a practical two-stage mmse based mimo detector for interference mitigation with non-cooperative...

RESEARCH Open Access

A practical two-stage MMSE based MIMOdetector for interference mitigation with non-cooperative interferersAnish Shah and Babak Daneshrad

Abstract

Wireless Multiple Input Multiple Output systems provide system designers with additional degrees of freedomThese can be used to increase throughput reliability or even combat spatial interference The classical MinimumMean Squared Error (MMSE) solution is the optimal linear estimator for these systems Its primary drawback is thatit requires an estimate of the channel response This is generally not an issue when interference is absentHowever in environments where interference power is stronger than the desired signal power this can becomedifficult to estimate The problem is even worse in packet-based systems which rely on training data to estimatethe channel before estimating the signal A strong interference will hinder the receiverrsquos ability to detect thepresence of the packet This makes it impossible to estimate the channel a critical component for the classicalMMSE estimator For this reason the classical solution is infeasible in real environments with stronger interferencesWe propose a two-stage system that uses practically obtainable channel state information We will show how thisapproach significantly improves packet detection and how the overall solution approaches the performance of theclassical MMSE estimator

Keywords MIMO MMSE Interference Mitigation

1 IntroductionThe unlicensed nature of the ISM band has allowed forrapid development and deployment of various wirelesstechnologies such as 80211 and bluetooth Since devicesare allowed to operate in the same band without pre-determined frequency or spatial planning they arebound to interfere with each other There have beenseveral attempts to mitigate this issue via higher layerprotocols Most of these involve some form of coopera-tive scheduling [12] Some work has been done to showthat time domain signal processing can be used to miti-gate the effects of narrowband interference [3-8] Theyhave shown in simulation how their techniques can sup-press interference on the data payload but have nottaken into account how interference affects other partsof the receiver The primary omission has been withrespect to synchronization This includes tasks such aspacket detection timing synchronization and channel

estimation Without the ability to perform these tasks itbecomes impossible to build a practical systemSome work has been done on MIMO-based interfer-

ence mitigation for cellular systems [910] Theseapproaches focus on reducing interference from neigh-boring cells or users by coordinating transmissionseither in time space or frequency They do not providea method for mitigating interference from a non-coop-erative external jammerThe iterative maximum likelihood algorithm described

in [81112] is very effective but computationally expen-sive making it difficult to implement for high dataratesystems They describe a turbo decoder approach tomitigate interference with an array of processors Turbodecoders have a computational complexity of O(l2k )where l is the block length and k is the constraint length[13] This method was proven on real systems but onlyfor low datarates It also requires the use of a turbocode in order for it to work The inability to work withan arbitrary FEC or modulation method makes theresult specific to the system that was demonstrated The

Correspondence anish2uclaeduDepartment of Electrical Engineering University of California Los AngelesLos Angeles USA

Shah and Daneshrad EURASIP Journal on Wireless Communicationsand Networking 2011 2011205httpjwcneurasipjournalscomcontent20111205

copy 2011 Shah and Daneshrad licensee Springer This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (httpcreativecommonsorglicensesby20) which permits unrestricted use distribution and reproduction inany medium provided the original work is properly cited

minimum interference method offers good performancein some scenarios but degrades when the interferencebecomes weak They address channel estimation in thepresence of interference but assume ideal packet detec-tion in the presence of this interferenceIt is our intention to demonstrate a method that can

be practically implemented on a real system As a designgoal we will ensure that our technique can operatewithout a priori knowledge of the nature or existence ofthe interference We will show how a two-stage MMSEMIMO estimator can be used to facilitate packet detec-tion as well as to provide superior bit error rate perfor-mance The first stage will be a pre-filter that operateson reduced Channel State Information (CSI) This pre-filter will suppress the interference to a level that allowsfor reliable packet detection and timing synchronizationThis will be followed by a secondary detection stagethat uses slightly more information to recover the trans-mitted data We will demonstrate how this allows thesynchronization tasks to be performed and providessimilar performance to an ideal MMSE MIMOestimatorThis paper will be organized as follows Section 2 will

describe the system model and provide derivations forthe filters that we are proposing Section 3 will discussthe simulation results Section 4 will validate some ofthe basic assumptions on a real-time hardware testbedFinally Section 5 will conclude this work

2 System modelFor our analysis we will use a typical MIMO systemwith multiple transmit and receive antennas (see Figure1) A pre-filter is used to improve synchronization per-formance We will examine two well-known algorithmsthat can be used as a pre-filter in addition to our pro-posed algorithm The filtered signal will be used by thesynchronization algorithm to determine whether a

packet is present and to estimate the symbol boundary(timing synchronization) This signal will then passthrough a secondary filter that will estimate the origin-ally transmitted signal The data payload of the packet isa simple uncoded QAM signal This was chosen so wemay directly evaluate the performance improvement ofour algorithm and avoid potential non-linear effectsfrom forward error correction schemes We used a stan-dard 80211a header [14] with well-known techniquesfor packet detection and timing synchronization from[15-17] It is our intention to show improvements inperformance as opposed to showing absolute perfor-mance For that reason we have chosen to use well-known training sequences as well as synchronizationalgorithms The performance improvements demon-strated in this work should be directly applicable to allpacket-based systems that require on packet detectionand timing synchronizationWe begin by defining some notation explicitly We

will use the superscript () to denote the complex conju-gate transpose (Hermitian) of a vector or a matrix Low-ercase boldface symbols (y) will be used to denotevectors and uppercase boldface (W) will be used todenote matrices The hat (x) will denote estimates ofsignals while a tilde (x) will be used to denote residualerror signals The trace operator for a matrix will bedenoted as Tr()First we will examine Rayleigh flat fading channels

the simplest class of channels These channels are mod-eled as a single impulse chosen from a Rayleigh distribu-tion A new channel will be chosen at random for eachpacket but remain constant throughout the duration ofthat packet We will discuss the ideal Minimum MeanSquared Error (MMSE) solution and show why it isimpractical in high interference scenarios We will thenreview the Sample Matrix Inverse (SMI) [18] as well asMaximal Signal to Interference plus Noise Ratio

Figure 1 System model


Page 2 of 14

(MSINR) [19] algorithms These are both well suited foruse as a pre-filter since neither require first-order infor-mation about the channel Each of these algorithms willuse a standard MMSE detector as the secondary filter todemodulate the data We will then discuss our proposedtwo-stage solution with its pre-filter and secondary filterWe will show how the combination of these filters isequivalent to the ideal linear MMSE solution Finallywe will extend each of these methods to cope with Ray-leigh frequency selective channels

21 Rayleigh flat fading channelsThe time domain received signal y(t) (1) is the linearcombination of the received signal of interest x(t) con-volved with its channel Hs additive white Gaussiannoise (AWGN) n(t) and the interference signal g(t)convolved with its channel Hi Since the channel is asingle impulse the convolution of the channel with thesignal is the same as multiplicationIn this work we will focus on linear estimators of the

form x = Wy for their simplicity and practicality ofimplementation The estimation error is given byy(t) = Hsx(t) + Hiγ (t) + n(t)

y(t) = Hsx(t) + Hiγ (t) + n(t) (1)

minW

E[xlowastx] = minW

E[Tr(xxlowast)] (2)

The linear estimator (W) that satisfies (2) will mini-mize the mean-squared error (MSE) of the estimator x This is equivalent to minimizing the trace of xxlowast Forthe ease of notation we define the covariance for thesignal of interest interference and additive white Gaus-sian noise as E[xx ] = Rx E[gg] = Rg and E[nn ] =Rn respectively The solution to (2) is the classicalMMSE solution given by Equation (3) [20]

WMMSE = RxyRminus1y

= RxHlowasts (HsRxHlowast

s + HiRγ Hlowasti + Rn)minus1

(3)

The classical MMSE estimator is very powerful butrequires first-order channel state information (CSI) forthe signal of interest (Hs ) Traditional packet based sys-tems transmit training data which the receiver can useto estimate (Hs ) This is fine when there is no interfer-ence present allowing packet detection and timing syn-chronization algorithms to work as expected It mayeven work when the interference is cooperative and canbe canceled using a cooperative scheme such as Walshcodes in a CDMA system If the interference is non-cooperative and stronger than the desired signal it maybe impossible to detect the packet This will cause thecommunications system to fail When the packet cannotbe detected and the symbol boundary cannot be

determined the channel cannot be estimated Thesepractical limitations render the classical approach infea-sible in many real scenariosWe propose a pre-filter based solely on second-order

statistics (HsRxHlowasts HiRγ Hlowast

i Rn) These statistics caneasily be estimated by averaging outer products ofreceived signals at different moments in time Interfer-ence mitigation algorithms that can operate with onlythese covariance estimates offer greater exibility forcommunications systems dealing with non-cooperativeinterferences211 Covariance estimatesAs long as the receiver can make reasonably accuratedecisions about the presence of the desired signal it cancalculate all of the necessary covariance matrices Figure2 shows the times at which two different covariancemeasurements can be made Time t1 indicates a time atwhich the packet is not being transmitted and time t2indicates the time during which the packet is beingtransmitted Let R1 (4) be the covariance measured dur-ing time t1 and R2 (5) be the covariance measured dur-ing time t2 The methods described for pre-filteringbelow will require only these quantities We will validatethis assumption with an example from a real-time hard-ware testbed showing how these determinations can bemade in Section 4

HiRγ Hlowasti + Rn = R1 (4)

HsRxHlowasts + HiRγ Hlowast

i + Rn = R2 (5)

Since the signal components are independent the cov-ariance of their sum is equal to the sum of their covar-iances This allows us to compute the covariance of thedesired signal as the difference between the R2 and R1

measurements (6)

HsRxHlowasts = R2 minus R1 (6)

We will describe a few alternatives for the pre-filter inthe following sections These will be important for boot-strapping the system using the available measurements(R2 and R1 )212 Sample matrix inverseAn example of an algorithm that relies only on second-order statistics is the Sample Matrix Inverse (SMI) [18]which has been shown to be very effective for

Figure 2 Timing for covariance estimation


Page 3 of 14

interference mitigation [21] This algorithm uses theinverse of the covariance of the interference + AWGNas its pre-filtering matrix (7)

WSMI = (HiRγ Hlowasti + Rn)minus1 (7)

The advantage of this algorithm is that the pre-filteronly needs knowledge of the covariance of the undesiredsignal components This can be particularly useful dur-ing the initialization of the communications system If astrong interference is present it may not be possible todetermine when the signal of interest is being trans-mitted This will make it impossible to take an accurateR2 measurement Instead the receiver can take severalR1 measurements and use the SMI as the pre-filter toimprove synchronization performanceSince the receiver will not know when the desired sig-

nal is present it may still take improper measurementsIt is therefore necessary to take consecutive measure-ments and apply the SMI until the desired signal can bedetected by the synchronization algorithm This equatesto a series of Bernoulli trials We know the likelihood ofx consecutive failures decays exponentially with x Thenumber of trials required is simply a function of thetime the desired signal occupies the band This caneasily be adjusted by the system designer to meet therequirements of the communication system In Section3 we will show how effective this algorithm is atimproving synchronization performance in the presenceof very strong interferences SMI can be used to boot-strap the system Once a good R1 measurement hasbeen taken the system will be able to determinewhether the desired signal is present or not It may notbe able to estimate the symbol boundary accurately butthis information will make it possible to take an R2

measurement and improve the pre-filter213 Maximal signal to interference and noise ratioThe Maximal Signal to Interference and Noise Ratio(MSINR) criterion seeks to maximize the signal powerwith respect to the interference + noise power This cri-terion is formulated by optimizing the power of each ofthe components in the received signal (8) The linearestimator is still computed as x = Wy resulting in itssecond-order statistics being described by (9)

E[yylowast] = HsRxHlowasts + HiRγ Hlowast

i + Rn (8)

E[xxlowast] = WHsRxHlowasts Wlowast + W(HiRγ Hlowast

i + Rn)Wlowast (9)

The MSINR criterion is given by (10) The pre-filterthat satisfies this criterion is the solution to the general-ized eigen-value problem and is given by (11) [19]

maxW

=Tr(WHsRxHlowast

s Wlowast)Tr(W(HiRγ Hlowast

i + Rn)Wlowast)(10)

WMSINR = HsRxHlowasts (HiRγ Hlowast

i + Rn)minus1 (11)

Instead of directly estimating the transmitted signalthis criterion will try to maximize its power relative tothe noise and interference Once again the demodula-tion can be done with a MMSE based decoder afterpacket detection timing synchronization and channelestimation have been completed This algorithmrequires the covariance of the desired signal as well asthe information used in the SMI Once the pre-filter isperforming well enough for synchronization to detectpackets the R2 measurement can be taken and the SMIpre-filter can be replaced with the MSINR pre-filter214 Two-stage MMSEConsider (3) for the MMSE Linear estimator The onlycomponent that is not a second-order statistic is Rx Hs

If we left multiply the MMSE estimator with thechannel matrix Hs we create an equation that is com-prised entirely of second-order statistics (12)

WS1 = HsWMMSE

= HsRxHlowasts (HsRxHlowast

s + HiRγ Hlowasti + Rn)minus1 (12)

This operation may introduce spatial interference bymixing the signal components from independent spatialstreams However if there is only one spatial streamthe result will be a spreading of the desired signal Thisis enough to allow many standard detection algorithmsto detect and synchronize with an incoming packetThis modified version of the MMSE estimator leads usto our two-stage approach to interference mitigationIn the first stage the pre-filter will be used to suppress

the interference as much as possible This suppressionmust be enough to facilitate packet detection timingsynchronization and channel estimation If these taskscan be performed reliably the estimated channel can beused in a secondary filter We use this to define a two-stage approach that achieves identical performance asthe classical linear MMSE estimator

WS2 = (Hlowasts Hs)minus1Hlowast

s (13)

In (12) we defined the pre-filter (WS1 ) using onlysecond-order statistics The second stage is a simplezero-forcing MIMO decoder (13) We are able to usethe first-order statistic Hs at this point because we willhave a channel estimate based on the training data fromthe packet header We will show how this estimate canbe obtained in (15)-(19)


Page 4 of 14

x = WS2WS1

= (Hlowasts Hs)minus1Hlowast

s HsWMMSEy

= WMMSEy

(14)

The zero-forcing decoder is used because Hs may notbe a square matrix If the matrix is not square it willnot be directly invertible This will happen anytimethere are fewer transmit streams than receive antennasEquation (14) shows how the application of these twofilters in series results in the original MMSE linear esti-mator Equations (13) and (14) together show how theMMSE estimator can be broken down into a two-stageprocess when ideal CSI is availableIn a real system however the channel matrix will

need to be estimated from the output of the pre-filter(WS1 ) The measured channel will be modified fromthe actual channel by the pre-filter The output of thepre-filter is given by (15)

xS1 = WS1y = HsWMMSEy (15)

22 Channel estimationMIMO training matrices (16) can be used to estimatethe combined effect of the channel and pre-filter fromxS1 The columns of the matrix correspond to spatialstreams and the rows correspond to symbols A subsetof this matrix can be used for systems that are smallerthan 4 times 4 This matrix pattern can also be extended toaccommodate systems with more antennas

P =

⎡⎢⎢⎣

a minusa a aa a minusa aa a a minusa

minusa a a a

⎤⎥⎥⎦ (16)

In a typical MIMO system the channel measurementis computed from the received training symbols Con-sider P = [p1 p2 p3 p4 ] where each pi corresponds to atransmission vector Each element in pi refers to thesymbol transmitted from that antenna for this vectorThe receiver can measure the received values for eachvector and construct a matrix with the estimates Thismeasurement is Z = Hs P In order to estimate thechannel Z is right multipliedby either the Hermitian ortranspose of the training matrix When this trainingmatrix is real-valued (a = 1) it does not matter which isused We will use the Hermitian since it will work forboth real and complex-valued training matrices Theresult of the right multiplication is given by (17)

PPlowast =

⎡⎢⎢⎣

aalowast 0 0 00 aalowast 0 00 0 aalowast 00 0 0 aalowast

⎤⎥⎥⎦ (17)

The ZS1 that will be estimated from xS1 is shown in(18) In order to estimate the original channel from thismodified version we use the inverse of the pre-filter(19)

ZS1 = WS1HsP (18)

Hs = (1α)(WS1)minus1ZS1Plowast (19)

23 Rayleigh frequency selective channelsEquation (3) implicitly assumes that the channel is non-dispersive This means that each entry in the channelmatrix is a constant complex value In order to modeldispersive channels we must extend this model to han-dle multipath

Hs = Hs0δ(t) + Hs1δ(t minus 1) + Hs2δ(t minus 2) + middot middot middot (20)

Hi = Hi0δ(t) + Hi1δ(t minus 1) + Hi2δ(t minus 2) + middot middot middot (21)

This can be done by modeling the channel as a seriesof complex impulses where the channel matrix for eachimpulse is composed of constant complex values (20)-(21) The length of the channel is determined by thedelay spread

yM =

⎡⎢⎢⎢⎣

y(t)y(t minus 1)

y(t minus M minus 1)

⎤⎥⎥⎥⎦ xM =

⎡⎢⎢⎢⎣

x(t)x(t minus 1)

x(t minus M minus 1)

⎤⎥⎥⎥⎦ (22)

γM =

⎡⎢⎢⎢⎣

γ (t)γ (t minus 1)

γ (t minus M minus 1)

⎤⎥⎥⎥⎦ nM =

⎡⎢⎢⎢⎣

n(t)n(t minus 1)

n(t minus M minus 1)

⎤⎥⎥⎥⎦(23)

HsMM =

⎡⎣

Hs0 Hs1 Hs2

0 Hs0 Hs1

0 0 Hs0

⎤⎦

HiMM =

⎡⎣

Hi0 Hi1 Hi20 Hi0 Hi10 0 Hi0

⎤⎦

(24)

In this scenario the MMSE estimator needs to bemodified to properly estimate the transmitted signalEquations (22) and (23) define new compound signalsthat are composed of M delayed versions of the originalsignals where M is the delay spread of the channelCorrespondingly we define new compound channelmatrices (24) composed of the channel matrices foreach impulse in the original dispersive channel For this


Page 5 of 14

example we will use M = 3 The entities defined in(22)-(24) are related by (25)

yM(t) = HsMMx(t) + HiMMγ M(t) + n(t) (25)

With these quantities defined we can re-examine thesolution to the MMSE criterion Since we are now try-ing to estimate x(t) from yM (t) the W that satisfies theMMSE criterion will be given by (26) We must alsodefine the covariance (27) of the signal components in(22) and (23) Assuming that the signals will be indepen-dent and identically distributed these covariancematrices will block diagonal as shown in (28)

WMMSE = RxyMRyMminus1 (26)

E[xM(t)xlowast

M(t)]

= RxM E[γM(t)γ lowast

M(t)]

= RγM

E[nM(t)nlowast

M(t)]

= RnM

(27)

RxM = diag (Rx Rx ) RγM = diag(Rγ Rγ

)

RnM = diag (Rn Rn )(28)

The cross-correlation of the desired x(t) with the com-pound yM (t) is given by (29) The covariance of yM (t)is straightforward and shown in (30) The resulting esti-mator is given by (31)

RxyM =[Rx 0 0

]HsMM

lowast (29)

RyM =

(HsMMRxMHsMMlowast+

HiMMRγMHiMMlowast + RnM)minus1

(30)

xMMSE(t) =[Rx 0 0

]HsMM

lowast

(HsMMRxMHsMMlowast+

HiMMRγM HiMMlowast + RnM )minus1yM(t)

(31)

Once again the MMSE estimator is very powerful butrequires first-order CSI (HsMM) for the signal of interestAs shown in the previous sections (4) and (5) we canestimate the second-order statistics by averaging theouter products of the compound received signals (22)-(23) This brings us back to the notion of building pre-filters using only second-order statistics We will nowconsider extensions of the previous algorithms for themore complex frequency selective channelThe SMI and MSINR approaches are easily extended

to work in this environment The pre-filters for theseapproaches are given by (32) and (33) respectively

WSMI = (HlowastiMM

RγMHiMM + RnM)minus1 (32)

WMSINR = HlowastsMM

RxMHsMM(HlowastiMM

RγM HiMM + RnM)minus1 (33)

Once again we examine the MMSE linear estimator(31) Similar to the flat fading scenario the only compo-nent that is not a second-order statistic is RxyM We candefine an estimator (34) that is composed only of sec-ond-order statistics

WS1 = HlowastsMM

RxMHsMM

(HlowastsMM

RxMHsMM+

HlowastiMM

RγMHiMM + RnM )minus1

(34)

WS2 =[Rx 0 0

]Hlowast

sMM(Hlowast

sMMRxMHsMM)minus1 (35)

WS1 will function as a pre-filter similar to pre-filterfrom the flat fading scenario (12) It will facilitate packetdetection and synchronization The second stage isdefined in (35) Equation (36) shows how the applicationof these two filters results in the original MMSE linearestimator This derivation is similar to the flat fadingscenario

x(t) = WS1WS2yM(t)

=[Rx 0 0

]Hlowast

sMM

(HlowastsMM

RxMHsMM )minus1HlowastsMM

RxMMHsMM

(HlowastsMM

RxMHsMM +Hlowast

iMMRγMHiMM + RnM )minus1

= WMMSEyM(t)

(36)

We have shown how this MMSE estimator can bebroken down into a two-stage process when ideal chan-nel state information is available In a real system thechannel matrix will need to be estimated from the out-put of the pre-filter (WS1 ) The measured channel willbe a modified version of the actual channel the signalwent through

xS1M = WS1y = HsMM WMMSEy (37)

The output of the pre-filter is given in (37) TheZS1MM that will be measured from xS1M is shown in(38) The dispersive channel can be estimated using M-sequences [2223] These sequences have strong auto-correlations at 0-offset and very low correlations for allother offsets In order to estimate the original channelfrom this modified version we use the inverse of thepre-filter (39)

ZS1MM = WS1HsMMP (38)

HsMM = (1α)(WS1)minus1ZS1MMPlowast (39)


Page 6 of 14

3 Simulation resultsThe algorithms described in Section 2 were simulated inMATLAB using a MIMO systems with 4 receive anten-nas This included the ideal MMSE solution SMIMSINR and the proposed two-stage MMSE solutionThe non-cooperative interference source was a singleantenna transmission convolved with its own channelThe interference signal was a white Gaussian noise sig-nal which is essentially a wideband signal The desiredsignal was modeled to have 2 or 3 independent spatialstreams The transmission started with a knownsequence to be used for packet detection and timingsynchronization We used the standard 80211a header[14] with well-known techniques for packet detectionand timing synchronization from [15-17] This was fol-lowed by training data to be used for channel estimationby the receiver The body of the packet was an uncodedbit stream modulated onto a QPSK constellation Inde-pendent Rayleigh fading channels were generated ran-domly for each trial for both the desired and undesiredsignals These channels remained constant throughoutthe duration of each trial

31 Rayleigh flat fading channelsRayleigh flat fading channels are the easiest channels tocompensate They consist of a single impulse and allowus to model the channel as a simple gain and phaseadjustment of the transmitted signal We begin our ana-lysis by considering the original goal of our approachwhich is to ensure packet synchronization can be per-formed It is necessary to examine this performancebefore we can investigate the bit error rate (BER) With-out packet detection the communications system willfail For our system to declare successful synchroniza-tion the receiver must correctly detect the presence ofthe packet as well as accurately determine the symbolboundary The symbol boundary is used to determinewhen the packet started and when each symbol beginsand ends Without this information the receiver isunable to estimate the channel since it does not knowwhen the training data begins and ends The estimatedchannel is used by the receiver to estimate the trans-mitted signal in the secondary filterTable 1 provides details on the legend entries for the

synchronization failure curves as well as the BER curvesthat will follow For the ideal MMSE solution we used

(3) in the pre-filter There is no need for a secondary fil-ter since the pre-filter has already provided the bestpossible estimate of the transmitted signal When testingSMI and MSINR an ideal MMSE estimator was used asthe secondary filter Since the signal had already beenperturbed by a pre-filter the MMSE solution used theperturbed version of the channel WS1 HsFigure 3 shows the synchronization performance at

-20 dB SIR for a two-antenna transmission scheme Asexpected the synchronization algorithm completely failsin the absence of pre-filtering All of the methodsdescribed for pre-filtering offer significant improve-ments It is clear that without a pre-filter the systemcannot survive in the presence of strong externalinterferencesThe pre-filter designed to work with our two-stage

approach provides almost the same performance as theSMI pre-filter They both outperform the MSINR andtheir relative performance gap becomes much smaller asthe SNR becomes larger While MSINR does not pro-vide the same level of synchronization performance asSMI we will see that it does in fact provide far superiorBER performance This is because the SMI algorithmonly has knowledge of the interference It has no infor-mation about the channel of the desired signal Thiscreates very deep nulls for the interference but cancause degradation of the desired signal As the channelsand transmission schemes become more complex theperformance of SMI will degrade We will see this occurin the BER performance for the flat fading channel aswell as the frequency selective channel Figure 4 showsthe synchronization performance of these algorithms asa function of SIR at 10 dB SNR We can see that SMI isthe most effective when the interference is strong Asthe interference becomes weaker and less of an issuethe harshness of the null becomes detrimental to theperformance of the system This can be seen by thecrossover of the MMSE2 and SMI curves at 2 dB SIRThe bit error rate for these algorithms is given in Fig-

ure 5 As described earlier the second-stage filter forestimating the transmitted bits is calculated from thechannel that was estimated during synchronization As abound we show the BER performance of the systemwith an ideal version of the classical MMSE solutionWhile this solution is impractical due to the lack of achannel estimate for the pre-filter it represents the best

Table 1 Legend entry descriptions

No pre-filt Pre-filtering is omitted

IM MMSE The ideal MMSE solution (3) is used in the pre-filter no secondary filter is required

MMSE2 Equation (12) is used in the pre-filter and Equation (13) is used for MIMO detection with ideal CSI

MSINR Equation (11) is used in the pre-filter

SMI Equation (7) is used in the pre-filter


Page 7 of 14

Figure 3 Synchronization failure rate (-20 dB SIR)

Figure 4 Synchronization failure versus SIR


Page 8 of 14

performance we can expect of a linear estimation sys-tem The performance of our two-stage algorithmapproaches that of the infeasible MMSE solution Theloss in performance is less than 05 dB We also notethat the two-stage solution consistently outperformsSMI and MSINR in these two scenarios The perfor-mance gap between the two-stage MMSE solution andMSINR grows as the complexity of the problem growsThe improvement is roughly 2 dB when 3 spatialstreams are transmitted We will see how this gapbecomes even larger with frequency selective channelsFigure 6 shows the performance of the system as a

function of SIR for both the 2 and 3 TX antenna casesWe can see the gains for the two-stage approach areconsistent across the entire SIR range We also noticethat the SMI and MSINR approaches do not fare wellwhen the interference gets weaker In fact the perfor-mance is worse with these pre-filters than it is with nopre-filter at all This is an issue that we had first notedwith synchronization performance for SMI in Figure 4

This crossover represents an undesirable loss in perfor-mance The IM MMSE and two-stage solution bothtrack the performance improvement of the unmodifiedsystem once they approach that curve This represents agraceful transition as the interference becomes weakerand eventually ceases to impact the performance of thesystem This is evident for both the 2 and 3 TX antennacases

32 Rayleigh frequency selective channelsNext we shift our attention to frequency selective chan-nels Again we begin by examining the synchronizationperformance to ensure that the pre-filtering operation isproviding a significant improvement Figure 7 shows thesynchronization performance at -5 dB SIR for a two-antenna transmission scheme The legend entries arestill defined by Table 1 from the previous section Theequations are replaced with those from the frequencyselective channel work in Section 23 For the idealMMSE solution Equation (3) is replaced by (26) TheSMI and MSINR pre-filters (7) and (11) are replaced by(32) and (33) respectively Finally the two-stage MMSEfilters (12) and (13) are replaced by (34) and (35) respec-tively The criteria for successful synchronization arealso the same as they were in the previous sectionOnce again we see how drastic the improvement in

synchronization performance becomes with use of ourpre-filter (Figure 7) Without the pre-filtering operationsynchronization fails completely The two-stage MMSEpre-filtering operation improves that success rate toover 99 when the SNR is greater than 10 dB This is avery significant improvement that contributes to the sta-bility and throughput of the communications systemThe alternatives available for the pre-filter are inferiorto the proposed two-stage solution The SMI solutionalso fails to outperform the two-stage solution in thiscomplex channelThe bit error rate for these algorithms with SIR = -5

db is shown in Figure 8 We can see the improvementin performance from the two-stage approach The per-formance of the system without a pre-filter is not goodenough to sustain reliable communications The two-stage approach provides performance within 05 dB ofthe bound given by the ideal MMSE solution It also sig-nificantly outperforms MSINR which is the nearestcompetitor There is a 2 dB improvement when trans-mitting with two-spatial streams and even greaterimprovement for 3 spatial streamsFigure 9 shows the performance as a function of the

SIR Just as we saw in Figure 6 the two-stage solutionconsistently outperforms the SMI and MSINR solutionsThe IM MMSE and two-stage solution also improve asthe interference gets weaker and ceases to dominate theperformance of the system

(a) 2 Spatial Streams

(b) 3 Spatial Streams

Figure 5 BER at - 20 dB SIR


Page 9 of 14

Figure 6 BER at 10 dB SNR



0 of 14

4 Hardware implementationThe SMI multi-antenna interference mitigation schemewas implemented on a hardware testbed for verificationThe purpose of this was to prove that this type of algo-rithm can work on real hardware in a real environmentMost importantly it showed that the method describedfor obtaining the R1 and R2 measurements in Section 2could be realized in a real systemWe chose the SMI algorithm since it required the few-

est calculations to implement The limited hardwareresources available on the FPGA prevented us fromimplementing one of the more complex pre-filters Inaddition to the limited resources we were unable tochange the existing MMSE MIMO OFDM estimatorThis meant we could not implement the stage-2 filterrequired for our two-stage solution The pre-filter wasadded to the existing MIMO OFDM cognitive radiotestbed [24] (See Figure 10) The transmitter and recei-ver on this testbed are completely contained in anFPGA The interference mitigation module was added

before the receiver so it could pre-filter the received sig-nal and improve the SINR before the existing receiverattempted to decode the packet (See Figure 11) Thereceived OFDM signal was demodulated by a standardMMSE MIMO estimator in the pre-existing receiverThis was a key advantage of the SMI and MSINR algo-rithms discussed in the previous sections

41 System overviewThe estimation of the covariance is a straightforwardaveraging of the outer product of the incoming signalThe only concern when estimating the covariance isthat the signal being received should contain only theinterference and noise This is required in order to com-pute an accurate R1 measurement A controller statemachine was designed to enable estimation of the covar-iance during time periods which are unlikely to containthe signal of interest The details of this state machineare omittedAn onboard microprocessor was used to calculate the

spatial filtering matrix W based on the input covariancematrix R This was computed on a microprocessor withdouble precision floating point arithmetic using well-known matrix inversion algorithms (Cholesky) A simpleprotocol was developed for passing matrices betweenthe host and FPGA to prevent data corruption Theinterval between passing R to the host and receiving aW back was 1 msFigure 12 shows a logic analyzer trace of the execution

of this state machine and its impact on the performanceof a packet based communications system The signalpower at the input and output of the filter is shown inthe first two traces These show when the interference ispresent at the input as well as when it is being success-fully mitigated at the output Near the bottom of the fig-ure we can see when good packets are being receivedAt the beginning of this trace there was no interferencepresent and every packet was successfully receivedAbout a quarter of the way into the trace the filter

input changes This is when the interference signalbecame active Unsurprisingly it prevented the systemfrom receiving packets The time required for the sys-tem to recover and receive packets is a function of thesystem design parameters In this case the most signifi-cant source of delay was the matrix inversion requiredto compute the pre-filter This computation took a con-siderable amount of time and dictated the rate at whichwe could update our pre-filterThe absolute bottom row on the trace indicates when

the controller is computing an R1 covariance estimateWhen the system is behaving well and decoding everypacket the estimates are taken immediately after thepacket ends The transmitter guaranteed this timewould be silent which makes it the optimal time to



Figure 8 BER at -5 dB SIR in frequency selective channels


Page 11 of 14

estimate the interference Note how these estimatesbecome less frequent when the interference turns onThe controller waits for a stimulus to begin estima-

tion If it does not receive the stimulus for a pre-defined time it assumes interference is preventing thereceiver from decoding packets It then switches to atimeout mode where it measures the covariance on afixed interval The gap between the last good estimateand the next attempt is a function of this timeout per-iod In this example the controller made a couple offailed attempts at covariance estimation while it was intimeout

Once it makes a good estimate it is able calculate thepre-filter The yellow W indicates the time at which thepre-filter is updated with good coefficients Theimprovement in performance is immediately visible atthe output of the filter In the second to last trace thegood packet indicators show successful reception ofpackets coinciding with the updated pre-filterIn this example the system recovered from the onset

of interference in 3 ms This time can be shortened byreducing the timeout period for the controller Anotherway to reduce the recovery time is to use a faster pro-cessor to compute the pre-filter from the covarianceestimateThis example validates the assumption that the recei-

ver can reasonably make an R1 measurement even inthe presence of strong interference

5 ConclusionWe have demonstrated a practically realizable two-stageMMSE based approach to interference mitigation andMIMO detection The advantage of our algorithm isthat it enables synchronization tasks such as packetdetection timing synchronization and channel estima-tion to be performed in the absence of complete chan-nel state information The pre-filtering operation usesinformation that can be easily estimated in the absence


Figure 10 Real-time MIMO OFDM testbed


Page 12 of 14

of training sequences The second-stage filter uses infor-mation from the pre-filter as well as channel estimatescomputed during synchronizationWe have shown how the synchronization performance

of this algorithm is superior to the classical approachwith no pre-filter We have also shown that the BERperformance is within a 05 dB of the ideal (yet infeasi-ble) classical MMSE solution We have demonstratedsignificant improvement over the existing algorithms forcomplex transmission schemes and channels We havealso demonstrated how the necessary statistics can beestimated and how the system can be built to achieve

good performance This has not only been done for Ray-leigh flat fading channels but for frequency selectivechannels as wellOur approach is significant because it lends itself to

practically realizable systems The use of second-orderstatistics in the pre-filter is something that can easily beimplemented on real-time hardware We have alsoshown how a system can be designed to make thenecessary measurements in the presence of a stronginterference This was demonstrated on a real-timehardware testbed with a non-cooperative interferencesource

Figure 11 Interference mitigation subsystem insertion for RX chain

Figure 12 Interference mitigation hardware execution


Page 13 of 14

51 Competing interestsThe authors declare that they have no competinginterests

Received 4 February 2011 Accepted 19 December 2011Published 19 December 2011

References1 J Peha Wireless communications and coexistence for smart environments

Pers Commun IEEE [see also IEEE Wireless Communications] 7(5) 66ndash68(2000)

2 C Chiasserini R Rao Coexistence mechanisms for interference mitigationbetween ieee 80211 wlans and bluetooth INFOCOM 2002 in Proceedingsof IEEE Twenty-First Annual Joint Conference of the IEEE Computer andCommunications Societies 2 590ndash598 (2002)

3 Z Wu C Nassar Narrowband interference rejection in ofdm via carrierinterferometry spreading codes IEEE Trans Wirel Commun 4(4) 1491ndash1505(2005)

4 B Le Floch M Alard C Berrou Coded orthogonal frequency divisionmultiplex [tv broadcasting] Proc IEEE 83(6) 982ndash996 (1995) doi1011095387096

5 D Wiegandt C Nassar High-throughput high-performance ofdm viapseudo-orthogonal carrier interferometry coding in Proceedings of 12th IEEEInternational Symposium on Personal Indoor and Mobile RadioCommunications 2 G-98ndashG-102 (2001)

6 D Wiegandt C Nassar High-throughput high-performance ofdm viapseudo-orthogonal carrier interferometry type 2 in Proceedings of the 5thInternational Symposium on Wireless Personal Multimedia Communications 2729ndash733 (2002)

7 A Coulson Narrowband interference in pilot symbol assisted OFDMsystems IEEE Trans Wirel Commun 3(6) 2277ndash2287 (2004) doi101109TWC2004837471

8 D Bliss Robust mimo wireless communication in the presence ofinterference using ad hoc antenna arrays Military CommunicationsConference 2003 MILCOM 2003 IEEE 2 1382ndash1385 (2003)

9 H Zhang H Dai Cochannel interference mitigation and cooperativeprocessing in downlink multicell multiuser mimo networks EURASIP J WirelCommun Netw 222ndash235 (2004) [Online] httpdxdoiorg101155S1687147204406148

10 J Andrews Interference cancellation for cellular systems a contemporaryoverview IEEE Wirel Commun 12(2) 19ndash29 (2005) doi101109MWC20051421925

11 D Bliss P Wu A Chan Multichannel multiuser detection of space-timeturbo codes experimental performance results in Conference Record of theThirty-Sixth Asilomar Conference on Signals Systems and Computers 21343ndash1348 (2002)

12 D Bliss A Chan N Chang Mimo wireless communication channelphenomenology IEEE Trans Antennas Propag 52(8) 2073ndash2082 (2004)doi101109TAP2004832363

13 J Costello A Banerjee C He P Massey A comparison of low complexityturbo-like codes in Conference Record of the Thirty-Sixth Asilomar Conferenceon Signals Systems and Computers 1 16ndash20 (2002)

14 IEEE Std 80211a-1999 Part 11 Wireless LAN Medium Access Control (MAC)and Physical Layer (PHY) Specifications IEEE (1999)

15 T Schmidl D Cox Robust frequency and timing synchronization for OFDMIEEE Trans Commun 45(12) 1613ndash1621 (1997) doi10110926650240

16 S Nandula K Giridhar Robust timing synchronization for OFDM basedwireless lan system TENCON 2003 Conference on ConvergentTechnologies for Asia-Pacific Region 4 1558ndash1561 (2003)

17 J Singh I Tolochko K Wang M Faulkner Timing synchronization for 80211aWLAN under multipath channels Proc ATNAC Melbourne Australia (2003)

18 A Steinhardt N Pulsone Subband stap processing the fifth generation inProceedings of the 2000 IEEE Sensor Array and Multichannel Signal ProcessingWorkshop (2000)

19 RJ Mallioux Phased Array Handbook (Artech House Norwood MA 1994)20 A Sayed Fundamentals of Adaptive Filtering (Wiley Hoboken NJ 2003)21 Anish Shah B Daneshrad Weijun Zhu Narrowband jammer resistance for

MIMO OFDM MILCOM San Diego CA (2008)

22 W Davies System Identification for Self-Adaptive Control (New York Wiley-Interscience 1970)

23 GT Buracas GM Boynton Efficient design of event-related FMRI experimentsusing m-sequences NeuroImage 163 (Part 1) 801ndash813 (2002) [Online]httpwwwsciencedirectcomsciencearticleB6WNP-46HDMPV-T2289fe67ae0ab5106dc08562a0c18e6a8

24 W Zhu B Daneshrad J Bhatia J Chen H-S Kim K Mohammed O Nasr SSasi A Shah M Tsai A real time mimo ofdm testbed for cognitive radio ampnetworking research in Proceedings of the 1st International Workshop onWireless Network Testbeds Experimental Evaluation amp Characterization serWiNTECH lsquo06 (ACM New York NY USA 2006) pp 115ndash116 [Online] httpdoiacmorg10114511609871161018

doi1011861687-1499-2011-205Cite this article as Shah and Daneshrad A practical two-stage MMSEbased MIMO detector for interference mitigation with non-cooperativeinterferers EURASIP Journal on Wireless Communicationsand Networking 2011 2011205

Submit your manuscript to a journal and benefi t from

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Submit your next manuscript at 7 springeropencom


Page 14 of 14

Abstract

1 Introduction

2 System model

21 Rayleigh flat fading channels

211 Covariance estimates

212 Sample matrix inverse

213 Maximal signal to interference and noise ratio

214 Two-stage MMSE

22 Channel estimation

23 Rayleigh frequency selective channels

3 Simulation results



4 Hardware implementation

41 System overview

5 Conclusion

51 Competing interests

References

minimum interference method offers good performancein some scenarios but degrades when the interferencebecomes weak They address channel estimation in thepresence of interference but assume ideal packet detec-tion in the presence of this interferenceIt is our intention to demonstrate a method that can

be practically implemented on a real system As a designgoal we will ensure that our technique can operatewithout a priori knowledge of the nature or existence ofthe interference We will show how a two-stage MMSEMIMO estimator can be used to facilitate packet detec-tion as well as to provide superior bit error rate perfor-mance The first stage will be a pre-filter that operateson reduced Channel State Information (CSI) This pre-filter will suppress the interference to a level that allowsfor reliable packet detection and timing synchronizationThis will be followed by a secondary detection stagethat uses slightly more information to recover the trans-mitted data We will demonstrate how this allows thesynchronization tasks to be performed and providessimilar performance to an ideal MMSE MIMOestimatorThis paper will be organized as follows Section 2 will

describe the system model and provide derivations forthe filters that we are proposing Section 3 will discussthe simulation results Section 4 will validate some ofthe basic assumptions on a real-time hardware testbedFinally Section 5 will conclude this work

2 System modelFor our analysis we will use a typical MIMO systemwith multiple transmit and receive antennas (see Figure1) A pre-filter is used to improve synchronization per-formance We will examine two well-known algorithmsthat can be used as a pre-filter in addition to our pro-posed algorithm The filtered signal will be used by thesynchronization algorithm to determine whether a

packet is present and to estimate the symbol boundary(timing synchronization) This signal will then passthrough a secondary filter that will estimate the origin-ally transmitted signal The data payload of the packet isa simple uncoded QAM signal This was chosen so wemay directly evaluate the performance improvement ofour algorithm and avoid potential non-linear effectsfrom forward error correction schemes We used a stan-dard 80211a header [14] with well-known techniquesfor packet detection and timing synchronization from[15-17] It is our intention to show improvements inperformance as opposed to showing absolute perfor-mance For that reason we have chosen to use well-known training sequences as well as synchronizationalgorithms The performance improvements demon-strated in this work should be directly applicable to allpacket-based systems that require on packet detectionand timing synchronizationWe begin by defining some notation explicitly We

will use the superscript () to denote the complex conju-gate transpose (Hermitian) of a vector or a matrix Low-ercase boldface symbols (y) will be used to denotevectors and uppercase boldface (W) will be used todenote matrices The hat (x) will denote estimates ofsignals while a tilde (x) will be used to denote residualerror signals The trace operator for a matrix will bedenoted as Tr()First we will examine Rayleigh flat fading channels

the simplest class of channels These channels are mod-eled as a single impulse chosen from a Rayleigh distribu-tion A new channel will be chosen at random for eachpacket but remain constant throughout the duration ofthat packet We will discuss the ideal Minimum MeanSquared Error (MMSE) solution and show why it isimpractical in high interference scenarios We will thenreview the Sample Matrix Inverse (SMI) [18] as well asMaximal Signal to Interference plus Noise Ratio

Figure 1 System model


of 14




y(t) = Hsx(t) + Hiγ (t) + n(t) (1)

minW

E[xlowastx] = minW

E[Tr(xxlowast)] (2)


WMMSE = RxyRminus1y



(3)







i + Rn = R2 (5)


measurements (6)





Page 3 of 14







i + Rn (8)


i + Rn)Wlowast (9)


maxW

=Tr(WHsRxHlowast


i + Rn)Wlowast)(10)


i + Rn)minus1 (11)



WS1 = HsWMMSE






s (13)



Page 4 of 14

x = WS2WS1


s HsWMMSEy

= WMMSEy

(14)





P =

⎡⎢⎢⎣


minusa a a a

⎤⎥⎥⎦ (16)


PPlowast =

⎡⎢⎢⎣


⎤⎥⎥⎦ (17)


ZS1 = WS1HsP (18)






yM =

⎡⎢⎢⎢⎣

y(t)y(t minus 1)


⎤⎥⎥⎥⎦ xM =

⎡⎢⎢⎢⎣

x(t)x(t minus 1)


⎤⎥⎥⎥⎦ (22)

γM =

⎡⎢⎢⎢⎣



⎤⎥⎥⎥⎦ nM =

⎡⎢⎢⎢⎣

n(t)n(t minus 1)


⎤⎥⎥⎥⎦(23)

HsMM =

⎡⎣

Hs0 Hs1 Hs2

0 Hs0 Hs1

0 0 Hs0

⎤⎦

HiMM =

⎡⎣


⎤⎦

(24)



Page 5 of 14





E[xM(t)xlowast

M(t)]


M(t)]

= RγM

E[nM(t)nlowast

M(t)]

= RnM

(27)


)



RxyM =[Rx 0 0

]HsMM

lowast (29)

RyM =

(HsMMRxMHsMMlowast+


(30)

xMMSE(t) =[Rx 0 0

]HsMM

lowast

(HsMMRxMHsMMlowast+


(31)



WSMI = (HlowastiMM


WMSINR = HlowastsMM

RxMHsMM(HlowastiMM



WS1 = HlowastsMM

RxMHsMM

(HlowastsMM

RxMHsMM+

HlowastiMM


(34)

WS2 =[Rx 0 0

]Hlowast

sMM(Hlowast



x(t) = WS1WS2yM(t)

=[Rx 0 0

]Hlowast

sMM

(HlowastsMM


RxMMHsMM

(HlowastsMM

RxMHsMM +Hlowast


= WMMSEyM(t)

(36)







Page 6 of 14















Page 7 of 14




Page 8 of 14












Page 9 of 14




Page 10 of 14













Page 11 of 14










Page 12 of 14








Page 13 of 14





































Page 14 of 14

Abstract

1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References




y(t) = Hsx(t) + Hiγ (t) + n(t) (1)

minW

E[xlowastx] = minW

E[Tr(xxlowast)] (2)


WMMSE = RxyRminus1y



(3)







i + Rn = R2 (5)


measurements (6)





of 14







i + Rn (8)


i + Rn)Wlowast (9)


maxW

=Tr(WHsRxHlowast


i + Rn)Wlowast)(10)


i + Rn)minus1 (11)



WS1 = HsWMMSE






s (13)



Page 4 of 14

x = WS2WS1


s HsWMMSEy

= WMMSEy

(14)





P =

⎡⎢⎢⎣


minusa a a a

⎤⎥⎥⎦ (16)


PPlowast =

⎡⎢⎢⎣


⎤⎥⎥⎦ (17)


ZS1 = WS1HsP (18)






yM =

⎡⎢⎢⎢⎣

y(t)y(t minus 1)


⎤⎥⎥⎥⎦ xM =

⎡⎢⎢⎢⎣

x(t)x(t minus 1)


⎤⎥⎥⎥⎦ (22)

γM =

⎡⎢⎢⎢⎣



⎤⎥⎥⎥⎦ nM =

⎡⎢⎢⎢⎣

n(t)n(t minus 1)


⎤⎥⎥⎥⎦(23)

HsMM =

⎡⎣

Hs0 Hs1 Hs2

0 Hs0 Hs1

0 0 Hs0

⎤⎦

HiMM =

⎡⎣


⎤⎦

(24)



Page 5 of 14





E[xM(t)xlowast

M(t)]


M(t)]

= RγM

E[nM(t)nlowast

M(t)]

= RnM

(27)


)



RxyM =[Rx 0 0

]HsMM

lowast (29)

RyM =

(HsMMRxMHsMMlowast+


(30)

xMMSE(t) =[Rx 0 0

]HsMM

lowast

(HsMMRxMHsMMlowast+


(31)



WSMI = (HlowastiMM


WMSINR = HlowastsMM

RxMHsMM(HlowastiMM



WS1 = HlowastsMM

RxMHsMM

(HlowastsMM

RxMHsMM+

HlowastiMM


(34)

WS2 =[Rx 0 0

]Hlowast

sMM(Hlowast



x(t) = WS1WS2yM(t)

=[Rx 0 0

]Hlowast

sMM

(HlowastsMM


RxMMHsMM

(HlowastsMM

RxMHsMM +Hlowast


= WMMSEyM(t)

(36)







Page 6 of 14















Page 7 of 14




Page 8 of 14












Page 9 of 14




Page 10 of 14













Page 11 of 14










Page 12 of 14








Page 13 of 14





































Page 14 of 14

Abstract

1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References







i + Rn (8)


i + Rn)Wlowast (9)


maxW

=Tr(WHsRxHlowast


i + Rn)Wlowast)(10)


i + Rn)minus1 (11)



WS1 = HsWMMSE






s (13)



of 14

x = WS2WS1


s HsWMMSEy

= WMMSEy

(14)





P =

⎡⎢⎢⎣


minusa a a a

⎤⎥⎥⎦ (16)


PPlowast =

⎡⎢⎢⎣


⎤⎥⎥⎦ (17)


ZS1 = WS1HsP (18)






yM =

⎡⎢⎢⎢⎣

y(t)y(t minus 1)


⎤⎥⎥⎥⎦ xM =

⎡⎢⎢⎢⎣

x(t)x(t minus 1)


⎤⎥⎥⎥⎦ (22)

γM =

⎡⎢⎢⎢⎣



⎤⎥⎥⎥⎦ nM =

⎡⎢⎢⎢⎣

n(t)n(t minus 1)


⎤⎥⎥⎥⎦(23)

HsMM =

⎡⎣

Hs0 Hs1 Hs2

0 Hs0 Hs1

0 0 Hs0

⎤⎦

HiMM =

⎡⎣


⎤⎦

(24)



Page 5 of 14





E[xM(t)xlowast

M(t)]


M(t)]

= RγM

E[nM(t)nlowast

M(t)]

= RnM

(27)


)



RxyM =[Rx 0 0

]HsMM

lowast (29)

RyM =

(HsMMRxMHsMMlowast+


(30)

xMMSE(t) =[Rx 0 0

]HsMM

lowast

(HsMMRxMHsMMlowast+


(31)



WSMI = (HlowastiMM


WMSINR = HlowastsMM

RxMHsMM(HlowastiMM



WS1 = HlowastsMM

RxMHsMM

(HlowastsMM

RxMHsMM+

HlowastiMM


(34)

WS2 =[Rx 0 0

]Hlowast

sMM(Hlowast



x(t) = WS1WS2yM(t)

=[Rx 0 0

]Hlowast

sMM

(HlowastsMM


RxMMHsMM

(HlowastsMM

RxMHsMM +Hlowast


= WMMSEyM(t)

(36)







Page 6 of 14















Page 7 of 14




Page 8 of 14












Page 9 of 14




Page 10 of 14













Page 11 of 14










Page 12 of 14








Page 13 of 14





































Page 14 of 14

Abstract

1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References

x = WS2WS1


s HsWMMSEy

= WMMSEy

(14)





P =

⎡⎢⎢⎣


minusa a a a

⎤⎥⎥⎦ (16)


PPlowast =

⎡⎢⎢⎣


⎤⎥⎥⎦ (17)


ZS1 = WS1HsP (18)






yM =

⎡⎢⎢⎢⎣

y(t)y(t minus 1)


⎤⎥⎥⎥⎦ xM =

⎡⎢⎢⎢⎣

x(t)x(t minus 1)


⎤⎥⎥⎥⎦ (22)

γM =

⎡⎢⎢⎢⎣



⎤⎥⎥⎥⎦ nM =

⎡⎢⎢⎢⎣

n(t)n(t minus 1)


⎤⎥⎥⎥⎦(23)

HsMM =

⎡⎣

Hs0 Hs1 Hs2

0 Hs0 Hs1

0 0 Hs0

⎤⎦

HiMM =

⎡⎣


⎤⎦

(24)



of 14





E[xM(t)xlowast

M(t)]


M(t)]

= RγM

E[nM(t)nlowast

M(t)]

= RnM

(27)


)



RxyM =[Rx 0 0

]HsMM

lowast (29)

RyM =

(HsMMRxMHsMMlowast+


(30)

xMMSE(t) =[Rx 0 0

]HsMM

lowast

(HsMMRxMHsMMlowast+


(31)



WSMI = (HlowastiMM


WMSINR = HlowastsMM

RxMHsMM(HlowastiMM



WS1 = HlowastsMM

RxMHsMM

(HlowastsMM

RxMHsMM+

HlowastiMM


(34)

WS2 =[Rx 0 0

]Hlowast

sMM(Hlowast



x(t) = WS1WS2yM(t)

=[Rx 0 0

]Hlowast

sMM

(HlowastsMM


RxMMHsMM

(HlowastsMM

RxMHsMM +Hlowast


= WMMSEyM(t)

(36)







Page 6 of 14















Page 7 of 14




Page 8 of 14












Page 9 of 14




Page 10 of 14













Page 11 of 14










Page 12 of 14








Page 13 of 14





































Page 14 of 14

Abstract

1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References





E[xM(t)xlowast

M(t)]


M(t)]

= RγM

E[nM(t)nlowast

M(t)]

= RnM

(27)


)



RxyM =[Rx 0 0

]HsMM

lowast (29)

RyM =

(HsMMRxMHsMMlowast+


(30)

xMMSE(t) =[Rx 0 0

]HsMM

lowast

(HsMMRxMHsMMlowast+


(31)



WSMI = (HlowastiMM


WMSINR = HlowastsMM

RxMHsMM(HlowastiMM



WS1 = HlowastsMM

RxMHsMM

(HlowastsMM

RxMHsMM+

HlowastiMM


(34)

WS2 =[Rx 0 0

]Hlowast

sMM(Hlowast



x(t) = WS1WS2yM(t)

=[Rx 0 0

]Hlowast

sMM

(HlowastsMM


RxMMHsMM

(HlowastsMM

RxMHsMM +Hlowast


= WMMSEyM(t)

(36)







of 14















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Abstract

1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References















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1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References




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1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References












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1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References




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1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References













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2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


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2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References








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1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References





































of 14

Abstract

1 Introduction

2 System model





214 Two-stage MMSE







41 System overview

5 Conclusion


References

a practical two-stage mmse based mimo detector for interference mitigation with non-cooperative...

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