research article analysis of synchronization impairments...

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)


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)


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)


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


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


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)


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


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


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


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


Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB



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


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


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


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


[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

Propagation




Navigation and Observation



DistributedSensor Networks



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)


Baseband processing

Baseband processing


DA AD

Ti Tjfi fi










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


sdot

sin [120587 (119891 + 119891119895minus 119891

119894minus 119896120575

119894) 119879

119895119873119904]

sin [120587 (119891 + 119891119895minus 119891

119894minus 119896120575

119894) 119879

119895]

(4)


119895119894(119896120575

119894+ 119891

119894minus 119891






119894 120591

119895 120593

119894 and 120593

119895may be


119894= 0 120591

119895= 0 120593

119894= 0 and



119895(119891) taken at

frequencies 119891 = 119897119873119904119879119895= 119897120575


index (minus1198731199042 le 119897 le 119873

1199042 minus 1) and 120575



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)


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)



defined119867119895119894(119896120575

119894+119891

119894minus119891

119888) asymp 119867

119895119894(119896120575

119894) ≜ 119867

119895119894(119896) because |119891

119894minus119891

119888|












119894= 119879

119895= 119879 (8) becomes


sdot1

119873119904

sin [120587 (119896 minus ]) + 120587 (119891119895minus 119891

119894) 119879119873

119904]

sin [(120587119873119904) (119896 minus ]) + 120587 (119891

119895minus 119891

119894) 119879]

(9)


119892+ (119873

119904minus 1)2)119879 has been











119906times 119873

119887downlink


W (119896) = H119867

(119896) [H (119896)H119867

(119896)]minus1

(10)



119873119887gt 119873









given by

119883119887(119896) =

119873119906

sum

119906=1

119908119887119906(119896) 119904

119906(119896) (11)





119895(119896) + 119904

119895(119896) (12)


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)





119895(119896) in (7) we obtain the


119880119895(119896) =

1198731199042minus1

sum

]=minus1198731199042

] =119896

119873119906

sum

119906=1

119904119906(])

119873119887

sum

119887=1

120573119895119887(119896 ])119867

119895119887(]) 119908

119887119906(]) (15)


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)


0120575119895 It







119895(119896)119904

lowast

119906(]) = 119864



119909119910the




E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904

119873119887

sum

1198871=1

119873119887

sum

1198872=1

E 1205731198951198871

120573lowast

1198951198872

E 1198671198951198871

1199081198871119895119867lowast

1198951198872

119908lowast

1198872119895

(17)



120573lowast

1198951198872


120573119895119887=

1

119873119904

sdot 119890minus1204852120587(119891

119895minus119891119887)119905119899 sdot

sin [120587 (119891119895minus 119891

119887) 119879119873

119904]

sin [120587 (119891119895minus 119891

119887) 119879]

(18)


119895minus 119891


120587(119891119895minus 119891



119904)(sin(119873

119904119909) sin(119909))| ≪ |1 minus

119890minus120485(214119899+1)119873

119904119909




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)




1= 119887

2


1198871

and 1198911198872


E 1205731198951198871

120573lowast

1198951198872

= E 1198901204852120587(119891

1198871minus119891119888)119905119899 sdot E 119890

minus1204852120587(1198911198872minus119891119888)119905119899 (20)


E 1205731198951198871

120573lowast

1198951198872

=10038161003816100381610038161003816E 119890

minus1204852120587(119891119887minus119891119888)11990511989910038161003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816int 119901

(119891119887minus119891119888)119890minus1204852120587(119891

119887minus119891119888)119905119899119889 (119891

119887minus 119891

119888)

1003816100381610038161003816100381610038161003816

2

(21)






E 1205731198951198871

120573lowast

1198951198872

=

1 1198871= 119887

2

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

1198871

= 1198872

(22)


(119891119887minus119891119888)and is






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)


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)


119873119887

sum

119887=1

119867119895119887119908119887119906= 120575

119895119906 (25)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904[1 minus (1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870U] (26)


119887=1E|119867

119895119887119908119887119895|2






119873119887minus 119873

119906

(27)



10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

= 119890minus412058721205902

1198911199052

119899 asymp 1 minus 41205872

1205902

1198911199052

119899 (28)


119891and 119905



E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904[1 minus 4120587

2

1205902

1198911199052

119899sdot (1 minus

1

119873119887minus 119873

119906

)] (29)


119906= 119873




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)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904(1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI (31)

where119870IUI = sum119873119906

119906=1119906 =119895sum119873119887

119887=1E|119867

119895119887119908119887119906|2


minus1) sdot

sum119873119887

119887=1E|119867

119895119887119908119887119906|2


119887gt 119873

119906 we find

119870IUI asymp119873119906minus 1

119873119887minus 119873

119906

(32)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904sdot 4120587

2

1205902

1198911199052

119899sdot119873119906minus 1

119873119887minus 119873

119906

(33)





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)


1198951198871

1199081198871119906119867lowast

1198951198872

119908lowast



sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) ≜

1198611 119887

1= 119887

2

1198612 119887

1= 1198872

(35)



2= 119887

1and 119887


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)




119891and 1205902


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


E 10038161003816100381610038161003816119880119895(119896)

10038161003816100381610038161003816

2

asymp1198641199041205872

31205752sdot (120590

2

119891sdot

119873119906

119873119887minus 119873

119906

+ 1205902

119891119895) (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



119904119879)


119904119879 (this results in 119873



IIR asymp1

12 (107119899 + 05)2sdot (

119873119906

119873119906minus 1

+ 1205852119873119887minus 119873

119906

119873119906minus 1

) (40)




119906= 2





SIR =

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

(1 minus10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI

minus1

119873119906minus 1

(41)


119901(119905119899)|2 given


SIR asymp1

412058721205902

1198911199052119899

sdot119873119887minus 119873

119906

119873119906minus 1

minus119873119887minus 119873

119906+ 1

119873119906minus 1

(42)



119906ge 2




119887gt 119873



SIRmax asymp1

412058721205902

1198911199052119899

sdot119873119887

119873119906minus 1

minus119873119887+ 1

119873119906minus 1

(43)


















119891=



119891and 120590

119891119895




119892=





0

Time (ms)

Mea

n po

wer

(dB)

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101


ICI Nu = 3 Osc = 1ppb 120585 = 10


ICI Nu = 6 Osc = 1ppb 120585 = 0




2





SIR

(dB)

0

10

20

30

40

50

60

70

80

90


20dB











Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB









7 Conclusions


Appendices



119873119887

119887=1E|119867

119895119887119908119887119906|2

we have



119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Baseband processing

Baseband processing


DA AD

Ti Tjfi fi










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


sdot

sin [120587 (119891 + 119891119895minus 119891

119894minus 119896120575

119894) 119879

119895119873119904]

sin [120587 (119891 + 119891119895minus 119891

119894minus 119896120575

119894) 119879

119895]

(4)


119895119894(119896120575

119894+ 119891

119894minus 119891






119894 120591

119895 120593

119894 and 120593

119895may be


119894= 0 120591

119895= 0 120593

119894= 0 and



119895(119891) taken at

frequencies 119891 = 119897119873119904119879119895= 119897120575


index (minus1198731199042 le 119897 le 119873

1199042 minus 1) and 120575



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)


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)



defined119867119895119894(119896120575

119894+119891

119894minus119891

119888) asymp 119867

119895119894(119896120575

119894) ≜ 119867

119895119894(119896) because |119891

119894minus119891

119888|












119894= 119879

119895= 119879 (8) becomes


sdot1

119873119904

sin [120587 (119896 minus ]) + 120587 (119891119895minus 119891

119894) 119879119873

119904]

sin [(120587119873119904) (119896 minus ]) + 120587 (119891

119895minus 119891

119894) 119879]

(9)


119892+ (119873

119904minus 1)2)119879 has been











119906times 119873

119887downlink


W (119896) = H119867

(119896) [H (119896)H119867

(119896)]minus1

(10)



119873119887gt 119873









given by

119883119887(119896) =

119873119906

sum

119906=1

119908119887119906(119896) 119904

119906(119896) (11)





119895(119896) + 119904

119895(119896) (12)


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)





119895(119896) in (7) we obtain the


119880119895(119896) =

1198731199042minus1

sum

]=minus1198731199042

] =119896

119873119906

sum

119906=1

119904119906(])

119873119887

sum

119887=1

120573119895119887(119896 ])119867

119895119887(]) 119908

119887119906(]) (15)


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)


0120575119895 It







119895(119896)119904

lowast

119906(]) = 119864



119909119910the




E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904

119873119887

sum

1198871=1

119873119887

sum

1198872=1

E 1205731198951198871

120573lowast

1198951198872

E 1198671198951198871

1199081198871119895119867lowast

1198951198872

119908lowast

1198872119895

(17)



120573lowast

1198951198872


120573119895119887=

1

119873119904

sdot 119890minus1204852120587(119891

119895minus119891119887)119905119899 sdot

sin [120587 (119891119895minus 119891

119887) 119879119873

119904]

sin [120587 (119891119895minus 119891

119887) 119879]

(18)


119895minus 119891


120587(119891119895minus 119891



119904)(sin(119873

119904119909) sin(119909))| ≪ |1 minus

119890minus120485(214119899+1)119873

119904119909




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)




1= 119887

2


1198871

and 1198911198872


E 1205731198951198871

120573lowast

1198951198872

= E 1198901204852120587(119891

1198871minus119891119888)119905119899 sdot E 119890

minus1204852120587(1198911198872minus119891119888)119905119899 (20)


E 1205731198951198871

120573lowast

1198951198872

=10038161003816100381610038161003816E 119890

minus1204852120587(119891119887minus119891119888)11990511989910038161003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816int 119901

(119891119887minus119891119888)119890minus1204852120587(119891

119887minus119891119888)119905119899119889 (119891

119887minus 119891

119888)

1003816100381610038161003816100381610038161003816

2

(21)






E 1205731198951198871

120573lowast

1198951198872

=

1 1198871= 119887

2

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

1198871

= 1198872

(22)


(119891119887minus119891119888)and is






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)


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)


119873119887

sum

119887=1

119867119895119887119908119887119906= 120575

119895119906 (25)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904[1 minus (1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870U] (26)


119887=1E|119867

119895119887119908119887119895|2






119873119887minus 119873

119906

(27)



10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

= 119890minus412058721205902

1198911199052

119899 asymp 1 minus 41205872

1205902

1198911199052

119899 (28)


119891and 119905



E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904[1 minus 4120587

2

1205902

1198911199052

119899sdot (1 minus

1

119873119887minus 119873

119906

)] (29)


119906= 119873




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)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904(1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI (31)

where119870IUI = sum119873119906

119906=1119906 =119895sum119873119887

119887=1E|119867

119895119887119908119887119906|2


minus1) sdot

sum119873119887

119887=1E|119867

119895119887119908119887119906|2


119887gt 119873

119906 we find

119870IUI asymp119873119906minus 1

119873119887minus 119873

119906

(32)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904sdot 4120587

2

1205902

1198911199052

119899sdot119873119906minus 1

119873119887minus 119873

119906

(33)





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)


1198951198871

1199081198871119906119867lowast

1198951198872

119908lowast



sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) ≜

1198611 119887

1= 119887

2

1198612 119887

1= 1198872

(35)



2= 119887

1and 119887


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)




119891and 1205902


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


E 10038161003816100381610038161003816119880119895(119896)

10038161003816100381610038161003816

2

asymp1198641199041205872

31205752sdot (120590

2

119891sdot

119873119906

119873119887minus 119873

119906

+ 1205902

119891119895) (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



119904119879)


119904119879 (this results in 119873



IIR asymp1

12 (107119899 + 05)2sdot (

119873119906

119873119906minus 1

+ 1205852119873119887minus 119873

119906

119873119906minus 1

) (40)




119906= 2





SIR =

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

(1 minus10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI

minus1

119873119906minus 1

(41)


119901(119905119899)|2 given


SIR asymp1

412058721205902

1198911199052119899

sdot119873119887minus 119873

119906

119873119906minus 1

minus119873119887minus 119873

119906+ 1

119873119906minus 1

(42)



119906ge 2




119887gt 119873



SIRmax asymp1

412058721205902

1198911199052119899

sdot119873119887

119873119906minus 1

minus119873119887+ 1

119873119906minus 1

(43)


















119891=



119891and 120590

119891119895




119892=





0

Time (ms)

Mea

n po

wer

(dB)

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101


ICI Nu = 3 Osc = 1ppb 120585 = 10


ICI Nu = 6 Osc = 1ppb 120585 = 0




2





SIR

(dB)

0

10

20

30

40

50

60

70

80

90


20dB











Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB









7 Conclusions


Appendices



119873119887

119887=1E|119867

119895119887119908119887119906|2

we have



119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





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)



defined119867119895119894(119896120575

119894+119891

119894minus119891

119888) asymp 119867

119895119894(119896120575

119894) ≜ 119867

119895119894(119896) because |119891

119894minus119891

119888|












119894= 119879

119895= 119879 (8) becomes


sdot1

119873119904

sin [120587 (119896 minus ]) + 120587 (119891119895minus 119891

119894) 119879119873

119904]

sin [(120587119873119904) (119896 minus ]) + 120587 (119891

119895minus 119891

119894) 119879]

(9)


119892+ (119873

119904minus 1)2)119879 has been











119906times 119873

119887downlink


W (119896) = H119867

(119896) [H (119896)H119867

(119896)]minus1

(10)



119873119887gt 119873









given by

119883119887(119896) =

119873119906

sum

119906=1

119908119887119906(119896) 119904

119906(119896) (11)





119895(119896) + 119904

119895(119896) (12)


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)





119895(119896) in (7) we obtain the


119880119895(119896) =

1198731199042minus1

sum

]=minus1198731199042

] =119896

119873119906

sum

119906=1

119904119906(])

119873119887

sum

119887=1

120573119895119887(119896 ])119867

119895119887(]) 119908

119887119906(]) (15)


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)


0120575119895 It







119895(119896)119904

lowast

119906(]) = 119864



119909119910the




E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904

119873119887

sum

1198871=1

119873119887

sum

1198872=1

E 1205731198951198871

120573lowast

1198951198872

E 1198671198951198871

1199081198871119895119867lowast

1198951198872

119908lowast

1198872119895

(17)



120573lowast

1198951198872


120573119895119887=

1

119873119904

sdot 119890minus1204852120587(119891

119895minus119891119887)119905119899 sdot

sin [120587 (119891119895minus 119891

119887) 119879119873

119904]

sin [120587 (119891119895minus 119891

119887) 119879]

(18)


119895minus 119891


120587(119891119895minus 119891



119904)(sin(119873

119904119909) sin(119909))| ≪ |1 minus

119890minus120485(214119899+1)119873

119904119909




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)




1= 119887

2


1198871

and 1198911198872


E 1205731198951198871

120573lowast

1198951198872

= E 1198901204852120587(119891

1198871minus119891119888)119905119899 sdot E 119890

minus1204852120587(1198911198872minus119891119888)119905119899 (20)


E 1205731198951198871

120573lowast

1198951198872

=10038161003816100381610038161003816E 119890

minus1204852120587(119891119887minus119891119888)11990511989910038161003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816int 119901

(119891119887minus119891119888)119890minus1204852120587(119891

119887minus119891119888)119905119899119889 (119891

119887minus 119891

119888)

1003816100381610038161003816100381610038161003816

2

(21)






E 1205731198951198871

120573lowast

1198951198872

=

1 1198871= 119887

2

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

1198871

= 1198872

(22)


(119891119887minus119891119888)and is






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)


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)


119873119887

sum

119887=1

119867119895119887119908119887119906= 120575

119895119906 (25)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904[1 minus (1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870U] (26)


119887=1E|119867

119895119887119908119887119895|2






119873119887minus 119873

119906

(27)



10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

= 119890minus412058721205902

1198911199052

119899 asymp 1 minus 41205872

1205902

1198911199052

119899 (28)


119891and 119905



E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904[1 minus 4120587

2

1205902

1198911199052

119899sdot (1 minus

1

119873119887minus 119873

119906

)] (29)


119906= 119873




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)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904(1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI (31)

where119870IUI = sum119873119906

119906=1119906 =119895sum119873119887

119887=1E|119867

119895119887119908119887119906|2


minus1) sdot

sum119873119887

119887=1E|119867

119895119887119908119887119906|2


119887gt 119873

119906 we find

119870IUI asymp119873119906minus 1

119873119887minus 119873

119906

(32)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904sdot 4120587

2

1205902

1198911199052

119899sdot119873119906minus 1

119873119887minus 119873

119906

(33)





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)


1198951198871

1199081198871119906119867lowast

1198951198872

119908lowast



sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) ≜

1198611 119887

1= 119887

2

1198612 119887

1= 1198872

(35)



2= 119887

1and 119887


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)




119891and 1205902


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


E 10038161003816100381610038161003816119880119895(119896)

10038161003816100381610038161003816

2

asymp1198641199041205872

31205752sdot (120590

2

119891sdot

119873119906

119873119887minus 119873

119906

+ 1205902

119891119895) (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



119904119879)


119904119879 (this results in 119873



IIR asymp1

12 (107119899 + 05)2sdot (

119873119906

119873119906minus 1

+ 1205852119873119887minus 119873

119906

119873119906minus 1

) (40)




119906= 2





SIR =

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

(1 minus10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI

minus1

119873119906minus 1

(41)


119901(119905119899)|2 given


SIR asymp1

412058721205902

1198911199052119899

sdot119873119887minus 119873

119906

119873119906minus 1

minus119873119887minus 119873

119906+ 1

119873119906minus 1

(42)



119906ge 2




119887gt 119873



SIRmax asymp1

412058721205902

1198911199052119899

sdot119873119887

119873119906minus 1

minus119873119887+ 1

119873119906minus 1

(43)


















119891=



119891and 120590

119891119895




119892=





0

Time (ms)

Mea

n po

wer

(dB)

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101


ICI Nu = 3 Osc = 1ppb 120585 = 10


ICI Nu = 6 Osc = 1ppb 120585 = 0




2





SIR

(dB)

0

10

20

30

40

50

60

70

80

90


20dB











Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB









7 Conclusions


Appendices



119873119887

119887=1E|119867

119895119887119908119887119906|2

we have



119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










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)





119895(119896) in (7) we obtain the


119880119895(119896) =

1198731199042minus1

sum

]=minus1198731199042

] =119896

119873119906

sum

119906=1

119904119906(])

119873119887

sum

119887=1

120573119895119887(119896 ])119867

119895119887(]) 119908

119887119906(]) (15)


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)


0120575119895 It







119895(119896)119904

lowast

119906(]) = 119864



119909119910the




E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904

119873119887

sum

1198871=1

119873119887

sum

1198872=1

E 1205731198951198871

120573lowast

1198951198872

E 1198671198951198871

1199081198871119895119867lowast

1198951198872

119908lowast

1198872119895

(17)



120573lowast

1198951198872


120573119895119887=

1

119873119904

sdot 119890minus1204852120587(119891

119895minus119891119887)119905119899 sdot

sin [120587 (119891119895minus 119891

119887) 119879119873

119904]

sin [120587 (119891119895minus 119891

119887) 119879]

(18)


119895minus 119891


120587(119891119895minus 119891



119904)(sin(119873

119904119909) sin(119909))| ≪ |1 minus

119890minus120485(214119899+1)119873

119904119909




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)




1= 119887

2


1198871

and 1198911198872


E 1205731198951198871

120573lowast

1198951198872

= E 1198901204852120587(119891

1198871minus119891119888)119905119899 sdot E 119890

minus1204852120587(1198911198872minus119891119888)119905119899 (20)


E 1205731198951198871

120573lowast

1198951198872

=10038161003816100381610038161003816E 119890

minus1204852120587(119891119887minus119891119888)11990511989910038161003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816int 119901

(119891119887minus119891119888)119890minus1204852120587(119891

119887minus119891119888)119905119899119889 (119891

119887minus 119891

119888)

1003816100381610038161003816100381610038161003816

2

(21)






E 1205731198951198871

120573lowast

1198951198872

=

1 1198871= 119887

2

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

1198871

= 1198872

(22)


(119891119887minus119891119888)and is






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)


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)


119873119887

sum

119887=1

119867119895119887119908119887119906= 120575

119895119906 (25)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904[1 minus (1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870U] (26)


119887=1E|119867

119895119887119908119887119895|2






119873119887minus 119873

119906

(27)



10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

= 119890minus412058721205902

1198911199052

119899 asymp 1 minus 41205872

1205902

1198911199052

119899 (28)


119891and 119905



E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904[1 minus 4120587

2

1205902

1198911199052

119899sdot (1 minus

1

119873119887minus 119873

119906

)] (29)


119906= 119873




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)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904(1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI (31)

where119870IUI = sum119873119906

119906=1119906 =119895sum119873119887

119887=1E|119867

119895119887119908119887119906|2


minus1) sdot

sum119873119887

119887=1E|119867

119895119887119908119887119906|2


119887gt 119873

119906 we find

119870IUI asymp119873119906minus 1

119873119887minus 119873

119906

(32)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904sdot 4120587

2

1205902

1198911199052

119899sdot119873119906minus 1

119873119887minus 119873

119906

(33)





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)


1198951198871

1199081198871119906119867lowast

1198951198872

119908lowast



sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) ≜

1198611 119887

1= 119887

2

1198612 119887

1= 1198872

(35)



2= 119887

1and 119887


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)




119891and 1205902


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


E 10038161003816100381610038161003816119880119895(119896)

10038161003816100381610038161003816

2

asymp1198641199041205872

31205752sdot (120590

2

119891sdot

119873119906

119873119887minus 119873

119906

+ 1205902

119891119895) (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



119904119879)


119904119879 (this results in 119873



IIR asymp1

12 (107119899 + 05)2sdot (

119873119906

119873119906minus 1

+ 1205852119873119887minus 119873

119906

119873119906minus 1

) (40)




119906= 2





SIR =

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

(1 minus10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI

minus1

119873119906minus 1

(41)


119901(119905119899)|2 given


SIR asymp1

412058721205902

1198911199052119899

sdot119873119887minus 119873

119906

119873119906minus 1

minus119873119887minus 119873

119906+ 1

119873119906minus 1

(42)



119906ge 2




119887gt 119873



SIRmax asymp1

412058721205902

1198911199052119899

sdot119873119887

119873119906minus 1

minus119873119887+ 1

119873119906minus 1

(43)


















119891=



119891and 120590

119891119895




119892=





0

Time (ms)

Mea

n po

wer

(dB)

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101


ICI Nu = 3 Osc = 1ppb 120585 = 10


ICI Nu = 6 Osc = 1ppb 120585 = 0




2





SIR

(dB)

0

10

20

30

40

50

60

70

80

90


20dB











Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB









7 Conclusions


Appendices



119873119887

119887=1E|119867

119895119887119908119887119906|2

we have



119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














E 1205731198951198871

120573lowast

1198951198872

=

1 1198871= 119887

2

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

1198871

= 1198872

(22)


(119891119887minus119891119888)and is






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)


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)


119873119887

sum

119887=1

119867119895119887119908119887119906= 120575

119895119906 (25)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904[1 minus (1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870U] (26)


119887=1E|119867

119895119887119908119887119895|2






119873119887minus 119873

119906

(27)



10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

= 119890minus412058721205902

1198911199052

119899 asymp 1 minus 41205872

1205902

1198911199052

119899 (28)


119891and 119905



E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904[1 minus 4120587

2

1205902

1198911199052

119899sdot (1 minus

1

119873119887minus 119873

119906

)] (29)


119906= 119873




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)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

= 119864119904(1 minus

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI (31)

where119870IUI = sum119873119906

119906=1119906 =119895sum119873119887

119887=1E|119867

119895119887119908119887119906|2


minus1) sdot

sum119873119887

119887=1E|119867

119895119887119908119887119906|2


119887gt 119873

119906 we find

119870IUI asymp119873119906minus 1

119873119887minus 119873

119906

(32)


E 10038161003816100381610038161003816119904119895

10038161003816100381610038161003816

2

asymp 119864119904sdot 4120587

2

1205902

1198911199052

119899sdot119873119906minus 1

119873119887minus 119873

119906

(33)





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)


1198951198871

1199081198871119906119867lowast

1198951198872

119908lowast



sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) ≜

1198611 119887

1= 119887

2

1198612 119887

1= 1198872

(35)



2= 119887

1and 119887


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)




119891and 1205902


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


E 10038161003816100381610038161003816119880119895(119896)

10038161003816100381610038161003816

2

asymp1198641199041205872

31205752sdot (120590

2

119891sdot

119873119906

119873119887minus 119873

119906

+ 1205902

119891119895) (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



119904119879)


119904119879 (this results in 119873



IIR asymp1

12 (107119899 + 05)2sdot (

119873119906

119873119906minus 1

+ 1205852119873119887minus 119873

119906

119873119906minus 1

) (40)




119906= 2





SIR =

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

(1 minus10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI

minus1

119873119906minus 1

(41)


119901(119905119899)|2 given


SIR asymp1

412058721205902

1198911199052119899

sdot119873119887minus 119873

119906

119873119906minus 1

minus119873119887minus 119873

119906+ 1

119873119906minus 1

(42)



119906ge 2




119887gt 119873



SIRmax asymp1

412058721205902

1198911199052119899

sdot119873119887

119873119906minus 1

minus119873119887+ 1

119873119906minus 1

(43)


















119891=



119891and 120590

119891119895




119892=





0

Time (ms)

Mea

n po

wer

(dB)

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101


ICI Nu = 3 Osc = 1ppb 120585 = 10


ICI Nu = 6 Osc = 1ppb 120585 = 0




2





SIR

(dB)

0

10

20

30

40

50

60

70

80

90


20dB











Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB









7 Conclusions


Appendices



119873119887

119887=1E|119867

119895119887119908119887119906|2

we have



119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












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)


1198951198871

1199081198871119906119867lowast

1198951198872

119908lowast



sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) ≜

1198611 119887

1= 119887

2

1198612 119887

1= 1198872

(35)



2= 119887

1and 119887


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)




119891and 1205902


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


E 10038161003816100381610038161003816119880119895(119896)

10038161003816100381610038161003816

2

asymp1198641199041205872

31205752sdot (120590

2

119891sdot

119873119906

119873119887minus 119873

119906

+ 1205902

119891119895) (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



119904119879)


119904119879 (this results in 119873



IIR asymp1

12 (107119899 + 05)2sdot (

119873119906

119873119906minus 1

+ 1205852119873119887minus 119873

119906

119873119906minus 1

) (40)




119906= 2





SIR =

10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

(1 minus10038161003816100381610038161003816F

119901(119905119899)10038161003816100381610038161003816

2

)119870IUI

minus1

119873119906minus 1

(41)


119901(119905119899)|2 given


SIR asymp1

412058721205902

1198911199052119899

sdot119873119887minus 119873

119906

119873119906minus 1

minus119873119887minus 119873

119906+ 1

119873119906minus 1

(42)



119906ge 2




119887gt 119873



SIRmax asymp1

412058721205902

1198911199052119899

sdot119873119887

119873119906minus 1

minus119873119887+ 1

119873119906minus 1

(43)


















119891=



119891and 120590

119891119895




119892=





0

Time (ms)

Mea

n po

wer

(dB)

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101


ICI Nu = 3 Osc = 1ppb 120585 = 10


ICI Nu = 6 Osc = 1ppb 120585 = 0




2





SIR

(dB)

0

10

20

30

40

50

60

70

80

90


20dB











Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB









7 Conclusions


Appendices



119873119887

119887=1E|119867

119895119887119908119887119906|2

we have



119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119906ge 2




119887gt 119873



SIRmax asymp1

412058721205902

1198911199052119899

sdot119873119887

119873119906minus 1

minus119873119887+ 1

119873119906minus 1

(43)


















119891=



119891and 120590

119891119895




119892=





0

Time (ms)

Mea

n po

wer

(dB)

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101


ICI Nu = 3 Osc = 1ppb 120585 = 10


ICI Nu = 6 Osc = 1ppb 120585 = 0




2





SIR

(dB)

0

10

20

30

40

50

60

70

80

90


20dB











Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB









7 Conclusions


Appendices



119873119887

119887=1E|119867

119895119887119908119887119906|2

we have



119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

Time (ms)

Mea

n po

wer

(dB)

minus80

minus70

minus60

minus50

minus40

minus30

minus20

minus10

10minus1 100 101


ICI Nu = 3 Osc = 1ppb 120585 = 10


ICI Nu = 6 Osc = 1ppb 120585 = 0




2





SIR

(dB)

0

10

20

30

40

50

60

70

80

90


20dB











Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB









7 Conclusions


Appendices



119873119887

119887=1E|119867

119895119887119908119887119906|2

we have



119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Number of users

SIR

(dB)

52 3 4 5 6

10

15

20

25

30

35

40

45

20 dB









7 Conclusions


Appendices



119873119887

119887=1E|119867

119895119887119908119887119906|2

we have



119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119906sdot 119870


119895119887and 119908

119887119906can


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)


119887gt 119873

119906 we reach

119870 asymp1

119873119887minus 119873

119906

119870min asymp1

119873119887

(A2)


ℎand



and 1198612


]=minus1198731199042] =119896

E1205731198951198871

(119896 ])120573lowast1198951198872


119895119887(119896 ]) is given in (9) The


119891




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)



120572(119896) = (1119873119904)(sin(119873

119904Φ(119896)) sin(Φ(119896))) with Φ(119896) = 120587119896119873

119904


2

119904)(sin2(119873

119904119909)sin2(119909)) asymp 1 minus

(13)1198732

1199041199092 (1198732


2

119904for119873




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)


= 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)


119895= 119891



ge 120590119891 (B6) is given

by

1198731199042minus1

sum

]=minus1198731199042

] =119896

E 1205731198951198871

(119896 ]) 120573lowast1198951198872

(119896 ]) asymp1205872

1205902

119891119895

31205752≜ 119861

2 (B7)


1and 119861





1198731199042minus1

119899=minus11987311990421198901204852Φ(119909)119899 where



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)



119903=0119860119903exp(1204852120587(119903119879)119909) be an





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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












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


119873minus1

sum

119896=0

100381610038161003816100381610038161003816120572(120601)

119896

100381610038161003816100381610038161003816

2

=1

Δint

119879

0

|120572 (119909)|2

119889119909 (C4)



119903in two ways

First if F119891(119909)(119903) = 1119879 int119879

0


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)


119904](119903) = sum

infin








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)



120572(120601)


119873minus1

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

120572(120601)

119896

119879

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

=1

119873

119873minus1

sum

119896=0

10038161003816100381610038161003816Γ(120601)

119903

10038161003816100381610038161003816

2

(C11)


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)


Disclosure




Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article analysis of synchronization impairments...

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