Efficient closed-loop schemes for MIMO-OFDM-based WLANs

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Efficient closed-loop schemes for MIMO-OFDM-based WLANs
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EURASIP Journal on Applied Signal Processing
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Zheng, Xiayu
Jiang, Yi
Li, Jian
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The single-input single-output (SISO) orthogonal frequency-division multiplexing (OFDM) systems for wireless local area networks (WLAN) defined by the IEEE 802.11a standard can support data rates up to 54Mbps. In this paper, we consider deploying two transmit and two receive antennas to increase the data rate up to 108 Mbps. Applying our recent multiple-input multipleoutput (MIMO) transceiver designs, that is, the geometric mean decomposition (GMD) and the uniform channel decomposition (UCD) schemes, we propose simple and efficient closed-loop MIMO-OFDM designs for much improved performance, compared to the standard singular value decomposition (SVD) based schemes as well as the open-loop V-BLAST (vertical Bell Labs layered space-time) based counterparts. In the explicit feedback mode, precoder feedback is needed for the proposed schemes. We show that the overhead of feedback can be made very moderate by using a vector quantization method. In the time-division duplex (TDD) mode where the channel reciprocity is exploited, our schemes turn out to be robust against the mismatch between the uplink and downlink channels. The advantages of our schemes are demonstrated via extensive numerical examples.
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Publication of this article was funded in part by the University of Florida Open-Access publishing Fund. In addition, requestors receiving funding through the UFOAP project are expected to submit a post-review, final draft of the article to UF's institutional repository, IR@UF, (www.uflib.ufl.edu/ufir) at the time of funding. The Institutional Repository at the University of Florida (IR@UF) is the digital archive for the intellectual output of the University of Florida community, with research, news, outreach and educational materials

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HindawiPublishingCorporation EURASIPJournalonAppliedSignalProcessing Volume2006,ArticleID91919,Pages 1 – 10 DOI10.1155/ASP/2006/91919EfcientClosed-LoopSchemesforMIMO-OFDM-BasedWLANsXiayuZheng,1YiJiang,2andJianLi11DepartmentofElectricalandComputerEngineering,UniversityofFlorida,Gainesville,FL32611-6130,USA2DepartmentofElectricalandComputerEngineering,UniversityofColorado,Boulder,CO80309-0425,USA Received28December2005;Revised18July2006;Accepted13August2006 Thesingle-inputsingle-output(SISO)orthogonalfrequency-divisionmultiplexing(OFDM)systemsforwirelesslocalareanetworks(WLAN)denedbytheIEEE802.11astandardcansupportdataratesupto54Mbps.Inthispaper,weconsiderdeploying twotransmitandtworeceiveantennastoincreasethedatarateupto108Mbps.Applyingourrecentmultiple-inputmultipleoutput(MIMO)transceiverdesigns,thatis,thegeometricmeandecomposition(GMD)andtheuniformchanneldecomposition (UCD)schemes,weproposesimpleande cientclosed-loopMIMO-OFDMdesignsformuchimprovedperformance,compared tothestandardsingularvaluedecomposition(SVD)basedschemesaswellastheopen-loopV-BLAST(verticalBellLabslayered space-time)basedcounterparts.Intheexplicitfeedbackmode,precoderfeedbackisneededfortheproposedschemes.Weshow thattheoverheadoffeedbackcanbemadeverymoderatebyusingavectorquantizationmethod.Inthetime-divisionduplex (TDD)modewherethechannelreciprocityisexploited,ourschemesturnouttoberobustagainstthemismatchbetweenthe uplinkanddownlinkchannels.Theadvantagesofourschemesaredemonstratedviaextensivenumericalexamples. Copyright2006XiayuZhengetal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense, whichpermitsunrestricteduse,distribution,andreproductio ninanymedium,providedtheoriginalworkisproperlycited.1.INTRODUCTION Thesingle-inputsingle-output(SISO)orthogonalfrequency-divisionmultiplexing(OFDM)systemsforwirelesslocal areanetworks(WLAN)denedbytheIEEE802.11astandardcansupportdataratesupto54Mbps[ 1 ].Improvingthedataratetoover100Mbpsisamajorgoalofthe next-generationWLANs[ 2 3 ].Themultiple-inputmultipleoutput(MIMO)communicationtechnologyiswidelyregardedasakeytoachievesuchahighdatarate. Assumingthatthechannelstateinformation(CSI)is availableatboththetransmitterandthereceiver,theMIMO channelcanbedecoupled,usingsingularvaluedecomposition(SVD),intomultipleorthogonalsubchannels(oreigenmodes)oneachsubcarrier[ 4 ].Tomaximizethechannel throughout,powerallocationandbitloadingshouldbeappliedtothesubchannelsinboththespatialandfrequency domains(see,e.g.,[ 5 ]andthereferencestherein).However, bitloadingisoftennotadoptedinpractice,suchasinthe IEEE802.11standards,duetoitscomplexity.Ifthesame constellationisusedacrossallthesubchannels,theweaker eigenmodescorrespondingtothesmallersingularvaluesof thechannelmatricestendtoexperiencedeeperfading[ 4 ], whichdegradestheoverallsystemperformancesignicantly. In[ 6 ],apowerallocationmethodwasproposedbasedon theminimummean-squarederror(MMSE)criterionforthe MIMOsystems.Thismethodtendstoputmorepoweron theweakersubchannels,whichmaycausesignicantcapacityloss. Inthispaperweproposesimpleande cientclosed-loop designsforMIMO-OFDM-basedWLANs.Wefocusonthe IEEE802.11astandard,althoughourschemesarealsoapplicabletootherstandardsincludingtheUSstandardIEEE 802.11gandtheEuropeanstandardHIPERLAN/2[ 7 ].Our schemescombinetherecentlyproposedgeometricmeandecomposition(GMD)anduniformchanneldecomposition (UCD)transceiverdesigns[ 8 9 ]withhorizontalencoding andsuccessive(noniterative)decoding.(Anideasimilarto GMDappearedintheindependentworkof[ 10 ].)GMDand UCDdecomposeeachMIMOchannelintomultipleequal gainsubchannelsforeachsubcarrier,whichallowsourdesignstoobviatetheneedofanypowerallocations.Thesimulationresultsshowthatourclosed-loopschemesenjoymultidBimprovementcomparedtothestandardsingularvalue decomposition(SVD)basedschemesaswellastheopenloopV-BLAST(verticalBelllabslayeredspace-time)based counterparts. Intheexplicitfeedbackmode,precoderfeedbackisrequiredfortheproposedschemes.Wepresentavectorquantizationalgorithmfore cientprecoderquantization.This quantizationalgorithmisinspiredbyanobservationofthe interestinglinkbetweena22unitarymatrixanda2D

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2EURASIPJournalonAppliedSignalProcessing Input data S/P Convolution encoder Convolution encoder Interleaving and data-mapping Interleaving and data-mapping N 22 precoders IFFT+CP and P/S IFFT+CP and P/S x1 kx1 kx2 kx2 k Figure 1:TransmitterdesignforMIMO-OFDM-basedWLAN.unitsphere.Weshowthatthe22unitaryprecodermatrix foreachfrequencysubcarriercanbequantizedby6bitswith verysmallperformancedegradations.Inthetime-division duplex(TDD)mode,wherethechannelreciprocityprinciple holds[ 3 ],ourschemesdonotrequireanyprecoderfeedback. Withasimplemodication,ourschemescanbemadequite robustagainsttheuplink-downlinkchannelmismatches. Theremainderofthispaperisorganizedasfollows. Section2 describesthe22MIMOchannelmodelwithspatialcorrelations. Section3 presentsourclosed-loopMIMO WLANsystemconguration,includingtheprecoderand equalizerdesigns,andthesuccessivesoftdecodingapproach. In Section4 ,weconsidertheexplicitfeedbackmodeand providetwoquantizationmethodsforprecoderfeedback, whereanewvectorquantizationalgorithmisproposed.In Section5 ,weconsidertheTDDmodeandshowthatthe proposedschemescanbemadequiterobustagainstthe uplink-downlinkchannelmismatches.Numericalexamples aregivenin Section6 todemonstratethee ectivenessofour schemes.Weconcludethepaperin Section7 Notation Weusebolduppercaseletterstodenotematricesandbold lowercaselettersforcolumnvectors.Weuse()Ttodenote thetransposeand()todenotetheHermitiantranspose.standsfortheEuclideannormandFfortheFrobeniusnorm. INisthe NN identitymatrix;det()isthedeterminantofamatrixand E []denotestheexpectationoperation. 2.CHANNELMODEL Considera22MIMOchannelwithspatialcorrelations, wherethechannelcanbemodeledas[ 11 ]: h ( t )=R1 / 2 rh11( t ) h12( t ) h21( t ) h22( t )R1 / 2 t,(1) with Rtand Rrquantifyingthespatialcorrelationsofthe channelfadingatthetransmitterandthereceiver,respectively,and hij( t )=LŠ1l=0h( l ) ijtŠlTs,1i j2,(2) denotingthefrequency-selectivechannellinkbetweenthe j thtransmitand i threceiveantennas(with L beingthechannellengthand Tsthesamplingperiod).Therandomvariables{h( l ) ij}2 i j=1, l=1, ... L ,areassumedtobeindependentlydistributedzero-mean,circularlysymmetriccomplex Gaussianvariables.Applying Nc-pointfastFouriertransform (FFT)to h ( t ),weobtainthechannelresponseinthefrequencydomain: H ( n )=LŠ1l=0hlTseŠj 2 nl/Nc,1nNc. (3) Wedenoteby Hk Hf ( k ) = H11f ( k )H12f ( k )H21f ( k )H22f ( k ) ,1kN (4) theatfadingchannelmatrixatthe k thdatasubcarrier ( Nc=64and N=48forIEEE802.11a),with f ( k ),1kN ,denotingthedatasubcarriermappingfunctionin[ 1 ]. 3.CLOSED-LOOPMIMOWLANSYSTEMDESIGN 3.1.Systemdescription OurMIMO-OFDMtransmitterschemeisshownin Figure1 Weadoptthehorizontalencodingmethod[ 12 ],wherethe twoparallelbranchesperformencoding,bitinterleaving, anddatamappingseparately.Let xikdenotethe i thencoded datasymbol, i=1,2,onthe k thsubcarrier,andlet xk=[ x1 kx2 k]T.Oneachofthe N datasubcarriers,thetransmitterappliesa22precodermatrix Pktoobtainxk=Pkxk, 1kN .Denotebyxikthe i thelementofxk.Thenxikis thesymboltobetransmittedinthe i thbranchatthe k thsubcarrier.ConsequentlyeachprecodedbranchisOFDMmodulatedusingan Nc-pointIFFTandisaddedwithacyclicprex(CP)beforetransmission.ThelengthofCPisassumedto belongerthanthechannellength L ,andthereforetheintersymbolinterference(ISI)canbecompletelyeliminatedatthe receiverside. Intheexplicitfeedbackmode,theprecoders{Pk}N k=1are calculatedandquantizedatthereceiver,andthenfedback fromthereceivertothetransmitter.IntheTDDmode,where thechannelreciprocityprincipleholds,oncethetransmitter

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XiayuZhengetal. 3 Output P/S Viterbi decoder (soft) Viterbi decoder (soft) Deinterleaving and soft-demapping Deinterleaving and soft-demapping Encoder and interleaving N 22 equalizers (soft-output) S/P andremove CP+FFT S/P andremove CP+FFT y1 ky1 ky2 ky2 k Figure 2:ReceiverdesignforMIMO-OFDM-basedWLAN.estimatesthereversechannel,thatis,theonefromthereceivertothetransmitter,viatrainingpilots,itcancalculate theprecoders Pk, k=1, ... N ,tobeusedintheforward channel,thatis,fromthetransmittertothereceiver. Assumingaccuratesynchronization,frequencyo setestimationandchannelestimation,thereceiverrstremoves theCPandappliesan Nc-pointFFTtoeachreceivedbranch asshownin Figure2 .Thenthereceivedsignalvectoratthe k thdatasubcarrieris yk=HkPkxk+ zk, k=1,2, ... N ,(5) where zk N (0, 2 zI )denotesthecircularlysymmetriccomplexGaussiannoise.Thekeycomponentsofourclosed-loop designsaretheprecoder Pkatthetransmitterandthecorrespondingequalizersatthereceiver,aswedescribenext. 3.2.Precoderandequalizerdesign WedesigntheprecoderandequalizerbasedonourGMD andUCDtransceiverdesignschemes[ 8 9 ].Bothschemes arebasedonthefollowingtheorem[ 8 ]. Theorem1. Anyrank K matrix H CMNwithsingularvalues H ,1H ,2H K> 0 canbedecomposedinto H=QRP,(6) where ()denotestheconjugatetranspose, R RKKisan uppertriangularmatrixwithequaldiagonalelements ri=(K n=1H n)1 /K, 1iK ,and Q CMKand P CNKaresemiunitarymatrices. Considerthechannelmodel( 5 ).Intheexplicitfeedback mode,theGMDscheme[ 8 ]startswiththeGMDmatrix decomposition Hk=QkRkPkatthereceiver,toobtain Pk, whichistheunitaryprecodertobefedbacktothetransmitter.Utilizingtheprecoder Pkatthetransmitterasin( 5 )leads tothefollowingreceiveddatavector: yk=QkRkxk+ zk, k=1,2, ... N. (7) Atthereceiver,multiplying ykby Qkyieldsyk=Rkxk+zk,(8) where Rk=QkHkPkisa22uppertriangularmatrixwith equal diagonalandzk N (0, 2 zI ).Theinformationsymbols in xkcanthenbedetectedsuccessivelystartingfromthesecondelementof xk(see Section3.3 ). TheUCDscheme[ 9 ]issomewhatmorecomplicated thanGMD.LikeGMD,theUCDschemehastwoimplementationsformsofwhichonecanberegardedasacombinationofalinearprecoderwithanMMSEV-BLASTequalizer. ComparedtoGMD,whichsu ersfromcapacitylossatlowto-moderateSNR,UCDisstrictlycapacitylosslessandcan achievetheoptimaldiversity-multiplexinggaintradeo [ 13 ]. Thedetailsareomittedhereduetolimitedspace. BothGMDandUCDobviatetheneedofbitloadingand powerallocationatthetransmitterandrequireonlythefeedbackoftheunitaryprecoders Pk, k=1, ... N .IntheTDD mode,theforwardchannelisestimatedatthetransmitter andthereforetheprecoders Pkcanbecalculatedatthetransmitter. 3.3.Successivesoftdecoding Notethat Rkisanuppertriangularmatrix.Asshownin Figure2 ,weadopttheschemesofdeinterleaving,softdemappingandthelow-complexitysoftViterbidecoderused in[ 2 ]foreachbranchseparately.Werstdetectthedatasequenceofthelowerbranchtogetthesoftinformation.Assumingsuccessfuldecodingofthedataofthelowerbranch, wecancanceltheinterferenceduetothelowerbranchcompletelybeforedecodingtheupperbranch,asisdenotedby thefeedbacklinkatthelowerpartof Figure2 .Theinterferencecancelationprocessofeachsubcarrier k usingGMDis outlinedasfollows. (1)Initialstage Calculate 2 2 k=E z2 k ( Rk22 2 =2 z Rk2 22,x2 k= y2 k Rk22, k=1, ... N (9) where( Rk)ij, i j=1,2,isthe( i j )thelementof Rkandy2 kis

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4EURASIPJournalonAppliedSignalProcessing thesecondentryofyk.Notethat 2 2 kalongwithx2 kprovides thesoftinformationforthelowerbranch.Wecandecodethe lowerbranchdatasequencebyusingthesoftViterbidecoder. (2)Cancellationstage Calculate 2 1 k=E z1 k Rk)11 2 =2 z Rk)2 11=2 2 k,x1 k= y1 kŠ Rk12 x2 k Rk11, k=1, ... N (10) where 2 1 kalongwithx1 kprovidesthesoftinformationfor theupperbranch.Here x2 kisthereconstructeddatasymbolsequenceobtainedfromtheViterbi-decoderofthelower branch.Notethat 2 1 k=2 2 kbecause Rkhasequaldiagonal.Giventhesoftinformationfortheupperbranch,wecan alsodecodetheupperbranchdatasequencebyusingthesoft Viterbidecoder. ForUCD,thesuccessivesoftdecodingprocedureissimilar.Because 2 1 k=2 2 k,thetwobrancheshavee ectively thesameoutputSNR.Incontrast,theSVD-basedorthe conventionalV-BLAST-basedmethodsleadtotwosubchannelswithunbalancedgains.Forthesystemswithaxed symbolconstellationacrossallthesubchannels,theweaker subchanneldominatestheoverallpacket-error-rate(PER) performance,althoughiterativedecodingbetweenthetwo branchesishelpfulforreducingthePERofV-BLAST[ 12 ]. 4.PRECODERQUANTIZATION Intheexplicitfeedbackmode,thechannelisestimatedatthe receiver.Wecomputetheprecoders Pk, k=1, ... N ,atthe receiverandfeedthembacktothetransmitter.Inthefollowing,wepresenttwoquantizationapproachestoreducethe overheadofprecoderfeedback. 4.1.Scalarquantization Asimplescalarquantizationschemeisasfollows.Notethat a22unitaryprecodercanberepresentedby P ( )= cos Šsin eŠjsin ejcos ,0< ,0< 2 (11) Denote Pn1, n2 = cos n1Šsin n1eŠjn 2sin n1ejn 2cos n1 ,(12) where n1=n1/N1,0n1N1Š1, n2=2 n2/N2, 0n2N2Š1,with N1and N2denotingthequantization levelsof n1and n2,respectively.Afterobtainingtheprecoder PkusingGMDorUCD,wequantize Pktothe“closest” (viaroundo )gridpointin( 12 ).Henceforeachsubcarrier k ,weonlyneedtofeedtheindex( n1, n2)backtothetransmitter,whichrequireslog2( N1N2)bits.Toreducethee ect ofquantizationerrorandimprovetherobustnessforGMD, insteadofapplyingtheoriginalequalizer Qkatthereceiver, weinsteaduseQkobtainedbytheQRdecomposition: HkPn1, n2QkRk, k=1, ... N. (13) Notethat P ( n1, n2)isknownatthereceiver.Wealsoneed toreplace RkbyRkinourinterferencecancelationstage. Clearly,when N1and N2arereasonablylarge,Rkisapproximatelyequalto RkandthetwodiagonalelementsofRkare almostthesame,thatis,thegainsofthetwobranchesremain almostthesame.However,larger N1and N2alsomeanmore feedbackoverhead.Inpractice,weneedtochoose N1and N2toachieveareasonabletradeo betweenfeedbackoverhead andperformance. Similarly,wecanapplytheMMSEV-BLASTalgorithm [ 14 ]to HkP ( n1, n2)toobtaintheequalizerwhenusing UCD. 4.2.Vectorquantization Vectorquantizationcanbeadoptedtofurtherreducethe overheadofprecoderfeedback.Wepresentageometricapproachtoperformvectorquantization.Supposewequantize theprecoder P ( )tobe P ( ),where( )correspond toanelementinacodebookknowntoboththetransmitterandreceiver.Insteadoftransmittingthedesireddatavector P ( ) xkatthetransmitter,where xkistheencodeddata vector,wetransmit P ( ) xk.Tooptimizethequantization scheme,weminimizethefollowingcostfunction: d=E P ( ) xkŠP ( ) xk 2 =ExkP ( )ŠP ( )P ( )ŠP ( )xk, (14) withrespectto and .Thiscostfunctionmeasurestheaveragedistortioncausedbytheniterateprecoderquantization. Heretheexpectationisover xk.Aftersomestraightforward algebra,weobtainP ( )ŠP ( )P ( )ŠP ( ) =2 I2Š2cos cos +sin sin cos( Š )I2. (15) Becausethevalueof E [xk2]doesnota ectourquantizationproblem,withoutlossofgenerality,let E [xk2]=1. Then d=2Š2cos cos +sin sin cos( Š ) 2Š2 (16) Inthefollowing,wegiveageometricinterpretationof Wenotethatthereisaone-to-oneandontomappingfrom theunitaryprecoderset{P ( ):0< ,0< 2 }

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XiayuZhengetal. 5 tothe2Dunitsphere{v R3:v=1}.Anypointonthe 2Dunitspherewithangles( )canberepresentedinthe Cartesiancoordinateas v=[ cos sin cos sin sin ]T, wheretherstelementof v isthe(1,1)-elementof P ( ) andthesecondandthirdelementsof v ,respectively,arethe realandimaginarypartsofthe(2,1)-elementof P ( ). Each P ( )correspondstoapoint v onthe2Dunitsphere. Similarly,anypointonthe2Dunitspherewithangles ( )canberepresentedbytheCartesiancoordinatev= cos sin cos sin sinT.Weseethat isjusttheinnerproductbetween v andv .Dene astheanglebetween v andv .Then =cos and d=(2Š2cos )=vŠ v2. (17) Basedonthisderivation,weconcludethatagoodcodebook{ vi}Nvi=1shouldbedistributedontheunitsphereasuniform aspossible. Weusethefollowingstepstodeterminethecodebook. First,wegenerateatrainingset{vn, n=1,2, ... Nt}viarandomlypicking Ntpointsonthe2Dunitsphere,where Ntis averylargenumber.Next,startingwithaninitialcodebook (obtainedviathesplittingmethod[ 15 ]),weiterativelyupdatethecodebook[ 15 ]untilnofurtherimprovementonthe minimumdistanceisobservedbasedonthefollowingcriteria. (1)Nearestneighborcondition(NNC):foragivencodebook{ vi}Nvi=1,assignavector vntothe i thregion Si= vn: vnŠ vi 2 vnŠ vj 2,j =i,(18) where Si, i=1,2, ... Nv,isthepartitionsetforthe i thcode vector. (2)Centroidcondition(CC):foragivenpartition Si,the updatedoptimumcodevectors{ vi}Nvi=1satisfyvi=argmin vi=1E vnŠ vi 2|vnSi, i=1,2, ... Nv. (19) Asshownintheappendix,thesolutiontotheaboveoptimizationproblemisvi= vi vi i=1,2, ... Nv,(20) where vi= vnSivn/vnSi1isthemeanvectorforthepartitionset Si, i=1,2, ... Nv. Hence,foreachsubcarrier k ,werstmaptheprecoder Pkasapoint v onthe2Dunitsphere.AccordingtotheNNC criterion,weobtainthequantizedvectorv fromthecodebookwithindex i .Theindex i isfedbacktothetransmitter toreconstructtheprecoder P ( ).Inthiscasetheoverhead offeedbackislog2( Nv)bitspersubcarrier. 5.ROBUSTTRANSCEIVERDESIGNINTHETDDMODE IntheTDDmode,thechannelreciprocitycanbeexploited toobviatetheneedofprecoderfeedbackinhighthroughputMIMOWLANsystem[ 3 ].However,thereisalwaysa mismatchbetweentheforwardchannel(fromtransmitter toreceiver)andreversechannel(fromreceivertotransmitter)duetochannelvariationsand/orampliermismatches, whichposesmajordi cultiesofutilizingtheconventional closed-loopschemes[ 16 ]. Ourclosed-loopschemescanbemodiedtoberobust againstthemismatchesandbebackwardcompatiblewiththe standardopen-loopV-BLASTreceiver[ 8 ].DenotebyHkthe forwardchannelassumedbythetransmitterandby Hkthe actualchannelmatrixatthe k thdatasubcarrier.Wemaydenotethechannelmismatchasfollows:Hk=Hk+ E ,1kN ,(21) where E isamatrixwhoseelementsareindependentlyand identicallydistributed(iid)complex-valuedGaussianvariableswithzero-meanandvariance 2=E [Hk2 F] / 4,and determinesthelevelofchannelmismatch.Atthetransmitter,theprecodersPk, k=1, ... N ,areobtainedbasedonHk, k=1, ... N .Thepilot(forchannelestimation)anddatasequencesarebothprecodedusingprecodersPk, k=1, ... N beforetransmission,whichleadstothefollowingreceived signalsinsteadof( 7 ): yk=HkPkxk+ zk, k=1,2, ... N. (22) Assumingperfectchannelestimationatthereceiver,theestimatedchannelmatrixonthe k thdatasubcarrieristhe“virtualchannel” HkPk.Asin Figure2 ,anequalizer Qkisapplied tothe k thsubcarriertoyieldyk= QkHkPkxk+ zk,(23) wheretheequalizer QkisobtainedfromtheQRdecompositionof HkPk,thatis, HkPk= Qk Rk.Henceyk= Rkxk+zk,(24) andwecanapplysuccessivesoftdecodingasdescribedin Section3.3 toretrievethetransmitteddataonthe k thdata subcarrier.Notethatthechannelgainsofthetwobranches areusuallyunbalancedduetothemismatchesbetweenHkand Hk.However,forsomesmall ,theoutputSNRsof thetwobranchesshouldbeclose,whichresultsinonly marginalperformanceloss,asshownwithnumericalexamplesin Section6 Similarly,forUCD,theprecoderPkiscalculatedaccordingtotheUCDprocedurebasedonHkandthereceiverinvolvesanMMSEV-BLASTequalizer. 6.NUMERICALEXAMPLES Wepresentseveralnumericalexamplestodemonstratethe superiorperformanceoftheproposedschemes.Thesystem parametersusedherearebasedontheIEEE802.11astandard.Forthetwotransmitandtworeceiveantennasystems, the64-QAMmodulationandthechannelcodingrateof R=3 / 4areused.Thetotalfrequencybandwidthis20MHz, whicharedividedinto64subcarriers,including48datasubcarriers.ForeachOFDMsymbolwithlength64thereisCP

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6EURASIPJournalonAppliedSignalProcessing withlength16whicharediscardedatthereceivertoremove ISI.Thereforethetotaldatarateis2log2643 / 448 / 642064 / (64+16)=108Mbps.Thechannelbetweeneach transmitandreceiveantennapairisgeneratedaccordingto theChayatmodel[ 17 ]with50nsroot-mean-squared(RMS) delayspread(herethesamplingperiodis Ts=50ns).We assumethatthechannelsareperfectlyestimatedatthereceiver.Thedataareformattedintopacketsconsistingof1000 informationbytes.AccordingtoIEEE802.11a,thegoalisto achievethepacket-error-rate(PER)of0 1. Forthepurposeofcomparison,wealsoimplementthe followingthreestandardschemes. TherstisasimpleSVD-basedscheme.Forthisscheme, boththetransmitterandreceiverapplyunitaryrotations todiagonalizethechannelmatrixateachsubcarrier,which yields248=96orthogonaldatasubchannels.Nobitallocationisinvolvedhere,sinceotherwise256-QAMorlarger constellationswouldbeused,whichwouldposedi culties inthehardwareimplementationsduetothephasenoiseissues,andsoforth.Theinputpowerisuniformlyallocated toallthe96datasubchannels.Oneencoder/decoderissu cientinthiscasebecausetheSVDcompletelyeliminatesthe interferencebetweensubchannelsandnosuccessivedecodingisneeded. Thesecondschemeissimilartotherst,exceptthatthe powerallocation(PA)algorithmof[ 6 ]isappliedateachsubcarrier.Becausethetwosubchannelsateachsubcarrieris usuallyhighlyunbalanced,thispowerallocationalgorithm tendstocompensatetheweakeronewithmorepower. Thethirdisanopen-loopMMSEV-BLAST-based scheme.JustliketheproposedGMDandUCDschemes,it appliestwoindependentencodersanddecodersforsuccessiveinterferencecancelation.Ofcourse,unlikeGMDand UCD,thetwodatabrancheshaveusuallyunbalancedchannelgains.Improvementcanbeachievedviaiterativedecodingasdescribedinthefollowing.Afterdecodingthelower branch,wecandecodetheupperbranchwiththeinuence fromthelowerbranchcanceled.Nowgiventhedecodeddata fromtheupperbranch,wecanobtainimproveddecodingof thelowerbranchbyremovingtheinuenceofthedatafrom theupperbranch.Thisprocedurecanbeiterated. Wealsoincludethechanneloutageprobabilitycurveas abenchmark.Channeloutageprobabilityisdenedasthe probabilitythattheinstantaneousmutualinformationofthe channel, I (SNR)=2048 6464 64+1648k=1logdetI2+ SNR 2 HkHk, (25) islessthan108.Thechanneloutageprobabilityisthelower boundofthePERperformanceofanyMIMOscheme.An information-theoreticallyoptimalschemecombinedwitha capacity-achievingcodeshouldbeabletoachievethiscurve. First,weconsiderchannelswithoutspatialcorrelation, thatis, Rr=I2and Rt=I2(cf.( 1 )). Figure3 showsthe PERperformancesoftheproposedGMD/UCDschemes,the closed-loopSVDwithandwithoutPA,andtheopen-loop 1416182022242628 104103102101100SNRPacketerrorrateOutageprobability MMSE MMSEite=3 SISO(54Mbps) SVD SVDwithPA GMD UCD Figure 3:PerformancecomparisonofMIMOWLAN(108Mbps) schemesforuncorrelatedchannelsintheabsenceofquantization errors.MMSEV-BLAST[ 18 ]basedscheme.Weassumeperfectprecoderfeedback.Itcanbeseenfrom Figure3 thattheproposedclosed-loopdesignshavemorethan4dBSNRimprovementovertheMMSEV-BLASTschemeatPERequal to0 1,althoughonecanhavea1dBgainbyapplyingiterativedecoding.TheSVD-basedmethodwithoutPAhasperformanceinferiortotheopen-loopMMSEV-BLAST-based scheme.TheschemebasedonSVDwithPAperformsbetter, butthereisstillmorethan3dBlosscomparedtotheGMD andUCDschemes.Thedashedlinerepresentstheperformanceofthe802.11asystemwithdatarate54Mbps.Itis remarkablethatcomparedwiththeSISOscheme,oursimple closed-loop22schemescandoublethedatarateandat thesametimesave2 5dBintotaltransmissionpower.Moreover,thePERcurvesoftheGMDandUCDschemeshavedecreasingslopesmuchsteeperthantheothermethods,which impliesmuchimproveddiversitygain.Thereisstillagapof about4 5dBbetweentheUCDschemeandtheoutageprobabilitycurve.Combinedwithacapacityachievingcode,such asaTurbocodeandalowdensityparitycheckcode(LDPC) [ 19 ],theproposedschemesshouldclosethegapfurther. Figure4 showsatypicalexampleoftheoutputSNRsof theeigen-subchannels(ŠŠandŠŠ)obtainedbySVD atthe48datasubcarriers.WeseethattheweakereigensubchannelshaveverylowoutputSNR(say,lessthan0dB). Theseweaksubchannelsmaycausetoomanydetectionerrorsfortheerrorcontrolcodetohandle.However,ateach subcarrier,theGMDandUCDschemesdecomposeaMIMO channelintotwo identical subchannels.TheoutputSNRs ofthesubchannelsofGMDandUCDarealsoshownin Figure4 .Wecanseethatthequalityofthesubchannelsof GMDandUCDaremuchmorestableacrossthesubcarriers.

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XiayuZhengetal. 7 01020304050105 0 5 10 15 20 25 30 IndexofdatasubcarrierOutputSNR(dB)Eigensubch.1 Eigensubch.2 GMD UCD Figure 4:OutputSNRsofthesubchannelsobtainedviaGMD, UCD,andSVD,withinputSNR=22dB.ThisgureprovidesinsightintothereasonwhyGMDand UCDperformsignicantlybetterthantheSVD-basedmethods.WecanalsoseethatUCDoutperformsGMDwhenthe channelisclosetosingular,liketheoneatthe30thsubcarrier. Inthesecondexample,weconsideraspatiallycorrelated channelwith Rr= 10 7 0 71, Rt= 10 3 0 31,(26) whilealltheotherparametersremainthesameastherst example.Theresultsaregivenin Figure5 .Comparedwith Figure3 ,in Figure5 alltheMIMO-OFDMschemessu er fromperformancedegradationsduetothespatialfadingcorrelations.However,therelativeadvantageoftheproposed closed-loopschemesisevenmoreprominentinthisscenario. Specically,theUCDschemehasamorethan4dBgainover theSVD-basedschemesandapproximatelya6dBgainover theopen-loopMMSEV-BLASTschemeatPERequalto0 1. Indeed,weexpectthattheeigen-subchannelsobtainedby SVDshouldhavemoredisparatechannelgainsinthepresenceoffadingcorrelations.Despitethefadingcorrelations, theproposedGMDandUCDsystemsatthe108Mbpsdata ratestillprovidebetterPERperformancethantheSISOsystemathalfthedatarate. Weconsidernextthee ectofquantizedprecoderon systemperformance.Weuse8-bitscalarquantizationwith N1=24and N2=24(cf. Section4.1 )and m -bitvectorquantizationwith Nv=2m, m=2,4,6,(cf. Section4.2 )toquantizetheprecoder Pkofeachdatasubcarrier.Figures 6 and 7 showthatthe6-bitvectorquantizationperformsequally wellasthe8-bitscalarquantization.Byusingthe6-bitvec101520253035 104103102101100SNRPacketerrorrateOutageprobability MMSE SISO(54Mbps) SVD SVDwithPA GMD UCD Figure 5:PerformancecomparisonofMIMOWLAN(108Mbps) schemesforcorrelatedchannelsintheabsenceofquantizationerrors. 202122232425262728 103102101100SNRPacketerrorrateMMSE GMD2bitsVQ GMD4bitsVQ GMD6bitsVQ GMD8bitsSQ GMDperfectfeedback Figure 6:Performancecomparisonoftheproposedclosed-loop schemesforuncorrelatedchannelswith8-bitscalarquantization andvariousvectorquantizationbitratespersubcarrierintheexplicitfeedbackmodeforGMD.torquantization,ourquantizedclosed-loopMIMOschemes su erfromlessthan0 3dBSNRlosscomparedtotheperfectfeedbackcaseatPER=0 1.Thissmalllossisnegligible comparedtothesignicantimprovementofourproposed schemeoverothers.Whenmorebitsareused,wecanfurther closethesmallgap.

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8EURASIPJournalonAppliedSignalProcessing 202122232425262728 103102101100SNRPacketerrorrateMMSE UCD2bitsVQ UCD4bitsVQ UCD6bitsVQ UCD8bitsSQ UCDperfectfeedback Figure 7:Performancecomparisonoftheproposedclosed-loop schemesforuncorrelatedchannelswith8-bitscalarquantization andvariousvectorquantizationbitratespersubcarrierintheexplicitfeedbackmodeforUCD. 20212223242526 103102101100SNRPacketerrorrateGMD =0 GMD =5% GMD =10% Figure 8:Performancecomparisonoftheproposedclosed-loop schemesforuncorrelatedchannelsunderchannelmismatcheswith di erenterrorparametersintheTDDmodeforGMD.Finally,weconsidertheTDDmode.Figures 8 and 9 show thatourclosed-loopschemesarequiterobustagainstthe mismatchesbetweenthechannelmatricesobtainedatthe transmitterandthereceiver.Ourclosed-loopschemessufferfromlessthan0 2dBlossatPER=0 1evenwhenthe errorparameterisashighas =0 1. 20212223242526 103102101100SNRPacketerrorrateUCD =0 UCD =5% UCD =10% Figure 9:Performancecomparisonoftheproposedclosed-loop schemesforuncorrelatedchannelsunderchannelmismatcheswith di erenterrorparametersintheTDDmodeforUCD.7.CONCLUSIONS Wehavepresentedsimpleande cientclosed-loopdesignsforMIMO-OFDM-basedWLANsasapromisingtechnologyforthenext-generationwirelessLANcommunications.BycombiningtherecentGMDandUCDtransceiver designsandthehorizontalencodingarchitecture,wecan achievemulti-dBimprovementovertheclosed-loopSVDbasedschemesandtheopen-loopMMSEV-BLASTarchitecture.Theadvantageofourschemesisevenmoreprominent whenthefadingchannelsarespatiallycorrelated.Wehave alsoproposedane cientalgorithmforthequantizationof 22unitaryprecoders.Usingonlya6-bitvectorquantizationateachdatasubcarrier,thesystemcanachieveperformanceveryclosetotheperfectprecoderfeedback,which representsaverymoderatefeedbackoverheadintheexplicit feedbackmode.IntheTDDmode,whenthechannelreciprocitymechanismisavailable,wecanmodifyourclosedloopdesignstoberobustagainstthemismatchesbetween theforwardchannelandreversechannel.Theextensivenumericalexperimentsvalidatethesuperiorperformanceofthe proposedschemes.Finally,weremarkthat,althoughourdiscussionsfocusonthe22system,ourschemescanbereadily extendedtothecaseofmoretransmitandmorereceiveantennas. APPENDIX Let vn [ vn 1vn 2vn 3]T=[ cos sin cos sin sin ]T, andvi [vi 1vi 2vi 3]T.Theoptimizationproblemin( 19 ) becomes minvnSi vnŠ vi 2s.t. vi 2=1 (A.1)

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XiayuZhengetal. 9 TheLagrangiancorrespondingtotheconstrainedoptimizationproblemis L vi, = vnSi3l=1vnlŠ vil2+ 3l=1v2 ilŠ1. (A.2) Fromtherstderivativeconditions: L vi, vil=0, l=1,2,3,(A.3) and vi=1,wehavevil= vnSivnl 3 l=1 vnSivnl2, l=1,2,3 (A.4) Dene vi=[ vi 1 vi 2 vi 3]T,where vil= vnSivnl/vnSi1, l=1,2,3.Thenweobtainvi= vi vi (A.5) ACKNOWLEDGMENTS ThisworkwassupportedinpartbytheNationalScience FoundationGrantCCR-0104887andtheworkofXiayu ZhengwasalsosupportedinpartbytheUniversityofFlorida AlumniFellowship. REFERENCES[1]IEEEStandard802.11a-1999,“Supplementtoinformation technology-telecommunicationsandinformationexchange betweensystems-localandmetropolitanareanetworksspecicrequirements—part11:wirelessLANmediumaccess control(MAC)andphysicallayer(PHY)specications:high speedphysicallayer(PHY)inthe5GHzband,”1999. [2]J.LiuandJ.Li,“AMIMOsystemwithbackwardcompatibility forOFDM-basedWLANs,” EurasipJournalonAppliedSignal Processing ,vol.2004,no.5,pp.696–706,2004. [3]IEEE802.11-04/0889r6,“TGnSyncproposaltechnicalspecication,”May2005. [4]S.Nanda,R.Walton,J.Ketchum,M.Wallace,andS.Howard, “Ahigh-performanceMIMOOFDMwirelessLAN,” IEEE CommunicationsMagazine ,vol.43,no.2,pp.101–109,2005. [5]P.Xia,S.Zhou,andG.B.Giannakis,“AdaptiveMIMO-OFDM basedonpartialchannelstateinformation,” IEEETransactions onSignalProcessing ,vol.52,no.1,pp.202–213,2004. [6]H.SampathandA.J.Paulraj,“Jointtransmitandreceiveoptimizationforhighdataratewirelesscommunicationusing multipleantennas,”in Proceedingsofthe33rdAsilomarConferenceonSignals,SystemsandComputers(ACSSC’99) ,vol.1, pp.215–219,PacicGrove,Calif,USA,October1999. [7]ETSI,“Broadbandradioaccessnetworks(BRAN);HIPERLAN type2:physical(PHY)layer,”ETSITS101475V1.2.2,February2005. [8]Y.Jiang,J.Li,andW.W.Hager,“Jointtransceiverdesignfor MIMOcommunicationsusinggeometricmeandecomposition,” IEEETransactionsonSignalProcessing ,vol.53,no.10, pp.3791–3803,2005. [9]Y.Jiang,J.Li,andW.W.Hager,“UniformchanneldecompositionforMIMOcommunications,” IEEETransactionsonSignal Processing ,vol.53,no.11,pp.4283–4294,2005. [10]J.-K.Zhang,A.Kav ci c,andK.M.Wong,“Equal-diagonal QRdecompositionanditsapplicationtoprecoderdesignfor successive-cancellationdetection,” IEEETransactionsonInformationTheory ,vol.51,no.1,pp.154–172,2005. [11]C.-N.Chuah,D.N.C.Tse,J.M.Kahn,andR.A.Valenzuela, “CapacityscalinginMIMOwirelesssystemsundercorrelatedfading,” IEEETransactionsonInformationTheory ,vol.48, no.3,pp.637–650,2002. [12]X.Li,H.Huang,G.J.Foschini,andR.A.Valenzuela,“Effectsofiterativedetectionanddecodingontheperformance ofBLAST,”in ProceedingsofIEEEGlobalTelecommunications Conference(GLOBECOM’00) ,vol.2,pp.1061–1066,San Francisco,Calif,USA,November-December2000. [13]L.ZhengandD.N.C.Tse,“Diversityandmultiplexing:afundamentaltradeo inmultiple-antennachannels,” IEEETransactionsonInformationTheory ,vol.49,no.5,pp.1073–1096, 2003. [14]B.Hassibi,“Afastsquare-rootimplementationforBLAST,” in Proceedingsofthe34thAsilomarConferenceonSignals,SystemsandComputers(ACSSC’00) ,vol.2,pp.1255–1259,PacicGrove,Calif,USA,October-November2000. [15]Y.Linde,A.Buzo,andR.Gray,“Analgorithmforvectorquantizerdesign,” IEEETransactionsonCommunications ,vol.28, no.1,pp.84–95,1980. [16]G.Lebrun,J.Gao,andM.Faulkner,“MIMOtransmission overatime-varyingchannelusingSVD,” IEEETransactionson WirelessCommunications ,vol.4,no.2,pp.757–764,2005. [17]N.Chayat,“Tentativecriteriaforcomparisonofmodulation methods,”document:IEEEP802.11-97/96,September1997. [18]G.J.Foschini,G.D.Golden,R.A.Valenzuela,andP.W.Wolniansky,“Simpliedprocessingforhighspectrale ciencywirelesscommunicationemployingmulti-elementarrays,” IEEE JournalonSelectedAreasinCommunications ,vol.17,no.11, pp.1841–1852,1999. [19]B.Lu,G.Yue,andX.Wang,“Performanceanalysisanddesign optimizationofLDPC-codedMIMOOFDMsystems,”IEEE TransactionsonSignalProcessing ,vol.52,no.2,pp.348–361, 2004. XiayuZheng receivedtheB.S.andM.S. degreesinelectricalengineeringandinformationsciencefromtheUniversityof ScienceandTechnologyofChina(USTC), Hefei,China,in2001and2004,respectively.HeiscurrentlypursuingthePh.D. degreewiththeDepartmentofElectrical andComputerEngineering,Universityof Florida,Gainesville.Hisresearchinterests areintheareasofsignalprocessingand wirelesscommunications. YiJiang receivedtheB.S.degreeinelectrical engineeringandinformationsciencefrom theUniversityofScienceandTechnologyof China(USTC),Hefei,China,in2001.He receivedtheM.S.andPh.D.degreesfrom theUniversityofFlorida,Gainesville,both inelectricalengineering,in2003and2005, respectively.IntheSummerof2005,he wasaResearchConsultantwithInformationScienceTechnologiesInc.(ISTI),Fort Collins,Colo.HeisnowaPostdocwiththeUniversityofColorado, Boulder.Hisresearchinterestsareintheareasofsignalprocessing, wirelesscommunications,andinformationtheory.

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10EURASIPJournalonAppliedSignalProcessing JianLi receivedtheM.S.andPh.D.degrees inelectricalengineeringfromTheOhio StateUniversity,Columbus,in1987and 1991,respectively.FromJuly1991toJune 1993,shewasanAssistantProfessorwith theDepartmentofElectricalEngineering, UniversityofKentucky,Lexington.Since August1993,shehasbeenwiththeDepartmentofElectricalandComputerEngineering,UniversityofFlorida,Gainesville, wheresheiscurrentlyaProfessor.Hercurrentresearchinterests includespectralestimation,statisticalandarraysignalprocessing, andtheirapplications.SheisaFellowofIEEEandaFellowofIEE. Shereceivedthe1994NationalScienceFoundationYoungInvestigatorAwardandthe1996O ceofNavalResearchYoungInvestigatorAward.ShehasbeenaMemberoftheEditorialBoardof SignalProcessing,apublicationoftheEuropeanAssociationfor SignalProcessing(EURASIP),since2005.SheispresentlyaMemberoftwooftheIEEESignalProcessingSocietytechnicalcommittees:theSignalProcessingTheoryandMethods(SPTM)Technical CommitteeandtheSensorArrayandMultichannel(SAM)TechnicalCommittee.