119906
(27)
If we further consider the case in which the CFOs of thebase stations are Gaussian iid with zero mean and variance1205902
119891 the Fourier transform in (22) becomes
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
= 119890minus412058721205902
1198911199052
119899 asymp 1 minus 41205872
1205902
1198911199052
119899 (28)
where the first order Taylor series expansion 119890minus119909 asymp 1 minus 119909 hasbeen used It is to be noted that this approximation is accuratefor typical values of 120590
119891and 119905
119899 Using approximations (27) and
(28) we can formulate themeanpower of the userrsquos self-signal(26) as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904[1 minus 4120587
2
1205902
1198911199052
119899sdot (1 minus
1
119873119887minus 119873
119906
)] (29)
from where we see that it decreases linearly with the basestationsrsquo CFO variance and quadratically with time Result(29) also shows that the mean power of the self-signal growswith the number of base stations and drops with the numberof users It is noteworthy that for 119873
119906= 119873
119887minus 1 the mean
power of the self-signal remains constant However thisshould not be used as a system design rule as for determiningthe system performance the degradation due to IUI and ICImust be considered as well
42 Power of the Interuser Interference The derivation of themean power of the IUI (14) follows similar steps as the onesin Section 41 Concretely
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119906
sum
119906=1
119906 =119895
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
sdot E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(30)
and noting that in this case always 119895 = 119906 in (25) we find that
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904(1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI (31)
where119870IUI = sum119873119906
119906=1119906 =119895sum119873119887
119887=1E|119867
119895119887119908119887119906|2
If all users undergoidentical channel statistics119870IUI simplifies to119870IUI = (119873119906
minus1) sdot
sum119873119887
119887=1E|119867
119895119887119908119887119906|2
whereas using the results of Appendix Afor the Rayleigh fading channel and119873
119887gt 119873
119906 we find
119870IUI asymp119873119906minus 1
119873119887minus 119873
119906
(32)
Using (32) and (28) in (31) we obtain
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904sdot 4120587
2
1205902
1198911199052
119899sdot119873119906minus 1
119873119887minus 119873
119906
(33)
Result (33) reveals that the IUI power grows linearly withthe base stationsrsquo CFO variance and quadratically with time
International Journal of Antennas and Propagation 7
Using more base stations decreases the IUI power whileadding more users results in increasing it Due to theapproximation used in (19) it can be said that the impactof the mobile usersrsquo CFO onto the IUI power level is lesssignificant than the impact of the base stationsrsquo CFOs
43 Power of the Intercarrier Interference Following similarsteps as before now for (15) the mean power of the ICI is
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
sdot E 1198671198951198871
(]) 1199081198871119906(])119867lowast
1198951198872
(]) 119908lowast1198872119906(])
(34)
Assuming identical channel statistics for all subcarriers thelast term in (34) does not depend on subcarrier index ] andbecomes E119867
1198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906 already seen in Sections 41
and 42 For convenience we define1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) ≜
1198611 119887
1= 119887
2
1198612 119887
1= 1198872
(35)
which is analyzed inAppendix B In (34) we separate the sumover 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871as in (23) and obtain
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot 119861
1sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
= 119864119904sdot (119861
1minus 119861
2) sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(36)
We use the expressions for 1198611and 119861
2in Appendix B which
provide accurate approximations for Gaussian distributedCFOs for the base stations and the mobile user 119895 with zeromean and variances 1205902
119891and 1205902
119891119895 respectively and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot 119870ICI + 120590
2
119891119895) (37)
Here 119870ICI = sum119873119906
119906=1sum119873119887
119887=1E|119867
119895119887119908119887119906|2
= 119870IUI minus 119870U + 1 is aconstant For aRayleigh fadingMIMOchannel we use in (37)the expression provided by Appendix A for119870ICI and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot
119873119906
119873119887minus 119873
119906
+ 1205902
119891119895) (38)
Expression (38) shows that the mean power of the userrsquos ICIcan be split into two additive parts the first part dependingon the base stationsrsquo CFO variance which is multiplied witha weighting factor 119870ICI according to the cluster size and thesecond part depending on the userrsquos own CFO Furthermore(38) reveals that the power of the ICI does not vary with timeand that it is inverse to the square of the OFDM subcarrierspacing
44 ICI-to-IUI Ratio and SIR The relative magnitudes of thepower terms derived so far in this section are analyzed nextA simple expression for the mean ICI-to-IUI power ratio ofa user 119895 (an interference-to-interference ratio IIR) can beobtained from (33) and (38)
IIR =
E 10038161003816100381610038161003816119880119895
10038161003816100381610038161003816
2
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp1
1212057521199052119899
sdot (119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (39)
with 119873119887gt 119873
119906ge 2 and 120585 = 120590
119891119895120590
119891defined as the ratio
between the standard deviations of the userrsquos and the basestationsrsquo CFOs By replacing 120575 = (119873
119904119879)
minus1 and discrete time119905119899= (107119899 + 05)119873
119904119879 (this results in 119873
119904≫ 1 and a cyclic
prefix equal to 007119873119904[33]) relation (39) becomes
IIR asymp1
12 (107119899 + 05)2sdot (
119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (40)
It is interesting to observe that in the approximation (40)the IIR does not depend on the OFDM system parametersbut only on the OFDM symbols index 119899
Regarding the ratio 120585 it has been shown in [14 21]that mobile terminals attain a carrier frequency accuracywhich is at least one order of magnitude below the accuracyof typical base station oscillators used in Third GenerationPartnership Project (3GPP) Long Term Evolution (LTE) [31]Assuming for instance 1205852 ≫ 1 the first additive term in (40)becomes much smaller than the second term indicating thatthe terminalsrsquo CFO is the main source of ICI Using 120585 = 0one can evaluate the IIR including only the ICI part due to thebase stationsrsquo CFOs in this case the ratio does not depend onthe base stationsrsquo CFOs at all Using a more practical value offor example 120585 = 10 the largest possible IIR after 119899 = 14
OFDM symbols (119905119899asymp 1ms) which occurs with 119873
119906= 2
users given 119873119887= 7 equals minus73 dB The IIR drops quickly
down already to minus273 dB for 119899 = 140 (119905119899asymp 10ms) These
quantitive results show that for typical JT CoMP scenariosthe IUI dominates over the ICI
Wewill nowneglect the ICI and define themean SIR (self-user signal-to-IUI ratio) by the ratio between (26) and (31)
SIR =
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
(1 minus10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI
minus1
119873119906minus 1
(41)
For a MIMO channel with iid Rayleigh fading entries (119870IUIgiven by (32)) and for iid Gaussian CFOs (|F
119901(119905119899)|2 given
by (28)) (41) can be simplified as
SIR asymp1
412058721205902
1198911199052119899
sdot119873119887minus 119873
119906
119873119906minus 1
minus119873119887minus 119873
119906+ 1
119873119906minus 1
(42)
8 International Journal of Antennas and Propagation
Expressions (41) and (42) are valid for 119873119887gt 119873
119906ge 2
Expression (42) reveals that for ZF precoding the mean SIRis in an approximately inverse relation to the base stationsrsquoCFO variance and to the square of time Furthermore it isshown that the SIR grows with the number of base stationsand drops with the number of jointly served users
45 TheValue of User Selection in JTCoMPand an SIRBoundThe SNR gains in JT CoMP are increased if a schedulerselects the appropriate users to be jointly served on thesame time and frequency resources As shown in [34] theselection criteria are based on the rule that all users gainfrom the cooperation in the cluster compared to the case ofnoncoordinated transmission Evaluated on a field scenariothe resulting singular value statistics after such user selectionwas found to be comparable to the one of a Rayleigh fadingchannel a fact that also relates to scenarios where users arelocated close to the cell edge that is in which gains throughcoordination are particularly large
Considering now an idealized scenario from the precod-ing point of view with orthogonal channel vectors of equalpower among the users scenario which has been analyzed in[5 11] an upper bound can be derived for the mean SIR usingZF precoding in JT CoMP Using the expression for 119870IUI asprovided in Appendix A for 119873
119887gt 119873
119906ge 2 we obtain the
following expression
SIRmax asymp1
412058721205902
1198911199052119899
sdot119873119887
119873119906minus 1
minus119873119887+ 1
119873119906minus 1
(43)
The distance between (42) and (43) reveals the potential forSIR enhancement if the usersrsquo selection process considers theorthogonality among their channel vectors
It should be clarified at this point that (42) and (43) do notconsider any gains from resource allocation the evaluationof which not in the scope of this work In an orthogonalfrequency division multiple access (OFDMA) system eachuser can be assigned a part of the spectrum in a way that hisperformance and the network performance can be optimizedas known from [35] and also shown in [36] for the multiuserdownlink
The signal model given in Section 3 for the impairedOFDM-based JT CoMP is valid for any OFDMA schemeUsing OFDMA in the downlink does not affect the signalstructure of a userrsquos received self-signal and the IUI TheICI also has the same structure and statistical propertiesas typically data are transmitted on all subcarriers which isrecommended for keeping frequency-flat distribution of theICI power [13] The degradation mechanisms due to multipleCFOs and SFOs as described in our model will be the samefor any system using OFDM
What indeed changes in OFDMA is the channel statis-tics for each user after resource allocation Therefore ifintended to evaluate the overall performance of a coordinatedmultipoint system using OFDMA the general mean powerexpressions (26) (31) and (37) would need to be evaluated forthe actual usersrsquo channel statistics This procedure practicallyimplies a calculation of constants119870U119870IUI and119870ICI
In this section the received power of a user was studiedin relation to the interference from other users and othersubcarriers in amulticellular multiuser downlink with coop-erative base stations Closed-form expressions and accurateapproximations for all relevant termswere derivedMoreoverit was demonstrated that interuser interference has the mostrelevant effect Finally the impact of the radio channel wasinvestigated and analytical SIR expressions were derived forzero-forcing precoding
5 Evaluation and Numerical Validation
In this section analytical results of Section 4 are evaluatedand verified by simulations Based on those results adequaterequirements are determined for the base stationsrsquo oscillatorsin CoMP systems
A JT CoMP scenario is considered where a cooperationcluster of 119873
119887= 7 base stations transmits jointly by using
zero-forcing precoding to 119873119906terminals on the same time
and frequency resource with 119873119887gt 119873
119906ge 2 Out-of-cluster
interference is not consideredAt time instant 119905 = 0 the base stations receive an update
of the downlink channel matrix and compute a precodingmatrix This will be used during the following 10ms timeat which a new channel update will be obtained and a newprecoder will be calculated This routine agrees with thecurrent 3GPP LTE channel and precoder updating cycle [31]As already mentioned in Section 3 it is assumed that theradio channel is perfectly estimated and available at the basestations at 119905 = 0 whereas channel aging effects are notconsidered either Between updating instants the phases ofthe local base stationsrsquo oscillators slowly drift away fromeach other because of the individual frequency offsets withrespect to the ideal carrier frequency 119891
119888 As a consequence
self-signal drops IUI grows and overall the SIR drops withtime In a practical system each user can use downlink pilotsymbols to estimate and equalize its own CFO and avoid ICIenhancement However mobiles cannot stop the IUI fromrising because the degradation is caused jointly by all themisaligned base station oscillators
The oscillator accuracy Osc typically specified in partsper million (ppm) or parts per billion (ppb) relates to thestandard deviation of the resulting carrier frequency as 120590
119891=
Osc sdot 119891119888 The CFO variation over time is assumed to be slow
enough to be considered static with respect to the signalingand precoder updating timescale Here Gaussian iid CFOsare randomly assigned to base stations and mobile users allwith zero mean and standard deviations given by 120590
119891and 120590
119891119895
respectivelyTypical 3GPP LTE parameters are used specified in [31]
The broadband OFDM signal has 119873119904= 2048 subcarriers
separated by 120575 = 15 kHzThe length of the cyclic prefix is 144samples resulting in OFDM symbols with a length of 119873
119892=
2192 samples The ideal carrier and sampling frequencies are119891119888= 265GHz and 1119879 = 3072MHz respectively The
energy per data symbol is set to 119864119904= 1
Figure 2 depicts the mean power of the IUI and ICI overtime passed from the last precoder update (in logarithmicscale) for 3 and 6 mobile users a base station oscillator
International Journal of Antennas and Propagation 9
0
Time (ms)
Mea
n po
wer
(dB)
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
10minus1 100 101
IUI Nu = 3 Osc = 1ppbICI Nu = 3 Osc = 1ppb 120585 = 0
ICI Nu = 3 Osc = 1ppb 120585 = 10
IUI Nu = 6 Osc = 1ppbICI Nu = 6 Osc = 1ppb 120585 = 0
ICI Nu = 6 Osc = 1ppb 120585 = 0
Figure 2 Mean power of interuser and intercarrier interferences ina Rayleigh fading channel Here 7 base stations serve 3 and 6 usersrespectively Analytical results are shown by lines while markersshow numerical evaluations
accuracy of 1 ppb and Rayleigh fading channel conditionsThe curves illustrate analytical results given by (33) and(38) whereas solid lines depict results for 3 users anddashed lines for 6 users respectivelyMarkers show respectivenumerical evaluations of themeanpower of (14) and (15) over10
3 independent realizations of the MIMO channel matrixentries with 120590
2
ℎ= 1 and of the basesrsquo CFOs Regarding
the ICI two scenarios are considered a scenario with 120585 =
0 that is perfectly synchronized mobile users in order toevaluate solely the ICI due to the base stationsrsquo CFOs (red)and a second scenario with nonsynchronized mobiles with aCFO accuracy of an order of magnitude lower than the basestationsrsquo one that is 120585 = 10 (black) where the total ICI isevaluated
First of all Figure 2 verifies the analytical expres-sions (includingmathematical approximations) by numericalmeans It is further observed that the IUI grows approxi-mately with the square of time and that it increases with thenumber of users Comparing the mean power levels of IUI(blue curves) with the ICI both caused by the base stationsrsquoCFOs (red curves for 120585 = 0) we see that independently of thenumber of users and for typical feedback delays between 2msand 10ms the IUI lies between 40 dB and 50 dB above thepower of ICIThis clearly indicates that the IUI is significantlystronger than the ICI caused by the base stationsrsquo CFOsConsidering now the second scenario with nonperfectlysynchronized mobile users (120585 = 10) it can be observed thatthe part of the ICI caused by the usersrsquo CFO overwhelms theICI caused by the base stationsrsquo CFOs According to (38) the(dominant) ICI due to a mobile userrsquos CFO does not depend
SIR
(dB)
0
10
20
30
40
50
60
70
80
90
Time (ms)10minus1 100 101
20dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 3 Mean SIR over time for Rayleigh fading channel and SIRupper bound Here 7 base stations using oscillators of accuracygiven by Osc serve jointly 6 users Analytical results are shown bylines simulations by markers
on the number of served users which is reflected in the factthat the black curves in Figure 2 evaluating the total ICI for120585 = 10 almost overlap for the cases of 3 and 6 users It isalso interesting to observe that the IUI power level due tothe cooperating base stationsrsquo CFOs is still around 20 dB to30 dB (for 3 users) and 30 dB to 40 dB (for 6 users) higherthan the total ICI power considering typical feedback delaysbetween 2ms and 10ms These results clearly show that inthe a cooperative network even with typically nonperfectlysynchronized mobile users the main interference source liesin the base stationsrsquo CFOs
Figure 3 shows the SIR over the time as defined in (41)for119873
119887= 7 base stations serving jointly119873
119906= 6mobile users
which is also the highest possible number for the Rayleighfading channel The influence of the base stationsrsquo oscillatoraccuracy is evaluated and the corresponding ICI is neglectedas it is significantly smaller than the IUI Disregarding the ICIalso means that the synchronicity level of the mobile users isnot relevant as it does not contribute the IUI
For the Rayleigh fading channel the analytical expression(42) is compared with the ratio between the numericallyevaluated mean power of (13) and (14) over 103 independentchannel and CFO realizations which validates our analysisincluding the accuracy of the mathematical approximationsSimilarly for the SIR upper bound expression the analyticalexpression (43) is compared with results from numericalevaluation From Figure 3 it is seen that a degradation of thebase station oscillatorsrsquo accuracy by one order of magnitudeincreases the IUI and thus decreases the SIR by around 20 dBFurthermore the SIR drops approximately quadratically withtime that is 20 dB per time decade In terms of accuracyrequirements it is found that for reaching in average anSNR of 20 dB 10ms after the precoder update (free-running)oscillators of 1 ppb are needed
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
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DistributedSensor Networks
International Journal of
6 International Journal of Antennas and Propagation
where 119901(119891119887minus119891119888)denotes the probability distribution function
(pdf) of the CFO (119891119887minus 119891
119888) and is independent on index 119887 It
is to be noted that the integral in (21) is a Fourier transformof the CFOrsquos pdf Hence we can write
E 1205731198951198871
120573lowast
1198951198872
=
1 1198871= 119887
2
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
1198871
= 1198872
(22)
where F119901(sdot) denotes the Fourier transform of 119901
(119891119887minus119891119888)and is
here a function of time For being a characteristic functionthat is the Fourier transform of a pdf it is guaranteedthat |F
119901(119905119899)| le 1 (see [32]) which means that the power
of the userrsquos self-signal (17) drops under imperfect carriersynchronization conditions
Result (22) can be used for developing (17) further byseparating the sum over 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= 119864119904sdot (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
) sdot
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119895
10038161003816100381610038161003816
2
+ 119864119904sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
(23)
The double sum in the last line in (23) can be formulated as
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
119867lowast
1198951198872
1199081198871119895119908lowast
1198872119895
= E
119873119887
sum
1198871=1
1198671198951198871
1199081198871119895sdot
119873119887
sum
1198872=1
119867lowast
1198951198872
119908lowast
1198872119895
(24)
which by considering the ZF condition
119873119887
sum
119887=1
119867119895119887119908119887119906= 120575
119895119906 (25)
is found to be equal to 1 Thus (23) can be written as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904[1 minus (1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870U] (26)
where we defined119870U = 1 minussum119873119887
119887=1E|119867
119895119887119908119887119895|2
Note that (25)cannot be applied to the sum terms in the first and third line of(23) because of the power indexThe constant119870U depends onthe precoder and the channel statistical properties As shownin Appendix A for ZF precoding given a channel matrix with119873119887gt 119873
119906and complex Gaussian independent and identically
distributed (iid) entries (Rayleigh fading channel) withzero mean and variance 1205902
ℎ119870U can be approximated as
119870U asymp 1 minus1
119873119887minus 119873
119906
(27)
If we further consider the case in which the CFOs of thebase stations are Gaussian iid with zero mean and variance1205902
119891 the Fourier transform in (22) becomes
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
= 119890minus412058721205902
1198911199052
119899 asymp 1 minus 41205872
1205902
1198911199052
119899 (28)
where the first order Taylor series expansion 119890minus119909 asymp 1 minus 119909 hasbeen used It is to be noted that this approximation is accuratefor typical values of 120590
119891and 119905
119899 Using approximations (27) and
(28) we can formulate themeanpower of the userrsquos self-signal(26) as
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904[1 minus 4120587
2
1205902
1198911199052
119899sdot (1 minus
1
119873119887minus 119873
119906
)] (29)
from where we see that it decreases linearly with the basestationsrsquo CFO variance and quadratically with time Result(29) also shows that the mean power of the self-signal growswith the number of base stations and drops with the numberof users It is noteworthy that for 119873
119906= 119873
119887minus 1 the mean
power of the self-signal remains constant However thisshould not be used as a system design rule as for determiningthe system performance the degradation due to IUI and ICImust be considered as well
42 Power of the Interuser Interference The derivation of themean power of the IUI (14) follows similar steps as the onesin Section 41 Concretely
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904sdot
119873119906
sum
119906=1
119906 =119895
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
120573lowast
1198951198872
sdot E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(30)
and noting that in this case always 119895 = 119906 in (25) we find that
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
= 119864119904(1 minus
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI (31)
where119870IUI = sum119873119906
119906=1119906 =119895sum119873119887
119887=1E|119867
119895119887119908119887119906|2
If all users undergoidentical channel statistics119870IUI simplifies to119870IUI = (119873119906
minus1) sdot
sum119873119887
119887=1E|119867
119895119887119908119887119906|2
whereas using the results of Appendix Afor the Rayleigh fading channel and119873
119887gt 119873
119906 we find
119870IUI asymp119873119906minus 1
119873119887minus 119873
119906
(32)
Using (32) and (28) in (31) we obtain
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp 119864119904sdot 4120587
2
1205902
1198911199052
119899sdot119873119906minus 1
119873119887minus 119873
119906
(33)
Result (33) reveals that the IUI power grows linearly withthe base stationsrsquo CFO variance and quadratically with time
International Journal of Antennas and Propagation 7
Using more base stations decreases the IUI power whileadding more users results in increasing it Due to theapproximation used in (19) it can be said that the impactof the mobile usersrsquo CFO onto the IUI power level is lesssignificant than the impact of the base stationsrsquo CFOs
43 Power of the Intercarrier Interference Following similarsteps as before now for (15) the mean power of the ICI is
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
sdot E 1198671198951198871
(]) 1199081198871119906(])119867lowast
1198951198872
(]) 119908lowast1198872119906(])
(34)
Assuming identical channel statistics for all subcarriers thelast term in (34) does not depend on subcarrier index ] andbecomes E119867
1198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906 already seen in Sections 41
and 42 For convenience we define1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) ≜
1198611 119887
1= 119887
2
1198612 119887
1= 1198872
(35)
which is analyzed inAppendix B In (34) we separate the sumover 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871as in (23) and obtain
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot 119861
1sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
= 119864119904sdot (119861
1minus 119861
2) sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(36)
We use the expressions for 1198611and 119861
2in Appendix B which
provide accurate approximations for Gaussian distributedCFOs for the base stations and the mobile user 119895 with zeromean and variances 1205902
119891and 1205902
119891119895 respectively and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot 119870ICI + 120590
2
119891119895) (37)
Here 119870ICI = sum119873119906
119906=1sum119873119887
119887=1E|119867
119895119887119908119887119906|2
= 119870IUI minus 119870U + 1 is aconstant For aRayleigh fadingMIMOchannel we use in (37)the expression provided by Appendix A for119870ICI and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot
119873119906
119873119887minus 119873
119906
+ 1205902
119891119895) (38)
Expression (38) shows that the mean power of the userrsquos ICIcan be split into two additive parts the first part dependingon the base stationsrsquo CFO variance which is multiplied witha weighting factor 119870ICI according to the cluster size and thesecond part depending on the userrsquos own CFO Furthermore(38) reveals that the power of the ICI does not vary with timeand that it is inverse to the square of the OFDM subcarrierspacing
44 ICI-to-IUI Ratio and SIR The relative magnitudes of thepower terms derived so far in this section are analyzed nextA simple expression for the mean ICI-to-IUI power ratio ofa user 119895 (an interference-to-interference ratio IIR) can beobtained from (33) and (38)
IIR =
E 10038161003816100381610038161003816119880119895
10038161003816100381610038161003816
2
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp1
1212057521199052119899
sdot (119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (39)
with 119873119887gt 119873
119906ge 2 and 120585 = 120590
119891119895120590
119891defined as the ratio
between the standard deviations of the userrsquos and the basestationsrsquo CFOs By replacing 120575 = (119873
119904119879)
minus1 and discrete time119905119899= (107119899 + 05)119873
119904119879 (this results in 119873
119904≫ 1 and a cyclic
prefix equal to 007119873119904[33]) relation (39) becomes
IIR asymp1
12 (107119899 + 05)2sdot (
119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (40)
It is interesting to observe that in the approximation (40)the IIR does not depend on the OFDM system parametersbut only on the OFDM symbols index 119899
Regarding the ratio 120585 it has been shown in [14 21]that mobile terminals attain a carrier frequency accuracywhich is at least one order of magnitude below the accuracyof typical base station oscillators used in Third GenerationPartnership Project (3GPP) Long Term Evolution (LTE) [31]Assuming for instance 1205852 ≫ 1 the first additive term in (40)becomes much smaller than the second term indicating thatthe terminalsrsquo CFO is the main source of ICI Using 120585 = 0one can evaluate the IIR including only the ICI part due to thebase stationsrsquo CFOs in this case the ratio does not depend onthe base stationsrsquo CFOs at all Using a more practical value offor example 120585 = 10 the largest possible IIR after 119899 = 14
OFDM symbols (119905119899asymp 1ms) which occurs with 119873
119906= 2
users given 119873119887= 7 equals minus73 dB The IIR drops quickly
down already to minus273 dB for 119899 = 140 (119905119899asymp 10ms) These
quantitive results show that for typical JT CoMP scenariosthe IUI dominates over the ICI
Wewill nowneglect the ICI and define themean SIR (self-user signal-to-IUI ratio) by the ratio between (26) and (31)
SIR =
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
(1 minus10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI
minus1
119873119906minus 1
(41)
For a MIMO channel with iid Rayleigh fading entries (119870IUIgiven by (32)) and for iid Gaussian CFOs (|F
119901(119905119899)|2 given
by (28)) (41) can be simplified as
SIR asymp1
412058721205902
1198911199052119899
sdot119873119887minus 119873
119906
119873119906minus 1
minus119873119887minus 119873
119906+ 1
119873119906minus 1
(42)
8 International Journal of Antennas and Propagation
Expressions (41) and (42) are valid for 119873119887gt 119873
119906ge 2
Expression (42) reveals that for ZF precoding the mean SIRis in an approximately inverse relation to the base stationsrsquoCFO variance and to the square of time Furthermore it isshown that the SIR grows with the number of base stationsand drops with the number of jointly served users
45 TheValue of User Selection in JTCoMPand an SIRBoundThe SNR gains in JT CoMP are increased if a schedulerselects the appropriate users to be jointly served on thesame time and frequency resources As shown in [34] theselection criteria are based on the rule that all users gainfrom the cooperation in the cluster compared to the case ofnoncoordinated transmission Evaluated on a field scenariothe resulting singular value statistics after such user selectionwas found to be comparable to the one of a Rayleigh fadingchannel a fact that also relates to scenarios where users arelocated close to the cell edge that is in which gains throughcoordination are particularly large
Considering now an idealized scenario from the precod-ing point of view with orthogonal channel vectors of equalpower among the users scenario which has been analyzed in[5 11] an upper bound can be derived for the mean SIR usingZF precoding in JT CoMP Using the expression for 119870IUI asprovided in Appendix A for 119873
119887gt 119873
119906ge 2 we obtain the
following expression
SIRmax asymp1
412058721205902
1198911199052119899
sdot119873119887
119873119906minus 1
minus119873119887+ 1
119873119906minus 1
(43)
The distance between (42) and (43) reveals the potential forSIR enhancement if the usersrsquo selection process considers theorthogonality among their channel vectors
It should be clarified at this point that (42) and (43) do notconsider any gains from resource allocation the evaluationof which not in the scope of this work In an orthogonalfrequency division multiple access (OFDMA) system eachuser can be assigned a part of the spectrum in a way that hisperformance and the network performance can be optimizedas known from [35] and also shown in [36] for the multiuserdownlink
The signal model given in Section 3 for the impairedOFDM-based JT CoMP is valid for any OFDMA schemeUsing OFDMA in the downlink does not affect the signalstructure of a userrsquos received self-signal and the IUI TheICI also has the same structure and statistical propertiesas typically data are transmitted on all subcarriers which isrecommended for keeping frequency-flat distribution of theICI power [13] The degradation mechanisms due to multipleCFOs and SFOs as described in our model will be the samefor any system using OFDM
What indeed changes in OFDMA is the channel statis-tics for each user after resource allocation Therefore ifintended to evaluate the overall performance of a coordinatedmultipoint system using OFDMA the general mean powerexpressions (26) (31) and (37) would need to be evaluated forthe actual usersrsquo channel statistics This procedure practicallyimplies a calculation of constants119870U119870IUI and119870ICI
In this section the received power of a user was studiedin relation to the interference from other users and othersubcarriers in amulticellular multiuser downlink with coop-erative base stations Closed-form expressions and accurateapproximations for all relevant termswere derivedMoreoverit was demonstrated that interuser interference has the mostrelevant effect Finally the impact of the radio channel wasinvestigated and analytical SIR expressions were derived forzero-forcing precoding
5 Evaluation and Numerical Validation
In this section analytical results of Section 4 are evaluatedand verified by simulations Based on those results adequaterequirements are determined for the base stationsrsquo oscillatorsin CoMP systems
A JT CoMP scenario is considered where a cooperationcluster of 119873
119887= 7 base stations transmits jointly by using
zero-forcing precoding to 119873119906terminals on the same time
and frequency resource with 119873119887gt 119873
119906ge 2 Out-of-cluster
interference is not consideredAt time instant 119905 = 0 the base stations receive an update
of the downlink channel matrix and compute a precodingmatrix This will be used during the following 10ms timeat which a new channel update will be obtained and a newprecoder will be calculated This routine agrees with thecurrent 3GPP LTE channel and precoder updating cycle [31]As already mentioned in Section 3 it is assumed that theradio channel is perfectly estimated and available at the basestations at 119905 = 0 whereas channel aging effects are notconsidered either Between updating instants the phases ofthe local base stationsrsquo oscillators slowly drift away fromeach other because of the individual frequency offsets withrespect to the ideal carrier frequency 119891
119888 As a consequence
self-signal drops IUI grows and overall the SIR drops withtime In a practical system each user can use downlink pilotsymbols to estimate and equalize its own CFO and avoid ICIenhancement However mobiles cannot stop the IUI fromrising because the degradation is caused jointly by all themisaligned base station oscillators
The oscillator accuracy Osc typically specified in partsper million (ppm) or parts per billion (ppb) relates to thestandard deviation of the resulting carrier frequency as 120590
119891=
Osc sdot 119891119888 The CFO variation over time is assumed to be slow
enough to be considered static with respect to the signalingand precoder updating timescale Here Gaussian iid CFOsare randomly assigned to base stations and mobile users allwith zero mean and standard deviations given by 120590
119891and 120590
119891119895
respectivelyTypical 3GPP LTE parameters are used specified in [31]
The broadband OFDM signal has 119873119904= 2048 subcarriers
separated by 120575 = 15 kHzThe length of the cyclic prefix is 144samples resulting in OFDM symbols with a length of 119873
119892=
2192 samples The ideal carrier and sampling frequencies are119891119888= 265GHz and 1119879 = 3072MHz respectively The
energy per data symbol is set to 119864119904= 1
Figure 2 depicts the mean power of the IUI and ICI overtime passed from the last precoder update (in logarithmicscale) for 3 and 6 mobile users a base station oscillator
International Journal of Antennas and Propagation 9
0
Time (ms)
Mea
n po
wer
(dB)
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
10minus1 100 101
IUI Nu = 3 Osc = 1ppbICI Nu = 3 Osc = 1ppb 120585 = 0
ICI Nu = 3 Osc = 1ppb 120585 = 10
IUI Nu = 6 Osc = 1ppbICI Nu = 6 Osc = 1ppb 120585 = 0
ICI Nu = 6 Osc = 1ppb 120585 = 0
Figure 2 Mean power of interuser and intercarrier interferences ina Rayleigh fading channel Here 7 base stations serve 3 and 6 usersrespectively Analytical results are shown by lines while markersshow numerical evaluations
accuracy of 1 ppb and Rayleigh fading channel conditionsThe curves illustrate analytical results given by (33) and(38) whereas solid lines depict results for 3 users anddashed lines for 6 users respectivelyMarkers show respectivenumerical evaluations of themeanpower of (14) and (15) over10
3 independent realizations of the MIMO channel matrixentries with 120590
2
ℎ= 1 and of the basesrsquo CFOs Regarding
the ICI two scenarios are considered a scenario with 120585 =
0 that is perfectly synchronized mobile users in order toevaluate solely the ICI due to the base stationsrsquo CFOs (red)and a second scenario with nonsynchronized mobiles with aCFO accuracy of an order of magnitude lower than the basestationsrsquo one that is 120585 = 10 (black) where the total ICI isevaluated
First of all Figure 2 verifies the analytical expres-sions (includingmathematical approximations) by numericalmeans It is further observed that the IUI grows approxi-mately with the square of time and that it increases with thenumber of users Comparing the mean power levels of IUI(blue curves) with the ICI both caused by the base stationsrsquoCFOs (red curves for 120585 = 0) we see that independently of thenumber of users and for typical feedback delays between 2msand 10ms the IUI lies between 40 dB and 50 dB above thepower of ICIThis clearly indicates that the IUI is significantlystronger than the ICI caused by the base stationsrsquo CFOsConsidering now the second scenario with nonperfectlysynchronized mobile users (120585 = 10) it can be observed thatthe part of the ICI caused by the usersrsquo CFO overwhelms theICI caused by the base stationsrsquo CFOs According to (38) the(dominant) ICI due to a mobile userrsquos CFO does not depend
SIR
(dB)
0
10
20
30
40
50
60
70
80
90
Time (ms)10minus1 100 101
20dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 3 Mean SIR over time for Rayleigh fading channel and SIRupper bound Here 7 base stations using oscillators of accuracygiven by Osc serve jointly 6 users Analytical results are shown bylines simulations by markers
on the number of served users which is reflected in the factthat the black curves in Figure 2 evaluating the total ICI for120585 = 10 almost overlap for the cases of 3 and 6 users It isalso interesting to observe that the IUI power level due tothe cooperating base stationsrsquo CFOs is still around 20 dB to30 dB (for 3 users) and 30 dB to 40 dB (for 6 users) higherthan the total ICI power considering typical feedback delaysbetween 2ms and 10ms These results clearly show that inthe a cooperative network even with typically nonperfectlysynchronized mobile users the main interference source liesin the base stationsrsquo CFOs
Figure 3 shows the SIR over the time as defined in (41)for119873
119887= 7 base stations serving jointly119873
119906= 6mobile users
which is also the highest possible number for the Rayleighfading channel The influence of the base stationsrsquo oscillatoraccuracy is evaluated and the corresponding ICI is neglectedas it is significantly smaller than the IUI Disregarding the ICIalso means that the synchronicity level of the mobile users isnot relevant as it does not contribute the IUI
For the Rayleigh fading channel the analytical expression(42) is compared with the ratio between the numericallyevaluated mean power of (13) and (14) over 103 independentchannel and CFO realizations which validates our analysisincluding the accuracy of the mathematical approximationsSimilarly for the SIR upper bound expression the analyticalexpression (43) is compared with results from numericalevaluation From Figure 3 it is seen that a degradation of thebase station oscillatorsrsquo accuracy by one order of magnitudeincreases the IUI and thus decreases the SIR by around 20 dBFurthermore the SIR drops approximately quadratically withtime that is 20 dB per time decade In terms of accuracyrequirements it is found that for reaching in average anSNR of 20 dB 10ms after the precoder update (free-running)oscillators of 1 ppb are needed
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
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DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 7
Using more base stations decreases the IUI power whileadding more users results in increasing it Due to theapproximation used in (19) it can be said that the impactof the mobile usersrsquo CFO onto the IUI power level is lesssignificant than the impact of the base stationsrsquo CFOs
43 Power of the Intercarrier Interference Following similarsteps as before now for (15) the mean power of the ICI is
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
sdot E 1198671198951198871
(]) 1199081198871119906(])119867lowast
1198951198872
(]) 119908lowast1198872119906(])
(34)
Assuming identical channel statistics for all subcarriers thelast term in (34) does not depend on subcarrier index ] andbecomes E119867
1198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906 already seen in Sections 41
and 42 For convenience we define1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) ≜
1198611 119887
1= 119887
2
1198612 119887
1= 1198872
(35)
which is analyzed inAppendix B In (34) we separate the sumover 119887
2into the cases 119887
2= 119887
1and 119887
2= 1198871as in (23) and obtain
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
= 119864119904sdot 119861
1sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
1198872=1198871
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
= 119864119904sdot (119861
1minus 119861
2) sdot
119873119906
sum
119906=1
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
+ 119864119904sdot 119861
2sdot
119873119906
sum
119906=1
119873119887
sum
1198871=1
119873119887
sum
1198872=1
E 1198671198951198871
1199081198871119906119867lowast
1198951198872
119908lowast
1198872119906
(36)
We use the expressions for 1198611and 119861
2in Appendix B which
provide accurate approximations for Gaussian distributedCFOs for the base stations and the mobile user 119895 with zeromean and variances 1205902
119891and 1205902
119891119895 respectively and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot 119870ICI + 120590
2
119891119895) (37)
Here 119870ICI = sum119873119906
119906=1sum119873119887
119887=1E|119867
119895119887119908119887119906|2
= 119870IUI minus 119870U + 1 is aconstant For aRayleigh fadingMIMOchannel we use in (37)the expression provided by Appendix A for119870ICI and reach
E 10038161003816100381610038161003816119880119895(119896)
10038161003816100381610038161003816
2
asymp1198641199041205872
31205752sdot (120590
2
119891sdot
119873119906
119873119887minus 119873
119906
+ 1205902
119891119895) (38)
Expression (38) shows that the mean power of the userrsquos ICIcan be split into two additive parts the first part dependingon the base stationsrsquo CFO variance which is multiplied witha weighting factor 119870ICI according to the cluster size and thesecond part depending on the userrsquos own CFO Furthermore(38) reveals that the power of the ICI does not vary with timeand that it is inverse to the square of the OFDM subcarrierspacing
44 ICI-to-IUI Ratio and SIR The relative magnitudes of thepower terms derived so far in this section are analyzed nextA simple expression for the mean ICI-to-IUI power ratio ofa user 119895 (an interference-to-interference ratio IIR) can beobtained from (33) and (38)
IIR =
E 10038161003816100381610038161003816119880119895
10038161003816100381610038161003816
2
E 10038161003816100381610038161003816119904119895
10038161003816100381610038161003816
2
asymp1
1212057521199052119899
sdot (119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (39)
with 119873119887gt 119873
119906ge 2 and 120585 = 120590
119891119895120590
119891defined as the ratio
between the standard deviations of the userrsquos and the basestationsrsquo CFOs By replacing 120575 = (119873
119904119879)
minus1 and discrete time119905119899= (107119899 + 05)119873
119904119879 (this results in 119873
119904≫ 1 and a cyclic
prefix equal to 007119873119904[33]) relation (39) becomes
IIR asymp1
12 (107119899 + 05)2sdot (
119873119906
119873119906minus 1
+ 1205852119873119887minus 119873
119906
119873119906minus 1
) (40)
It is interesting to observe that in the approximation (40)the IIR does not depend on the OFDM system parametersbut only on the OFDM symbols index 119899
Regarding the ratio 120585 it has been shown in [14 21]that mobile terminals attain a carrier frequency accuracywhich is at least one order of magnitude below the accuracyof typical base station oscillators used in Third GenerationPartnership Project (3GPP) Long Term Evolution (LTE) [31]Assuming for instance 1205852 ≫ 1 the first additive term in (40)becomes much smaller than the second term indicating thatthe terminalsrsquo CFO is the main source of ICI Using 120585 = 0one can evaluate the IIR including only the ICI part due to thebase stationsrsquo CFOs in this case the ratio does not depend onthe base stationsrsquo CFOs at all Using a more practical value offor example 120585 = 10 the largest possible IIR after 119899 = 14
OFDM symbols (119905119899asymp 1ms) which occurs with 119873
119906= 2
users given 119873119887= 7 equals minus73 dB The IIR drops quickly
down already to minus273 dB for 119899 = 140 (119905119899asymp 10ms) These
quantitive results show that for typical JT CoMP scenariosthe IUI dominates over the ICI
Wewill nowneglect the ICI and define themean SIR (self-user signal-to-IUI ratio) by the ratio between (26) and (31)
SIR =
10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
(1 minus10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
)119870IUI
minus1
119873119906minus 1
(41)
For a MIMO channel with iid Rayleigh fading entries (119870IUIgiven by (32)) and for iid Gaussian CFOs (|F
119901(119905119899)|2 given
by (28)) (41) can be simplified as
SIR asymp1
412058721205902
1198911199052119899
sdot119873119887minus 119873
119906
119873119906minus 1
minus119873119887minus 119873
119906+ 1
119873119906minus 1
(42)
8 International Journal of Antennas and Propagation
Expressions (41) and (42) are valid for 119873119887gt 119873
119906ge 2
Expression (42) reveals that for ZF precoding the mean SIRis in an approximately inverse relation to the base stationsrsquoCFO variance and to the square of time Furthermore it isshown that the SIR grows with the number of base stationsand drops with the number of jointly served users
45 TheValue of User Selection in JTCoMPand an SIRBoundThe SNR gains in JT CoMP are increased if a schedulerselects the appropriate users to be jointly served on thesame time and frequency resources As shown in [34] theselection criteria are based on the rule that all users gainfrom the cooperation in the cluster compared to the case ofnoncoordinated transmission Evaluated on a field scenariothe resulting singular value statistics after such user selectionwas found to be comparable to the one of a Rayleigh fadingchannel a fact that also relates to scenarios where users arelocated close to the cell edge that is in which gains throughcoordination are particularly large
Considering now an idealized scenario from the precod-ing point of view with orthogonal channel vectors of equalpower among the users scenario which has been analyzed in[5 11] an upper bound can be derived for the mean SIR usingZF precoding in JT CoMP Using the expression for 119870IUI asprovided in Appendix A for 119873
119887gt 119873
119906ge 2 we obtain the
following expression
SIRmax asymp1
412058721205902
1198911199052119899
sdot119873119887
119873119906minus 1
minus119873119887+ 1
119873119906minus 1
(43)
The distance between (42) and (43) reveals the potential forSIR enhancement if the usersrsquo selection process considers theorthogonality among their channel vectors
It should be clarified at this point that (42) and (43) do notconsider any gains from resource allocation the evaluationof which not in the scope of this work In an orthogonalfrequency division multiple access (OFDMA) system eachuser can be assigned a part of the spectrum in a way that hisperformance and the network performance can be optimizedas known from [35] and also shown in [36] for the multiuserdownlink
The signal model given in Section 3 for the impairedOFDM-based JT CoMP is valid for any OFDMA schemeUsing OFDMA in the downlink does not affect the signalstructure of a userrsquos received self-signal and the IUI TheICI also has the same structure and statistical propertiesas typically data are transmitted on all subcarriers which isrecommended for keeping frequency-flat distribution of theICI power [13] The degradation mechanisms due to multipleCFOs and SFOs as described in our model will be the samefor any system using OFDM
What indeed changes in OFDMA is the channel statis-tics for each user after resource allocation Therefore ifintended to evaluate the overall performance of a coordinatedmultipoint system using OFDMA the general mean powerexpressions (26) (31) and (37) would need to be evaluated forthe actual usersrsquo channel statistics This procedure practicallyimplies a calculation of constants119870U119870IUI and119870ICI
In this section the received power of a user was studiedin relation to the interference from other users and othersubcarriers in amulticellular multiuser downlink with coop-erative base stations Closed-form expressions and accurateapproximations for all relevant termswere derivedMoreoverit was demonstrated that interuser interference has the mostrelevant effect Finally the impact of the radio channel wasinvestigated and analytical SIR expressions were derived forzero-forcing precoding
5 Evaluation and Numerical Validation
In this section analytical results of Section 4 are evaluatedand verified by simulations Based on those results adequaterequirements are determined for the base stationsrsquo oscillatorsin CoMP systems
A JT CoMP scenario is considered where a cooperationcluster of 119873
119887= 7 base stations transmits jointly by using
zero-forcing precoding to 119873119906terminals on the same time
and frequency resource with 119873119887gt 119873
119906ge 2 Out-of-cluster
interference is not consideredAt time instant 119905 = 0 the base stations receive an update
of the downlink channel matrix and compute a precodingmatrix This will be used during the following 10ms timeat which a new channel update will be obtained and a newprecoder will be calculated This routine agrees with thecurrent 3GPP LTE channel and precoder updating cycle [31]As already mentioned in Section 3 it is assumed that theradio channel is perfectly estimated and available at the basestations at 119905 = 0 whereas channel aging effects are notconsidered either Between updating instants the phases ofthe local base stationsrsquo oscillators slowly drift away fromeach other because of the individual frequency offsets withrespect to the ideal carrier frequency 119891
119888 As a consequence
self-signal drops IUI grows and overall the SIR drops withtime In a practical system each user can use downlink pilotsymbols to estimate and equalize its own CFO and avoid ICIenhancement However mobiles cannot stop the IUI fromrising because the degradation is caused jointly by all themisaligned base station oscillators
The oscillator accuracy Osc typically specified in partsper million (ppm) or parts per billion (ppb) relates to thestandard deviation of the resulting carrier frequency as 120590
119891=
Osc sdot 119891119888 The CFO variation over time is assumed to be slow
enough to be considered static with respect to the signalingand precoder updating timescale Here Gaussian iid CFOsare randomly assigned to base stations and mobile users allwith zero mean and standard deviations given by 120590
119891and 120590
119891119895
respectivelyTypical 3GPP LTE parameters are used specified in [31]
The broadband OFDM signal has 119873119904= 2048 subcarriers
separated by 120575 = 15 kHzThe length of the cyclic prefix is 144samples resulting in OFDM symbols with a length of 119873
119892=
2192 samples The ideal carrier and sampling frequencies are119891119888= 265GHz and 1119879 = 3072MHz respectively The
energy per data symbol is set to 119864119904= 1
Figure 2 depicts the mean power of the IUI and ICI overtime passed from the last precoder update (in logarithmicscale) for 3 and 6 mobile users a base station oscillator
International Journal of Antennas and Propagation 9
0
Time (ms)
Mea
n po
wer
(dB)
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
10minus1 100 101
IUI Nu = 3 Osc = 1ppbICI Nu = 3 Osc = 1ppb 120585 = 0
ICI Nu = 3 Osc = 1ppb 120585 = 10
IUI Nu = 6 Osc = 1ppbICI Nu = 6 Osc = 1ppb 120585 = 0
ICI Nu = 6 Osc = 1ppb 120585 = 0
Figure 2 Mean power of interuser and intercarrier interferences ina Rayleigh fading channel Here 7 base stations serve 3 and 6 usersrespectively Analytical results are shown by lines while markersshow numerical evaluations
accuracy of 1 ppb and Rayleigh fading channel conditionsThe curves illustrate analytical results given by (33) and(38) whereas solid lines depict results for 3 users anddashed lines for 6 users respectivelyMarkers show respectivenumerical evaluations of themeanpower of (14) and (15) over10
3 independent realizations of the MIMO channel matrixentries with 120590
2
ℎ= 1 and of the basesrsquo CFOs Regarding
the ICI two scenarios are considered a scenario with 120585 =
0 that is perfectly synchronized mobile users in order toevaluate solely the ICI due to the base stationsrsquo CFOs (red)and a second scenario with nonsynchronized mobiles with aCFO accuracy of an order of magnitude lower than the basestationsrsquo one that is 120585 = 10 (black) where the total ICI isevaluated
First of all Figure 2 verifies the analytical expres-sions (includingmathematical approximations) by numericalmeans It is further observed that the IUI grows approxi-mately with the square of time and that it increases with thenumber of users Comparing the mean power levels of IUI(blue curves) with the ICI both caused by the base stationsrsquoCFOs (red curves for 120585 = 0) we see that independently of thenumber of users and for typical feedback delays between 2msand 10ms the IUI lies between 40 dB and 50 dB above thepower of ICIThis clearly indicates that the IUI is significantlystronger than the ICI caused by the base stationsrsquo CFOsConsidering now the second scenario with nonperfectlysynchronized mobile users (120585 = 10) it can be observed thatthe part of the ICI caused by the usersrsquo CFO overwhelms theICI caused by the base stationsrsquo CFOs According to (38) the(dominant) ICI due to a mobile userrsquos CFO does not depend
SIR
(dB)
0
10
20
30
40
50
60
70
80
90
Time (ms)10minus1 100 101
20dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 3 Mean SIR over time for Rayleigh fading channel and SIRupper bound Here 7 base stations using oscillators of accuracygiven by Osc serve jointly 6 users Analytical results are shown bylines simulations by markers
on the number of served users which is reflected in the factthat the black curves in Figure 2 evaluating the total ICI for120585 = 10 almost overlap for the cases of 3 and 6 users It isalso interesting to observe that the IUI power level due tothe cooperating base stationsrsquo CFOs is still around 20 dB to30 dB (for 3 users) and 30 dB to 40 dB (for 6 users) higherthan the total ICI power considering typical feedback delaysbetween 2ms and 10ms These results clearly show that inthe a cooperative network even with typically nonperfectlysynchronized mobile users the main interference source liesin the base stationsrsquo CFOs
Figure 3 shows the SIR over the time as defined in (41)for119873
119887= 7 base stations serving jointly119873
119906= 6mobile users
which is also the highest possible number for the Rayleighfading channel The influence of the base stationsrsquo oscillatoraccuracy is evaluated and the corresponding ICI is neglectedas it is significantly smaller than the IUI Disregarding the ICIalso means that the synchronicity level of the mobile users isnot relevant as it does not contribute the IUI
For the Rayleigh fading channel the analytical expression(42) is compared with the ratio between the numericallyevaluated mean power of (13) and (14) over 103 independentchannel and CFO realizations which validates our analysisincluding the accuracy of the mathematical approximationsSimilarly for the SIR upper bound expression the analyticalexpression (43) is compared with results from numericalevaluation From Figure 3 it is seen that a degradation of thebase station oscillatorsrsquo accuracy by one order of magnitudeincreases the IUI and thus decreases the SIR by around 20 dBFurthermore the SIR drops approximately quadratically withtime that is 20 dB per time decade In terms of accuracyrequirements it is found that for reaching in average anSNR of 20 dB 10ms after the precoder update (free-running)oscillators of 1 ppb are needed
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
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DistributedSensor Networks
International Journal of
8 International Journal of Antennas and Propagation
Expressions (41) and (42) are valid for 119873119887gt 119873
119906ge 2
Expression (42) reveals that for ZF precoding the mean SIRis in an approximately inverse relation to the base stationsrsquoCFO variance and to the square of time Furthermore it isshown that the SIR grows with the number of base stationsand drops with the number of jointly served users
45 TheValue of User Selection in JTCoMPand an SIRBoundThe SNR gains in JT CoMP are increased if a schedulerselects the appropriate users to be jointly served on thesame time and frequency resources As shown in [34] theselection criteria are based on the rule that all users gainfrom the cooperation in the cluster compared to the case ofnoncoordinated transmission Evaluated on a field scenariothe resulting singular value statistics after such user selectionwas found to be comparable to the one of a Rayleigh fadingchannel a fact that also relates to scenarios where users arelocated close to the cell edge that is in which gains throughcoordination are particularly large
Considering now an idealized scenario from the precod-ing point of view with orthogonal channel vectors of equalpower among the users scenario which has been analyzed in[5 11] an upper bound can be derived for the mean SIR usingZF precoding in JT CoMP Using the expression for 119870IUI asprovided in Appendix A for 119873
119887gt 119873
119906ge 2 we obtain the
following expression
SIRmax asymp1
412058721205902
1198911199052119899
sdot119873119887
119873119906minus 1
minus119873119887+ 1
119873119906minus 1
(43)
The distance between (42) and (43) reveals the potential forSIR enhancement if the usersrsquo selection process considers theorthogonality among their channel vectors
It should be clarified at this point that (42) and (43) do notconsider any gains from resource allocation the evaluationof which not in the scope of this work In an orthogonalfrequency division multiple access (OFDMA) system eachuser can be assigned a part of the spectrum in a way that hisperformance and the network performance can be optimizedas known from [35] and also shown in [36] for the multiuserdownlink
The signal model given in Section 3 for the impairedOFDM-based JT CoMP is valid for any OFDMA schemeUsing OFDMA in the downlink does not affect the signalstructure of a userrsquos received self-signal and the IUI TheICI also has the same structure and statistical propertiesas typically data are transmitted on all subcarriers which isrecommended for keeping frequency-flat distribution of theICI power [13] The degradation mechanisms due to multipleCFOs and SFOs as described in our model will be the samefor any system using OFDM
What indeed changes in OFDMA is the channel statis-tics for each user after resource allocation Therefore ifintended to evaluate the overall performance of a coordinatedmultipoint system using OFDMA the general mean powerexpressions (26) (31) and (37) would need to be evaluated forthe actual usersrsquo channel statistics This procedure practicallyimplies a calculation of constants119870U119870IUI and119870ICI
In this section the received power of a user was studiedin relation to the interference from other users and othersubcarriers in amulticellular multiuser downlink with coop-erative base stations Closed-form expressions and accurateapproximations for all relevant termswere derivedMoreoverit was demonstrated that interuser interference has the mostrelevant effect Finally the impact of the radio channel wasinvestigated and analytical SIR expressions were derived forzero-forcing precoding
5 Evaluation and Numerical Validation
In this section analytical results of Section 4 are evaluatedand verified by simulations Based on those results adequaterequirements are determined for the base stationsrsquo oscillatorsin CoMP systems
A JT CoMP scenario is considered where a cooperationcluster of 119873
119887= 7 base stations transmits jointly by using
zero-forcing precoding to 119873119906terminals on the same time
and frequency resource with 119873119887gt 119873
119906ge 2 Out-of-cluster
interference is not consideredAt time instant 119905 = 0 the base stations receive an update
of the downlink channel matrix and compute a precodingmatrix This will be used during the following 10ms timeat which a new channel update will be obtained and a newprecoder will be calculated This routine agrees with thecurrent 3GPP LTE channel and precoder updating cycle [31]As already mentioned in Section 3 it is assumed that theradio channel is perfectly estimated and available at the basestations at 119905 = 0 whereas channel aging effects are notconsidered either Between updating instants the phases ofthe local base stationsrsquo oscillators slowly drift away fromeach other because of the individual frequency offsets withrespect to the ideal carrier frequency 119891
119888 As a consequence
self-signal drops IUI grows and overall the SIR drops withtime In a practical system each user can use downlink pilotsymbols to estimate and equalize its own CFO and avoid ICIenhancement However mobiles cannot stop the IUI fromrising because the degradation is caused jointly by all themisaligned base station oscillators
The oscillator accuracy Osc typically specified in partsper million (ppm) or parts per billion (ppb) relates to thestandard deviation of the resulting carrier frequency as 120590
119891=
Osc sdot 119891119888 The CFO variation over time is assumed to be slow
enough to be considered static with respect to the signalingand precoder updating timescale Here Gaussian iid CFOsare randomly assigned to base stations and mobile users allwith zero mean and standard deviations given by 120590
119891and 120590
119891119895
respectivelyTypical 3GPP LTE parameters are used specified in [31]
The broadband OFDM signal has 119873119904= 2048 subcarriers
separated by 120575 = 15 kHzThe length of the cyclic prefix is 144samples resulting in OFDM symbols with a length of 119873
119892=
2192 samples The ideal carrier and sampling frequencies are119891119888= 265GHz and 1119879 = 3072MHz respectively The
energy per data symbol is set to 119864119904= 1
Figure 2 depicts the mean power of the IUI and ICI overtime passed from the last precoder update (in logarithmicscale) for 3 and 6 mobile users a base station oscillator
International Journal of Antennas and Propagation 9
0
Time (ms)
Mea
n po
wer
(dB)
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
10minus1 100 101
IUI Nu = 3 Osc = 1ppbICI Nu = 3 Osc = 1ppb 120585 = 0
ICI Nu = 3 Osc = 1ppb 120585 = 10
IUI Nu = 6 Osc = 1ppbICI Nu = 6 Osc = 1ppb 120585 = 0
ICI Nu = 6 Osc = 1ppb 120585 = 0
Figure 2 Mean power of interuser and intercarrier interferences ina Rayleigh fading channel Here 7 base stations serve 3 and 6 usersrespectively Analytical results are shown by lines while markersshow numerical evaluations
accuracy of 1 ppb and Rayleigh fading channel conditionsThe curves illustrate analytical results given by (33) and(38) whereas solid lines depict results for 3 users anddashed lines for 6 users respectivelyMarkers show respectivenumerical evaluations of themeanpower of (14) and (15) over10
3 independent realizations of the MIMO channel matrixentries with 120590
2
ℎ= 1 and of the basesrsquo CFOs Regarding
the ICI two scenarios are considered a scenario with 120585 =
0 that is perfectly synchronized mobile users in order toevaluate solely the ICI due to the base stationsrsquo CFOs (red)and a second scenario with nonsynchronized mobiles with aCFO accuracy of an order of magnitude lower than the basestationsrsquo one that is 120585 = 10 (black) where the total ICI isevaluated
First of all Figure 2 verifies the analytical expres-sions (includingmathematical approximations) by numericalmeans It is further observed that the IUI grows approxi-mately with the square of time and that it increases with thenumber of users Comparing the mean power levels of IUI(blue curves) with the ICI both caused by the base stationsrsquoCFOs (red curves for 120585 = 0) we see that independently of thenumber of users and for typical feedback delays between 2msand 10ms the IUI lies between 40 dB and 50 dB above thepower of ICIThis clearly indicates that the IUI is significantlystronger than the ICI caused by the base stationsrsquo CFOsConsidering now the second scenario with nonperfectlysynchronized mobile users (120585 = 10) it can be observed thatthe part of the ICI caused by the usersrsquo CFO overwhelms theICI caused by the base stationsrsquo CFOs According to (38) the(dominant) ICI due to a mobile userrsquos CFO does not depend
SIR
(dB)
0
10
20
30
40
50
60
70
80
90
Time (ms)10minus1 100 101
20dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 3 Mean SIR over time for Rayleigh fading channel and SIRupper bound Here 7 base stations using oscillators of accuracygiven by Osc serve jointly 6 users Analytical results are shown bylines simulations by markers
on the number of served users which is reflected in the factthat the black curves in Figure 2 evaluating the total ICI for120585 = 10 almost overlap for the cases of 3 and 6 users It isalso interesting to observe that the IUI power level due tothe cooperating base stationsrsquo CFOs is still around 20 dB to30 dB (for 3 users) and 30 dB to 40 dB (for 6 users) higherthan the total ICI power considering typical feedback delaysbetween 2ms and 10ms These results clearly show that inthe a cooperative network even with typically nonperfectlysynchronized mobile users the main interference source liesin the base stationsrsquo CFOs
Figure 3 shows the SIR over the time as defined in (41)for119873
119887= 7 base stations serving jointly119873
119906= 6mobile users
which is also the highest possible number for the Rayleighfading channel The influence of the base stationsrsquo oscillatoraccuracy is evaluated and the corresponding ICI is neglectedas it is significantly smaller than the IUI Disregarding the ICIalso means that the synchronicity level of the mobile users isnot relevant as it does not contribute the IUI
For the Rayleigh fading channel the analytical expression(42) is compared with the ratio between the numericallyevaluated mean power of (13) and (14) over 103 independentchannel and CFO realizations which validates our analysisincluding the accuracy of the mathematical approximationsSimilarly for the SIR upper bound expression the analyticalexpression (43) is compared with results from numericalevaluation From Figure 3 it is seen that a degradation of thebase station oscillatorsrsquo accuracy by one order of magnitudeincreases the IUI and thus decreases the SIR by around 20 dBFurthermore the SIR drops approximately quadratically withtime that is 20 dB per time decade In terms of accuracyrequirements it is found that for reaching in average anSNR of 20 dB 10ms after the precoder update (free-running)oscillators of 1 ppb are needed
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Electrical and Computer Engineering
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Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 9
0
Time (ms)
Mea
n po
wer
(dB)
minus80
minus70
minus60
minus50
minus40
minus30
minus20
minus10
10minus1 100 101
IUI Nu = 3 Osc = 1ppbICI Nu = 3 Osc = 1ppb 120585 = 0
ICI Nu = 3 Osc = 1ppb 120585 = 10
IUI Nu = 6 Osc = 1ppbICI Nu = 6 Osc = 1ppb 120585 = 0
ICI Nu = 6 Osc = 1ppb 120585 = 0
Figure 2 Mean power of interuser and intercarrier interferences ina Rayleigh fading channel Here 7 base stations serve 3 and 6 usersrespectively Analytical results are shown by lines while markersshow numerical evaluations
accuracy of 1 ppb and Rayleigh fading channel conditionsThe curves illustrate analytical results given by (33) and(38) whereas solid lines depict results for 3 users anddashed lines for 6 users respectivelyMarkers show respectivenumerical evaluations of themeanpower of (14) and (15) over10
3 independent realizations of the MIMO channel matrixentries with 120590
2
ℎ= 1 and of the basesrsquo CFOs Regarding
the ICI two scenarios are considered a scenario with 120585 =
0 that is perfectly synchronized mobile users in order toevaluate solely the ICI due to the base stationsrsquo CFOs (red)and a second scenario with nonsynchronized mobiles with aCFO accuracy of an order of magnitude lower than the basestationsrsquo one that is 120585 = 10 (black) where the total ICI isevaluated
First of all Figure 2 verifies the analytical expres-sions (includingmathematical approximations) by numericalmeans It is further observed that the IUI grows approxi-mately with the square of time and that it increases with thenumber of users Comparing the mean power levels of IUI(blue curves) with the ICI both caused by the base stationsrsquoCFOs (red curves for 120585 = 0) we see that independently of thenumber of users and for typical feedback delays between 2msand 10ms the IUI lies between 40 dB and 50 dB above thepower of ICIThis clearly indicates that the IUI is significantlystronger than the ICI caused by the base stationsrsquo CFOsConsidering now the second scenario with nonperfectlysynchronized mobile users (120585 = 10) it can be observed thatthe part of the ICI caused by the usersrsquo CFO overwhelms theICI caused by the base stationsrsquo CFOs According to (38) the(dominant) ICI due to a mobile userrsquos CFO does not depend
SIR
(dB)
0
10
20
30
40
50
60
70
80
90
Time (ms)10minus1 100 101
20dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 3 Mean SIR over time for Rayleigh fading channel and SIRupper bound Here 7 base stations using oscillators of accuracygiven by Osc serve jointly 6 users Analytical results are shown bylines simulations by markers
on the number of served users which is reflected in the factthat the black curves in Figure 2 evaluating the total ICI for120585 = 10 almost overlap for the cases of 3 and 6 users It isalso interesting to observe that the IUI power level due tothe cooperating base stationsrsquo CFOs is still around 20 dB to30 dB (for 3 users) and 30 dB to 40 dB (for 6 users) higherthan the total ICI power considering typical feedback delaysbetween 2ms and 10ms These results clearly show that inthe a cooperative network even with typically nonperfectlysynchronized mobile users the main interference source liesin the base stationsrsquo CFOs
Figure 3 shows the SIR over the time as defined in (41)for119873
119887= 7 base stations serving jointly119873
119906= 6mobile users
which is also the highest possible number for the Rayleighfading channel The influence of the base stationsrsquo oscillatoraccuracy is evaluated and the corresponding ICI is neglectedas it is significantly smaller than the IUI Disregarding the ICIalso means that the synchronicity level of the mobile users isnot relevant as it does not contribute the IUI
For the Rayleigh fading channel the analytical expression(42) is compared with the ratio between the numericallyevaluated mean power of (13) and (14) over 103 independentchannel and CFO realizations which validates our analysisincluding the accuracy of the mathematical approximationsSimilarly for the SIR upper bound expression the analyticalexpression (43) is compared with results from numericalevaluation From Figure 3 it is seen that a degradation of thebase station oscillatorsrsquo accuracy by one order of magnitudeincreases the IUI and thus decreases the SIR by around 20 dBFurthermore the SIR drops approximately quadratically withtime that is 20 dB per time decade In terms of accuracyrequirements it is found that for reaching in average anSNR of 20 dB 10ms after the precoder update (free-running)oscillators of 1 ppb are needed
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Civil EngineeringAdvances in
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Electrical and Computer Engineering
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SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 International Journal of Antennas and Propagation
Number of users
SIR
(dB)
52 3 4 5 6
10
15
20
25
30
35
40
45
20 dB
SIRmax Osc = 01ppbSIRRayleigh Osc = 01ppb
SIRmax Osc = 1ppbSIRRayleigh Osc = 1ppb
Figure 4 Mean usersrsquo SIR 10ms after most recent precoder updatefor the Rayleigh fading and upper bound Here 7 base stationsserve jointly from 2 to 6 users Analytical results are shown by linessimulations by markers
Figure 4 illustrates the mean SIR as function of thenumber of users which are served by 7 base stations at thesame time and frequency resource for a time instant of 10msafter the most recent precoder update Such a time delay isrelatively large for a practical system and thus these resultsshould be interpreted more as a worst case rather than anaverage situationHere analytical results based on (41) for theRayleigh fading channel and the SIR upper bound are verifiedby numerical simulations It is observed that given the samesynchronization conditions serving jointlymore users resultsin a lower SIR compared to serving fewer users It is alsoobserved that as the number of users grows the distancebetween the SIR in Rayleigh fading and its upper boundbecomes larger Practically this means that themore the usersthat are jointly served the higher the potential benefits if setsof jointly served users are appropriately formed as alreadydiscussed in Section 45
6 Recommendations for CoMP System Design
In this section synchronization requirements for OFDM-based JT CoMP and ways to meet them are discussed Itcan be seen from Figure 4 that if for example an averageSIR of at least 25 dB will be guaranteed at all times betweenprecoder updates and for the maximum allowed numberof users then the accuracy of the base stationsrsquo oscillatorsneeds to be 01 ppb Note that base stations typically containalready sufficiently precise oven-controlled crystal oscillators(OCXOs) however their frequencies need to be locked toa common reference which can be for example providedby the GPS The satellites of the GPS system are equippedwith Rubidium or Cesium oscillators thus providing a highly
reliable reference below 1 120583s in terms of absolute time error[37] At each base station equipped with a GPS receiver thelocal oscillator will be then phase-locked to the incomingtime signal from the GPS and will also follow its carrierfrequency Practical synchronization of distributed stations ina JT CoMP testbed is described in [27]
Other schemes also for base stations with no GPSconnection suggest that base stations are connected viaEthernet to the backhaul network over which and by usinga network synchronization protocol such as IEEE1588 [38] acommon clock signal can be provided to themThis referencesignal can be then used for time synchronization as wellas for recovering a carrier frequency reference at each siteHowever the precision of such protocols could be probablynot sufficient for JT CoMP Therefore additional over-the-air synchronization procedures can be deployed such as theprotocol proposed in [28] Distributed base stations incor-porate a suitable frequency estimator for example the ML-based estimator described in [14] and additionally exchangepilots for enhancing the network synchronicity The protocolcan be also applied to networks with a large number of nodesand does not require expensive oscillators
7 Conclusions
An exact signal model for multiuser multicellular systemsusing OFDM was derived for transmissions impaired byindividual carrier and sampling frequency offsets on everytransmitter and receiver branch The model was specializedto the downlink of systems with cooperative base stationswhere precoding with the inverse of the channel matrixwas considered It was analyzed how carrier frequencymisalignments among the cooperative base stations decreasethe power of the self-userrsquos signal and cause interuser andintercarrier interference Closed-form exact expressions andaccurate approximations were derived for the mean powerof the above signals and it was shown that for practicalpurposes the interuser interference dominates on the inter-carrier interference Analytic expressions were also derivedfor the mean SIR for Rayleigh fading channel conditions aswell as an SIR upper bound for cooperative systems usingzero-forcing precoding The mean SIR decreases quadrati-cally with time and is inversely proportional to the variance ofthe base stationsrsquo frequency offsets It grows with the numberof base stations and drops with the number of users jointlyserved by them on the same time and frequency resourceFrom a practical point of view when a high SIR is targetedsynchronization requirements can be fulfilled by using at thebase stations OCXOs locked either to a precise GPS referenceor to an accurate clock signal provided through the backhaulnetwork
Appendices
A Analysis of 119870U 119870IUI and 119870ICI
The constants 119870U 119870IUI and 119870ICI are introduced in the maintext in (26) (31) and (38) respectively For identical channelstatistics for all users and 119870 ≜ sum
119873119887
119887=1E|119867
119895119887119908119887119906|2
we have
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
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Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
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Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 11
119870U = 1 minus 119870 119870IUI = (119873119906minus 1) sdot 119870 and 119870ICI = 119873
119906sdot 119870
The following derivation gives an approximation for 119870 inRayleigh fading The correlation between 119867
119895119887and 119908
119887119906can
be considered as weak because taking into account how thepseudoinverse is calculated there is only minor contributionof element119867
119895119887to 119908
119887119906 Thus
119870 =
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887119908119887119906
10038161003816100381610038161003816
2
=
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
| 119867119895119887
asymp
119873119887
sum
119887=1
E 10038161003816100381610038161003816119867119895119887
10038161003816100381610038161003816
2
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
= 1205902
ℎ
119873119887
sum
119887=1
E 1003816100381610038161003816119908119887119906
1003816100381610038161003816
2
(A1)
where Esdot | 119860 denotes the conditional expectation givenevent119860 Using the analytical expressions provided by [39] forthe sum term in the right-hand side of (A1) for ZF precodingwith119873
119887gt 119873
119906 we reach
119870 asymp1
119873119887minus 119873
119906
119870min asymp1
119873119887
(A2)
The first expression for 119870 in (A2) refers to iid complexGaussian random entries with zeromean and variance1205902
ℎand
can be used to obtain expressions for119870U119870IUI and119870ICI while119870min is used for their boundsThose can then be used in (26)(31) and (38) in the main text
B Analysis of 1198611
and 1198612
We analyze sum1198731199042minus1
]=minus1198731199042] =119896
E1205731198951198871
(119896 ])120573lowast1198951198872
(119896 ]) which is givenin (35) in the main text while 120573
119895119887(119896 ]) is given in (9) The
base stationsrsquo and 119895th mobile userrsquos CFOs are considered asiid Gaussian-distributed with zero mean and variance 1205902
119891
and 1205902119891119895 respectively
For the first case 1198871= 119887
2= 119887 we proceed as follows
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
=
1198731199042minus1
sum
]=minus1198731199042
10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2
(B1)
= 1 minus10038161003816100381610038161003816120573119895119887(119896 119896)
10038161003816100381610038161003816
2 (B2)
= 1 minus1
1198732
119904
sdot
sin2 (120587 (119891119895minus 119891
119887) 119879119873
119904)
sin2 (120587 (119891119895minus 119891
119887) 119879)
(B3)
asymp
1205872
(119891119895minus 119891
119887)2
31205752
=
1205872
[(119891119895minus 119891
119888) minus (119891
119887minus 119891
119888)]2
31205752
(B4)
From (B1) to (B2) we used the result (C1) of Appendix Cwhich shows that the power of 119873
119904samples of a function
120572(119896) = (1119873119904)(sin(119873
119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873
119904
taken at frequencies 119896 + 120601 is equal to 1 for any offset 120601 isin RImposing (9) into (B2) gives (B3) For typical values theTaylor series expansion (1119873
2
119904)(sin2(119873
119904119909)sin2(119909)) asymp 1 minus
(13)1198732
1199041199092 (1198732
119904minus1 asymp 119873
2
119904for119873
119904≫ 1) is accurate andwas used
in (B3) to reach (B4) Note that 120575 = (119873119904119879)
minus1 is the subcarrierspacing Taking the expectation of (B4) and considering theindependence between 119891
119895and 119891
119887 we reach
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 10038161003816100381610038161003816120573119895119887(119896 ])
10038161003816100381610038161003816
2
asymp
1205872
(1205902
119891+ 120590
2
119891119895)
31205752≜ 119861
1 (B5)
For the second case 1198871
= 1198872 we have
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ])
=1
1198732
119904
sdot10038161003816100381610038161003816F
119901(119905119899)10038161003816100381610038161003816
2
sdot
1198731199042minus1
sum
]=minus1198731199042
] =119896
10038161003816100381610038161003816100381610038161003816100381610038161003816
Esin [120587 (119896 minus ]) + 120587(119891
119895minus 119891
119887)119879119873
119904]
sin [(120587119873119904) (119896 minus ]) + 120587 (119891
119895minus 119891
119887) 119879]
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
(B6)
Expression (B6) was numerically evaluated and it was foundthat for 119891
119895= 119891
119888(perfectly synchronized mobiles) it is
significantly smaller than (B5) For 120590119891119895
ge 120590119891 (B6) is given
by
1198731199042minus1
sum
]=minus1198731199042
] =119896
E 1205731198951198871
(119896 ]) 120573lowast1198951198872
(119896 ]) asymp1205872
1205902
119891119895
31205752≜ 119861
2 (B7)
Expressions (B5) and (B7) provide accurate approximationsfor 119861
1and 119861
2for Gaussian-distributed CFOs Those results
allow for taking the step from (36) to (37) in the main text
C Power of a Periodic Band-Limited FunctionSampled with an Offset
Consider the function 120572(119909) = (1119873119904) sum
1198731199042minus1
119899=minus11987311990421198901204852Φ(119909)119899 where
Φ(119909) = 120587119909119873119904 It can be seen that 120572(119909) is a periodic band-
limited function with period 119879 = 119873119904and Nyquist frequency
1Δ = 1 and that 120572(119909) = sin(119873119904Φ(119909))119873
119904sin(Φ(119909)) We
want to show that
119873119904minus1
sum
119896=0
1003816100381610038161003816120572 (119896 + 120601)1003816100381610038161003816
2
=
119873119904minus1
sum
119896=0
|120572 (119896)|2
= 1 (C1)
for offset 120601 isin R To prove this we present the followinglemma
Lemma C1 Let 120572(119909) = sum119873minus1
119903=0119860119903exp(1204852120587(119903119879)119909) be an
arbitrary periodic band-limited function of period 119879 with
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
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Active and Passive Electronic Components
Control Scienceand Engineering
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Civil EngineeringAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Electrical and Computer Engineering
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Advances inOptoElectronics
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Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 International Journal of Antennas and Propagation
Fourier coefficients119860119903 Let one define 120574(120601)(119909) by sampling 120572(119909)
with frequency 1Δ and offset 120601
120574(120601)
(119909) = 120572 (119909 + 120601)(
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ)) (C2)
=
infin
sum
119896=minusinfin
120572(120601)
119896120575 (119909 minus 119896Δ) (C3)
where 120572(120601)119896
= 120572(119896Δ+120601) and 120575(sdot) is the delta distributionThenif 1Δ = 119873119879 (which is the Nyquist frequency for 120572(119909)) theenergy contained in one period of 120574(120601) is given by
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1
Δint
119879
0
|120572 (119909)|2
119889119909 (C4)
and therefore is independent of 120601
Proof 120574(120601) is a periodic function for which we calculateFourier coefficients Γ(120601)
119903in two ways
First if F119891(119909)(119903) = 1119879 int119879
0
119891(119909) expminus1204852120587119903119909119879119889119909denotes the Fourier transform for a periodic function 119891(119909)using (C3) one obtains
Γ(120601)
119903= [F 120572 (119909 + 120601) (119904) lowastF
infin
sum
119896=minusinfin
120575 (119909 minus 119896Δ) (119904)] (119903)
(C5)
= [exp 1204852120587 119904119879120601119860
119904] lowast [
1
Δ
infin
sum
119906=minusinfin
120575119906119873
119904] (119903) (C6)
=exp 1204852120587 (119903119879) 120601
Δ119860119903 (C7)
where 119903 isin Z [119886119904lowast 119887
119904](119903) = sum
infin
119904=minusinfin119886119904119887119903minus119904
denotes the discretelinear convolution 120575119909
119904is a sequence of numbers that has its
119904th entry equal to 1 if 119904 = 119909 and zero otherwise and 119903 is thesmallest positive integer that satisfies 119903 equiv 119903 mod 119873 If 119903 isin
0 119873 minus 1 then 119903 = 119903Next using (C3) and the fact that 119879 = 119873Δ the Fourier
coefficient Γ(120601)119896
can also be found as
Γ(120601)
119903=1
119879int
119879minusΔ2
minusΔ2
120574(120601)
(119909) 119890minus1204852120587(119903119879)119909
119889119909 (C8)
=1
119879
infin
sum
119896=minusinfin
int
(119873minus12)Δ
minusΔ2
120572(120601)
119896120575 (119909 minus 119896Δ) 119890
minus1204852120587(119903119879)119909
119889119909 (C9)
=1
119879
119873minus1
sum
119896=0
120572(120601)
119896119890minus1204852120587(119903119896119873)
(C10)
where (C8) holds because the integral of a periodic functionover its full period is translation invariant Equation (C10)implies that Γ(120601)
119903is the discrete Fourier transform (DFT) of
120572(120601)
119896119879 and therefore the DFT Parseval identity says that
119873minus1
sum
119896=0
10038161003816100381610038161003816100381610038161003816100381610038161003816
120572(120601)
119896
119879
10038161003816100381610038161003816100381610038161003816100381610038161003816
2
=1
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
(C11)
Finally using (C7) with 119903 = 119903 and (C11) it can be proven that
119873minus1
sum
119896=0
100381610038161003816100381610038161003816120572(120601)
119896
100381610038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
10038161003816100381610038161003816Γ(120601)
119903
10038161003816100381610038161003816
2
=1198792
119873
119873minus1
sum
119896=0
100381610038161003816100381610038161003816100381610038161003816
exp 1204852120587(119903119879)120601Δ
119860119903
100381610038161003816100381610038161003816100381610038161003816
2
= 119873
119873minus1
sum
119896=0
1003816100381610038161003816119860119903
1003816100381610038161003816
2
=119873
119879int
119879
0
|120572 (119909)|2
119889119909
(C12)
which is what we wanted to prove in (C1) Above we usedthe Parseval identity for Fourier series
Disclosure
This work was done during KonstantinosManolakis doctoralstudies at the Technische Universitat Berlin Germany
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the DeutscheForschungsgemeinschaft (DFG) for support within projectCoMP impairments (JU 27933-1) and CoMP mobil (JU27934-1) This work was also supported by projects fromCONICYT Departamento de Relaciones InternacionalesldquoPrograma de Cooperacion Cientıfica InternacionalrdquoCONICYTDFG-622 and FONDECYT 1110370 FernandoRosas would also like to acknowledge support of the F+fellowship from KU Leuven and the SBO project SINSfunded by the Agency for Innovation by Science andTechnology IWT (Belgium) Finally the authors would liketo thank Dr Malte Schellmann from the Huawei EuropeanResearch Center for his useful comments and suggestions
References
[1] M Karakayali G J Foschini and R A Valenzuela ldquoNetworkcoordination for spectrally efficient communications in cellularsystemsrdquo IEEE Wireless Communications vol 13 no 4 pp 56ndash61 2006
[2] G J Foschini K Karakayali and R A Valenzuela ldquoCoordi-nating multiple antenna cellular networks to achieve enormousspectral efficiencyrdquo IEE Proceedings Communications vol 153no 4 pp 548ndash555 2006
[3] D Gesbert M Kountouris R W Heath C-B Chae and TSalzer ldquoShifting the MIMO paradigmrdquo IEEE Signal ProcessingMagazine vol 24 no 5 pp 36ndash46 2007
[4] QH Spencer A L Swindlehurst andMHaardt ldquoZero-forcingmethods for downlink spatial multiplexing inmultiuserMIMO
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 13
channelsrdquo IEEE Transactions on Signal Processing vol 52 no 2pp 461ndash471 2004
[5] A Goldsmith S A Jafar N Jindal and S Vishwanath ldquoCapac-ity limits of MIMO channelsrdquo IEEE Journal on Selected Areas inCommunications vol 21 no 5 pp 684ndash702 2003
[6] R Irmer H Droste P Marsch et al ldquoCoordinated multipointconcepts performance and field trial resultsrdquo IEEE Communi-cations Magazine vol 49 no 2 pp 102ndash111 2011
[7] F Boccardi B Clerckx A Ghosh et al ldquoMultiple-antennatechniques in LTE-advancedrdquo IEEECommunicationsMagazinevol 50 no 3 pp 114ndash121 2012
[8] D Lee H Seo B Clerckx et al ldquoCoordinated multipoint trans-mission and reception in LTE-advanced deployment scenariosand operational challengesrdquo IEEE Communications Magazinevol 50 no 2 pp 148ndash155 2012
[9] V Jungnickel K Manolakis W Zirwas et al ldquoThe role of smallcells coordinated multipoint and massive MIMO in 5Grdquo IEEECommunications Magazine vol 52 no 5 pp 44ndash51 2014
[10] X Tao X Xu and Q Cui ldquoAn overview of cooperativecommunicationsrdquo IEEE Communications Magazine vol 50 no6 pp 65ndash71 2012
[11] G J Foschini and M J Gans ldquoOn limits of wireless communi-cations in a fading environment when usingmultiple antennasrdquoWireless Personal Communications vol 6 no 3 pp 311ndash3351998
[12] Q H Spencer C B Peel A L Swindlehurst and M HaardtldquoAn introduction to the multi-user MIMO downlinkrdquo IEEECommunications Magazine vol 42 no 10 pp 60ndash67 2004
[13] M Speth S A Fechtel G Fock and H Meyr ldquoOptimumreceiver design for wireless broad-band systems using OFDMPart Irdquo IEEE Transactions on Communications vol 47 no 11 pp1668ndash1677 1999
[14] C Oberli ldquoML-based tracking algorithms for MIMO-OFDMrdquoIEEE Transactions onWireless Communications vol 6 no 7 pp2630ndash2639 2007
[15] L Haring S Bieder A Czylwik and T Kaiser ldquoEstimationalgorithms of multiple channels and carrier frequency offsetsin application to multiuser OFDM systemsrdquo IEEE Transactionson Wireless Communications vol 9 no 3 pp 865ndash870 2010
[16] M Schellmann and V Jungnickel ldquoMultiple CFOs in OFDM-SDMA uplink interference analysis and compensationrdquoEURASIP Journal onWireless Communications and Networkingvol 2009 Article ID 909075 14 pages 2009
[17] K Manolakis C Oberli and V Jungnickel ldquoSynchronizationrequirements for OFDM-based cellular networks with coor-dinated base stations preliminary resultsrdquo in Proceedings ofthe 15th International OFDM-Workshop Hamburg GermanySeptember 2010
[18] P Marsch and G Fettweis Eds Coordinated Multi-Point inMobile Communications From Theory to Practice CambridgeUniversity Press 2011
[19] R Mudumbai D R Brown U Madhow and H V PoorldquoDistributed transmit beamforming challenges and recentprogressrdquo IEEE Communications Magazine vol 47 no 2 pp102ndash110 2009
[20] T Koivisto and V Koivunen ldquoImpact of time and frequencyoffsets on cooperative multi-user MIMO-OFDM systemsrdquo inProceedings of the IEEE 20th International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRCrsquo09) September 2009
[21] T Koivisto and V Koivunen ldquoLow complexity estimation ofmultiple frequency offsets using optimized training signalsrdquo inProceedings of the IEEE 11th International Symposium on SpreadSpectrum Techniques and Applications (ISITA rsquo10) pp 81ndash86Taichung Taiwan October 2010
[22] B W Zarikoff and J K Cavers ldquoMultiple frequency offsetestimation for the downlink of coordinated MIMO systemsrdquoIEEE Journal on Selected Areas in Communications vol 26 no6 pp 901ndash912 2008
[23] Y R Tsai H Y Huang Y C Chen and K J Yang ldquoSimulta-neous multiple carrier frequency offsets estimation for coor-dinated multipoint transmission in OFDM systemsrdquo IEEETransactions on Wireless Communications vol 12 no 9 pp4558ndash4568 2013
[24] BW Zarikoff and J K Cavers ldquoCoordinatedmulti-cell systemscarrier frequency offset estimation and correctionrdquo IEEE Jour-nal on SelectedAreas inCommunications vol 28 no 9 pp 1490ndash1501 2010
[25] V Kotzsch and G Fettweis ldquoInterference analysis in time andfrequency asynchronous network MIMO OFDM systemsrdquo inProceedings of the IEEEWireless Communications and Network-ing Conference (WCNC rsquo10) pp 1ndash6 April 2010
[26] D R Brown III and H V Poor ldquoTime-slotted round-tripcarrier synchronization for distributed beamformingrdquo IEEETransactions on Signal Processing vol 56 no 11 pp 5630ndash56432008
[27] V Jungnickel T Wirth M Schellmann T Haustein andW Zirwas ldquoSynchronization of cooperative base stationsrdquo inProceedings of the IEEE International Symposium on WirelessCommunication Systems (ISWCS rsquo08) pp 329ndash334 October2008
[28] R Rogalin O Y Bursalioglu H Papadopoulos et al ldquoScalablesynchronization and reciprocity calibration for distributedmul-tiuser MIMOrdquo IEEE Transactions on Wireless Communicationsvol 13 no 4 pp 1815ndash1831 2014
[29] O Simeone U Spagnolini Y Bar-Ness and S H StrogatzldquoDistributed synchronization in wireless networksrdquo IEEE SignalProcessing Magazine vol 25 no 5 pp 81ndash97 2008
[30] C Oberli ldquoErratum to lsquoML-based tracking algorithms forMIMO-OFDMrsquo IEEE Transactions on Wireless Communica-tions vol 6 no 7 pp 2630ndash2639 2007rdquo September 2007
[31] 3rd Generation Partnership Project TS 36300 V1170 EvolvedUniversal Terrestrial Radio Access (E-UTRA) and Evolved Uni-versal Terrestrial Radio Access Network (E-UTRAN) OverallDescription 3GPP Standards 2013
[32] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill New York NY USA 2nd edition 1991
[33] F Rosas L Herrera C Oberli K Manolakis and V Jung-nickel ldquoDownlink performance limitations of cellular systemswith coordinated base stations and mismatched precoderrdquo IETCommunications vol 8 no 1 pp 77ndash82 2014
[34] M Lossow S Jaeckel V Jungnickel and V Braun ldquoEfficientMAC protocol for JT CoMP in small cellsrdquo in Proceedings ofthe IEEE 2nd International Workshop on Small Cell WirelessNetworks (SmallNets rsquo13) June 2013
[35] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposium on InformationTheory pp 1394ndash1398 July 2006
[36] S Shi M Schubert and H Boche ldquoRate optimization formultiuser MIMO systems with linear processingrdquo IEEE Trans-actions on Signal Processing vol 56 no 8 pp 4020ndash4030 2008
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 International Journal of Antennas and Propagation
[37] M A Lombardi L M Nelson A N Novick and V SZhang ldquoTime and frequency measurements using the globalpositioning systemrdquoCal Lab International Journal ofMetrologyvol 8 no 3 pp 26ndash33 2001
[38] IEEE ldquoIEEE standard for a precision clock synchronizationprotocol for networked measurement and control systemsrdquoTech Rep 1588-2008 IEEE Standards Association 2008
[39] K Manolakis C Oberli and V Jungnickel ldquoRandom matricesand the impact of imperfect channel knowledge on cooperativebase stationsrdquo in Proceedings of the IEEE 14th InternationalWorkshop on Signal Processing Advances in Wireless Communi-cations (SPAWC rsquo13) pp 480ndash484 Darmstadt Germany June2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of