A MIMO system with backward compatibility for OFDM-based WLANs

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A MIMO system with backward compatibility for OFDM-based WLANs
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EURASIP Journal on Applied Signal Processing
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Liu, Jianhua
Li, Jian
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Orthogonal frequency division multiplexing (OFDM) has been selected as the basis for the new IEEE 802.11a standard for highspeed wireless local area networks (WLANs).We consider doubling the transmission data rate of the IEEE 802.11a system by using two transmit and two receive antennas. We propose a preamble design for this multi-input multi-output (MIMO) system that is backward compatible with its single-input single-output (SISO) counterpart as specified by the IEEE 802.11a standard. Based on this preamble design, we devise a sequential method for the estimation of the carrier frequency offset (CFO), symbol timing, and MIMO channel response. We also provide a simple soft detector based on the unstructured least square approach to obtain the soft information for the Viterbi decoder. This soft detector is very simple since it decouples the multidimensional QAM symbol detection into multiple one-dimensional QAM symbol—and further PAM symbol—detections. Both the sequential parameter estimation method and the soft detector can provide excellent overall system performance and are ideally suited for real-time implementations. The effectiveness of our methods is demonstrated via 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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EURASIPJournalonAppliedSignalProcessing2004:5,696c 2004HindawiPublishingCorporationAMIMOSystemwithBackwardCompatibilityforOFDM-BasedWLANsJianhuaLiuDepartmentofElectricalandComputerEngineering,UniversityofFlorida,P.O.Box116130,Gainesville,FL32611-6130,USAEmail:jhliu@dsp.u.eduJianLiDepartmentofElectricalandComputerEngineering,UniversityofFlorida,P.O.Box116130,Gainesville,FL32611-6130,USAEmail:li@dsp.u.eduReceived16December2002;Revised28June2003Orthogonalfrequencydivisionmultiplexing(OFDM)hasbeenselectedasthebasisforthenewIEEE802.11astandardforhigh-speedwirelesslocalareanetworksWLANs).WeconsiderdoublingthetransmissiondatarateoftheIEEE802.11asystembyusingtwotransmitandtworeceiveantennas.Weproposeapreambledesignforthismulti-inputmulti-outputMIMO)systemthatisbackwardcompatiblewithitssingle-inputsingle-outputSISO)counterpartasspeciedbytheIEEE802.11astandard.Basedonthispreambledesign,wedeviseasequentialmethodfortheestimationofthecarrierfrequencyo setCFO),symboltiming,andMIMOchannelresponse.WealsoprovideasimplesoftdetectorbasedontheunstructuredleastsquareapproachtoobtainthesoftinformationfortheViterbidecoder.ThissoftdetectorisverysimplesinceitdecouplesthemultidimensionalQAMsymboldetectionintomultipleone-dimensionalQAMsymbolandfurtherPAMsymboldetections.Boththesequentialparameterestimationmethodandthesoftdetectorcanprovideexcellentoverallsystemperformanceandareideallysuitedforreal-timeimplementations.Thee ectivenessofourmethodsisdemonstratedvianumericalexamples.Keywordsandphrases:MIMOsystem,OFDM,WLAN,symboltiming,carriersynchronization,channelestimation.1.INTRODUCTIONOrthogonalfrequencydivisionmultiplexing(OFDM)hasbeenselectedasthebasisforseveralnewhigh-speedwirelesslocalareanetworkWLAN)standards[1 ],includingIEEE802.11a[2 ],IEEE802.11g,andHIPERLAN/2.IEEE802.11gandHIPERLAN/2areverysimilartoIEEE802.11aintermsofsignalgenerationanddetection/decoding.WeuseIEEE802.11atoexemplifyourpresentationinthispaper.TheOFDM-basedWLANsystem,asspeciedbytheIEEE802.11astandard,usespacket-basedtransmission.Eachpacket,asshowninFigure1,consistsofanOFDMpacketpreamble,asignaleld,andanOFDMdataeld.Thepream-blecanbeusedtoestimatethechannelparameterssuchasthecarrierfrequencyo setCFO),symboltiming,aswellaschannelresponse.TheseparametersareneededforthedatasymboldetectionintheOFDMdataeld.Thepreamblede-signadoptedbythestandardisspecicallytailoredtothesingle-inputsingle-outputSISO)systemcasewhereboththetransmitterandreceiverdealwithasinglesignal.Thisstandardsupportsadatarateupto54Mbps.Transmissiondatarateshigherthan54Mbpsareofpar-ticularimportanceforfutureWLANs.Deployingmultipleantennasatboththetransmitterandreceiverisapromisingwaytoachieveahightransmissiondatarateformultipath-richwirelesschannelswithoutincreasingthetotaltransmis-sionpowerorbandwidth[3 ].Thecorrespondingsystem,asshowninFigure2,isreferredtoasamulti-inputmulti-output(MIMO)wirelesscommunicationsystem,whereM and N intheguredenotethenumbersoftransmitandre-ceiveantennas,respectively.AmongthevariouspopularMIMOwirelesscommuni-cationschemes,theBLASTBellLabsLayeredSpaceTime)approaches[4 5 ]areparticularlyattractive.BLASTattemptstoachievethepotentiallylargechannelcapacityo eredbytheMIMOsystem[6 7 ].InBLASTsystems,thedatastreamisdemultiplexedintoindependentsubstreamsthatarereferredtoaslayers.Theselayersaretransmittedsi-multaneously,thatis,onelayerpertransmitantenna.Atthereceiver,themultiplelayerscanbedetected,forexam-ple,throughsuccessivedetectionviaaninterferencecan-cellationandnullingalgorithm(ICNA)[5 ].Thedetection

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AMIMOSystemwithBackwardCompatibilityforOFDM-BasedWLANs697 t 10 t 9 t 8 t 7 t 6 t 5 t 4 t 3 t 2 t 1 10 0 8 = 8 s1 6+2 3 2 = 8 s OFDMpacketpreambleOFDMdataeldGI2T1 T 2 GISignalSignaleld 0 8+3 2 = 4 s0 8+3 2 = 4 s GI OFDM symbol GI OFDM symbol Figure1:PacketstructureoftheIEEE802.11astandard. H MN ReceiverTransmitter. Figure2:DiagramofaMIMOsystem.canalsobedoneviathespheredecoding(SPD)algorithm[ 8 ]. OurfocushereinisondoublingthedatarateoftheSISOsystemasspeciedbytheIEEE802.11astandardbyusingtwotransmitandtworeceiveantennasreferredtoastheMIMOsysteminthesequel)basedontheBLASTscheme.WeproposeapreambledesignforthisMIMOsystemthatisbackwardcompatiblewithitsSISOcounterpartasspeci-bytheIEEE802.11astandard.Thatis,aSISOreceivercanperformCFO,symboltiming,andchannelresponsees-timationbasedontheproposedpreambledesignanddetectuptothesignaleld.TheSISOreceiveristheninformed,byusing,forexample,thereservedbitinthesignaleld,thatatransmissionisaSISOornot.Ourpreambledesigncanbeusedwithtwotransmitandanynumberofreceiveantennas.However,wemainlyfocusonthetworeceiveantennacaseherein.BasedonourMIMOpreambledesign,weproposeasequentialmethod,ideallysuitedforreal-timeimplementa-tions,toestimatetheCFO,symboltiming,andMIMOchan-nelresponse.TheconvolutionalcodespeciedintheIEEE802.11astandardwillalsobeusedinourMIMOsystemforchan-nelcoding.Asaresult,softinformationfromtheMIMOde-tectorisneededbytheViterbidecodertoimprovethede-codingperformance.Boththee cientICNAandSPDalgo-rithmso eronlyhardoutput.SoftoutputcanbeinferredwiththeICNA-basedalgorithmforiterativedetectionanddecoding[9 ].However,thisalgorithmiscomputationallyex-tremelyheavyexponentionalintermsoftransmitantennanumberM aswellastheconstellationsize.Althoughreducedcomplexityversionswerealludedtoin[9 ],thecostsinper-formancedegradationbyusingtheseversionswerenotclear.Thespacetimebit-interleavedcodedmodulationSTBICM)approach[10 ]candeliversoftoutput,inboththeiterativeandnoniterativemodes,butitisalsocomputationallyex-tremelyheavy.AlistspheredecoderLSD)algorithm[11 ] wasrecentlyproposedtoreducethecomputationalcomplex-ityofSTBICMwithasmallperformancedegradation.How-ever,LSDisstillverycomplicatedandhardtoimplementinrealtimeforOFDM-basedMIMOWLANapplicationsduetothehighdatarate.WepresenthereinasimpleMIMOsoftdetector,basedontheunstructuredleastsquareLS)t-tingapproach.ThisLS-basedsoftdetectorisideallysuitedforreal-timeimplementationssinceitdecouplesthemultidi-mensionalquadratureamplitudemodulation(QAM)sym-boldetectionintomultipleone-dimensionalQAMsymboldetections.Weshowthattherealandimaginarypartsofthenoiseofthedecoupleddetectionoutputareindepen-dentofeachother.Hence,theQAMsymboldetectioncanbefurthersimpliedintotwopulseamplitudemodulation(PAM)symboldetections.Asaresult,thisLS-basedsoftde-tectorisordersofmagnitudemorecomputationallye cient thanLSD;yet,thee ciencyisachievedatacostofasmallperformancedegradation,duetotheaforementionedde-coupling.TheLS-baseddetectorcanalsobeseenasrelatedtothezero-forcingortothelineardecorrelatingdetector[ 12 ]. Theremainderofthispaperisorganizedasfollows.Section2describestheMIMOsystem.Thenewpreamblede-signisgiveninthissection.Section3presentsoursequen-tialmethodforCFO,symboltiming,andMIMOchannelresponseestimation.TheMIMOsoftdetectorisprovidedin Section4.NumericalexamplesaregiveninSection5to demonstratethee ectivenessoftheproposedmethods.Fi-nally,weendourpaperwithcommentsandconclusionsinSection6. 2.SYSTEMDESIGNOurMIMOsystemcloselyresemblesitsSISOcounterpartasspeciedbytheIEEE802.11astandard.WerstgiveabriefoverviewoftheIEEE802.11aSISOsystembeforeweproceedtodescribeourMIMOsystem.2.1.IEEE802.11astandardFigure1showsthepacketstructureasspeciedbytheIEEE802.11astandard.ThenominalbandwidthoftheOFDMsignalis20MHzandthein-phase/quadrature(I/Q)sam-plingintervalt S is50nanoseconds.Inthiscase,thenum-berofsamplesN S = 64foranOFDMdatasymbolisequaltothenumberofsubcarriers.TheOFDMpacketpreambleconsistsoftenidenticalshortOFDMtrainingsymbolsti i = 1,2,... ,10,eachofwhichcontainsN C = 16samples,andtwoidenticallongOFDMtrainingsymbolsTi i = 1,2,eachofwhichcontainsN S = 64samples.Betweentheshortand

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698EURASIPJournalonAppliedSignalProcessing longOFDMtrainingsymbols,thereisalongguardinterval(GI2)consistingof2N C = 32datasamples.GI2isthecyclicprCP)forthelongOFDMtrainingsymbolT1 ,thatis,itistheexactreplicaofthelast2N C samplesofT1 TheinformationcarryingdataareencodedintheOFDMdataeld.ThebinarysourcedatasequenceisrstscrambledandthenconvolutionallyencodedbyanindustrialstandardconstraintlengthK = 7,rate1/2encoder,whichhasgenera-tionpolynomialsg 0 = (133) 8 and g 1 = (171) 8 .Theencodedoutputisthenpuncturedaccordingtothedataraterequire-mentandissegmentedintoblocksoflengthN CBPS (num-berofcodedbitsperOFDMsymbol,eachofwhichcorre-spondstoanOFDMdatasymbol.ThebinarydataineachblockisrstinterleavedamongthesubcarriersreferredtoasthefrequencydomainFD)interleavinginthesequel)andthenmappedingroupsoflog2 A bits)intoA -QAMsymbols,whichareusedtomodulatethedi erentdatacarryingsub-carriers.EachOFDMdatasymbolintheOFDMdataeldemploysN S = 64subcarriers,48ofwhichareusedfordatasymbolsand4forpilotsymbols.Therearealso12nullsub-carrierswithoneinthecenterandtheother11onthetwoendsofthefrequencyband.TheOFDMdatasymbols,eachofwhichconsistsofN S = 64samples,areobtainedviatakingtheinversefastFouriertransformIFFTofthedatasymbols,pilotsymbols,andnullsontheseN S subcarriers.ToeliminatetheintersymbolinterferenceISI),eachOFDMdatasymbolisprecededbyaCPorGI,whichcontainsthelastN C samples oftheOFDMdatasymbol.Thesignaleldcontainstheinformationincludingthetransmissiondatarateanddatalengthofthepacket.Theinformationiscontainedin16binarybits.ThereisalsoareservedbitwhichcanbeusedtodistinguishtheMIMOfromSISOtransmissions)andaparitycheckbit.These18bits,paddedwith6zeros,arethenencoded(bythesameen-coderasfortheOFDMdataeld)toobtaina48-bitbinarysequence.Theencodedsequenceistheninterleavedamongsubcarriersandusedtomodulatethe48datacarryingsub-carriersusingBPSK.Thesignaleldconsistsof64samplesandisobtainedviatakingtheIFFTofthese48BPSKsym-bols,4pilotsymbols,and12nulls.Also,thereisaCPoflength N C toseparatethepreamblefromthesignaleld.2.2.SISOdatamodelToestablishthedatamodel,considerrstthegenerationofanOFDMdatasymbolintheOFDMdataeld.Letx SISO = [ x 1 1 x 1 2 x 1 N S ] T beavectorofN S datasymbols,where T denotesthetransposeandx 1 n S n S = 1,2,... N S isthesymbolmodulatingthen S thsubcarrierandisequalto0fornullsubcarriers,1or 1forpilotsubcarriers,andin C fordatacarryingsubcarriers.HereC isanitecon-stellation,suchasBPSK,QPSK,16-QAM,or64-QAM.LetW N S C N S N S bethefastFouriertransform(FFTmatrix.ThentheOFDMdatasymbols correspondingtox SISO isob-tainedbytakingtheIFFTofx SISO .Thatis,s = W H N S x SISO /N S where H denotestheconjugatetranspose.ToeliminatetheISI,eachOFDMdatasymbolisprecededbyaCPorGIs C formedusings Leth (t) t = p p t p t S (1) denotethetime-domainanaloguechannelimpulseresponseofthefrequency-selectivetime-invariantfadingchannel,where p and p t S ,0 p N C p Z ,arethecomplexgainandtimedelayofthep thpath,respectively.Leth (t) = h (t) 0 h (t) 1 h (t) N S 1 T (2) betheequivalentniteimpulseresponse(FIR)lterresponseof h (t) t ),thatis,ifh = W N S h (t) = [ h 1 h 2 h N S ] T is thesampledfrequencydomainchannelresponse,thenforn S = 1,2,... N S h n S = p p e p t S = 2 n S 1) / N S t S (3) The l thelementofh (t) l = 0,1,... N S 1,canbewrittenash (t) l = p p e l +( N S 1) p /N S sin p sin p l /N S ,(4whichincludestheleakagee ectduetothefrequencydo-mainsampling[13 ]. BydiscardingtherstN C samplesatthereceiver(as-sumingacorrectsymboltiming),thenoise-freeandCFOfreereceivedsignalvectorz ne SISO C N S 1 ,duetosamplingthereceivedsignal,isthecircularconvolutionofh (t) and s HencetheFFToutputofthereceiveddatavectorz SISO = z ne SISO + e SISO ,wheree SISO N 0 ,( 2 /N S I N S istheaddi-tivezero-meanwhitecircularlysymmetriccomplexGaussiannoisewithvariance 2 ,canbewrittenas[14 ] y SISO = W N S z SISO = diag { h } x SISO + W N S e SISO C N S 1 (5) Thedatamodelin5 canalsorepresenttheOFDMsymbolsinthesignaleldandthepreamble.Equation5 canalsobewrittenasy SISO = diag { x SISO } h + W N S e SISO (6) Notethat5 isusefulforsymboldetectionwhereas6 )isusedforchannelestimation.Forthesakeofsimulationsimplicity,theequivalentchannelh (t) isoftenapproximatedbyanexponentiallyde-cayingFIRlterwithlengthL F [ 14 ],denotedash (t) L F = h (t) 0 h (t) 1 h (t) L F 1 T (7) Inthiscase,thereceivedsignalcanbeeasilysimulatedastheconvolutionofthechannelh (t) L F andthetransmittedsignal.Lett r betherootmeansquare(RMS)delayspreadingtimeand t n = t r /t S .ThenL F 10 t n +1,where x denotesthesmallestintegernotlessthanx .Forl F = 0,1,... L F 1,wehaveh (t) l F N 0, 1 e 1 /t n e l F /t n (8) ThischannelmodelisreferredtoastheChayatmodel[15 ].

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AMIMOSystemwithBackwardCompatibilityforOFDM-BasedWLANs699 t 10 t 9 t 8 t 7 t 6 t 5 t 4 t 3 t 2 t 1 t 10 t 9 t 8 t 7 t 6 t 5 t 4 t 3 t 2 t 1 GI2 GI2 T 1 T 1 T 2 T 2 10 0 8 = 8 s 1 6+2 3 2 = 8 s 1 6+2 3 2 = 8 s Longtrainingsymbolblock2Longtrainingsymbolblock1ShorttrainingsymbolsIEEE802. 11acompatiblepacketpreambleOFDMdataeldGI2T1 T 2 GI2 T 1 T 2 GI GI 0 8+3 2 = 4 s 0 8+3 2 = 4 s OFDMsymbolOFDMsymbolOFDMsymbolOFDMsymbolGIGI GI GI SignalSignalSignalld Figure3:ProposedMIMOpreambleandsignaleld)structure.Notethatoursymboltimingestimationmethod,whichwillbepresentedinSection3,worksequallywellforthechannelmodelsgivenbyboth4 )and7 ).OurMIMOchan-nelresponseestimationmethodalsoworksequallywellforbothmodels.Weuse7 togeneratechannelstosimplifyoursimulations.2.3.MIMOpreambledesignFortheIEEE802.11aSISOsystem,theshortOFDMtrainingsymbolscanbeusedtodetectthearrivalofthepacket,al-lowtheautomaticgaincontrol(AGC)tostabilize,computeacoarseCFOestimate,andobtainacoarsesymboltiming,whereasthelongOFDMtrainingsymbolscanbeusedtocal-culateaneCFOestimate,rethecoarsesymboltiming,andestimatetheSISOchannelresponse.TheMIMOsystemconsideredhereinhastwotransmitandtworeceiveantennassuchasacrosseddipolepairforboththetransmitterandthereceiver).Twopacketsaretrans-mittedsimultaneouslyfromthetwotransmitantennas.Wedesigntwopreambles,oneforeachtransmitantenna.Weas-sumethatthereceiverantennaoutputssu erfromthesameCFOandhasthesamesymboltiming.Tobebackwardcom-patiblewiththeSISOsystem,weusethesameshortOFDMtrainingsymbolsasintheSISOpreambleforbothoftheMIMOtransmitantennas,asshowninFigure3. AsforthelongOFDMtrainingsymbols,theyshouldbedesignedtosupporttheMIMOchannelresponseestimation.MIMOchannelresponseestimationhasattractedmuchre-searchinterestlately.Orthogonaltrainingsequencestendtogivethebestperformancesee,e.g.,[16 ]andthereferencestherein).Wealsoadoptthisideaoforthogonaltrainingse-quencesinourpreambledesign.Intheinterestofbackwardcompatibility,weusethesameT1 andT2 aswellasGI2)asfortheSISOsystemforbothoftheMIMOtransmitantennasbeforethesignaleld,asshowninFigure3.Afterthesignaleld,weuseT1 andT2 (andGI2)foronetransmitantenna,and T 1 and T 2 (and GI2)fortheother.Thisway,whenthesimultaneouslytransmittedpacketsarereceivedbyasin-gleSISOreceiver,theSISOreceivercansuccessfullydetectuptothesignaleld,whichisdesignedtobethesameforbothtransmitantennas.ThereservedbitinthesignaleldcantelltheSISOreceivertostopitsoperationwheneveraMIMOtransmissionfollowsorotherwisetocontinueitsoperation.ThelongOFDMtrainingsymbolsbeforeandafterthesig-naleldareusedintheMIMOreceiversforchannelesti-mation.AlthoughtheemploymentofanadditionalpairoflongOFDMtrainingsymbolscanincreasetheoverhead,thecorrespondinglossofe ciencyisnotsignicantforlargerpacket.ThereservedbitinthesignaleldcanalsoinformtheMIMOreceiverthatthetransmissionisaSISOone.Whenthisoccurs,theMIMOreceivercanmodifyitschannelesti-mationandthedatabitdetectionstepsslightly,asdetailedattheendofSections3 and 4 ,respectively.OtherMIMOpreambledesignoptionswithbackwardcompatibilityarepossible.Forexample,byexploitingthetransmit/receivediversities,wemaygetimprovedsymboltimingorCFOcorrection.However,theseimprovementsdonotnecessarilyresultinimprovedpacketerrorratePER).Hence,wepreferthestraightforwardMIMOpreamblede-signshowninFigure3. 2.4.MIMOdatamodelTostayasclosetotheIEEE802.11astandardaspossible,weuseinourMIMOsystemthesamescrambler,convolutionalencoder,puncturer,FDinterleaver,symbolmapper,pilotse-quence,andCPasspeciedinthestandard.Toimprovedi-versity,weaddasimplespatialinterleavertoscattereverytwoconsecutivebitsacrossthetwotransmitantennas.ThisspatialinterleavingisperformedbeforetheFDinterleaving.Considerthen S thsubcarrier(fornotationalconvenience,wedropthenotationaldependenceonn S below).ConsiderthecaseofN receiveantennas.(NotethatconsideringthegeneralcaseofN receiveantennasdoesnotaddextradi cultiesforthediscussionsbelow.)LetH denotetheMIMOchannelmatrixforthen S thsubcarrier:H = h 1,1 h 1,2 h 2,1 h 2,2 h N ,1 h N ,2 C N 2 ,(9whereh n m denotesthechannelgainfromthem thtransmitantennatothen threceiveantennaforthen S thsubcarrier.Lety denoteareceiveddatavectorforthen S thsubcarrier,whichcanbewrittenasy = Hx + e C N 1 ,10)wherex = [ x 1 x 2 ] T isthe2 1QAMsymbolvectorsentonthe n S thsubcarrierande N 0 2 I N istheadditivewhitecircularlysymmetriccomplexGaussiannoisewithvariance 2 .InSection4,wewillprovideasoftdetectorbasedonthismodel.

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700EURASIPJournalonAppliedSignalProcessing 3.CFO,SYMBOLTIMING,ANDCHANNELESTIMATIONInthissection,wepresentoursequentialCFO,symboltim-ing,andMIMOchannelresponseestimationapproachbasedonourMIMOpreambledesign.TheCFOcanbeesti-matedfromthesamplesoftwoconsecutivedatablocksduetotheperiodicinputs(theshortOFDMtrainingsymbolst 1 ... ,t 10 ).BecauseofthefactthattheCFOcanbeout-sidetheunambiguousrangemeasurablebythelongOFDMtrainingsymbols,wehavetoestimatetheCFOintwosteps:(a)acoarseCFOestimationusingtheshortOFDMtrain-ingsymbolsandthenb)aneCFOestimation,todeter-minetheresidueofthecoarseCFOcorrection,usingthelongOFDMtrainingsymbols.AfterestimatingandaccountingfortheCFO,wecanobtainthesymboltiming.Weestimatethesymboltimingalsointwosteps:thecoarsesymboltim-ingandnesymboltiming.TheformerisobtainedbyusingthelaterportionoftheshortOFDMtrainingsymbolsinthepacketpreamble.Thesymboltimingisobtainedbyus-ingthelongOFDMtrainingsymbolsbeforethesignaleld.Finally,weobtaintheMIMOchannelresponseestimate.Theparameterestimatesareobtainedintheorderpresentedbe-low.3.1.CoarseCFOestimationLetz n l = z ne n l )+ e n l ), n = 1, ... N ,denotethel thtimesampleofthesignalreceivedfromthen threceiveantenna,startingfromthemomentthatthereceiverAGChasbecomestationarythereceiverAGCisassumedtobecomestation-aryatleastbeforereceivingthelasttwoshortOFDMtrainingsymbolsandremainstationarywhilereceivingtheremainderofthepacket).InthepresenceofCFO,wehave[17 ] z ne n l + N C = z ne n l e j 2 N C n = 1, ... N ,11)where isthenormalizedCFOwithrespecttothesamplingfrequency),whichwestillrefertoasCFOforconvenience.Foreachreceiveantennaoutput,considerthecorrelationbe-tweentwoconsecutivenoise-freereceiveddatablocks,eachofwhichisoflengthN C .ThenthesumofthecorrelationsforallreceiveantennascanbewrittenasN n = 1 k + N C 1 l = k z ne n l z ne n l + N C = e j 2 N C N n = 1 N C 1 l = 0 z ne n l 2 Pe j 2 N C (12) where denotesthecomplexconjugateandk isanynon-negativeintegersuchthatz ne n k +2 N C 1)isasampleofthen threceiveantennaoutputduetotheinputtransmitan-tennaoutput)beingasampleoftheshortOFDMtrainingsymbolsoftheMIMOpacketpreamble.LetP S = N n = 1 N C 1 l = 0 z n l z n l + N C = Pe j 2 N C + e P (13) wheree P isduetothepresenceofthenoise.WecalculatethecoarseCFOas[18 ] C 1 2 N C P S ,14)where x denotestakingtheargumentofx WenextcorrecttheCFOusing C togetthedatasamplesz C n l ), n = 1,2,... N ,asfollows:z C n l = z n l e j 2 C (15) Correspondingly,wehaveP C S = P S e j 2 N C C (16) Inthesequel,weonlyconsidertheCFOcorrecteddatagivenabove.Fornotationalconvenience,wedropthesuperscriptof z C n l ), n = 1,2,... N 3.2.CoarsesymboltimingestimationNowwecanuseacorrelationmethod,modiedbasedontheapproachpresentedin[17 ]toestimatethecoarsesymboltiming.ThesymboltimingisreferredtoasthestartingtimesampleduetotheinputbeingthelongOFDMtrainingsym-bolT1 (beforethesignaleld).OncethestartingtimesampleduetothelongOFDMtrainingsymbolT1 isdetermined,wecandeterminethestartingtimesampleduetoeveryOFDMdatasymbolthereafter.AccordingtothespecicationoftheIEEE802.11astandardandthesamplingrateof20MHz,thetruesymboltimingT 0 is193,asshowninFigure4. From13 )and16 ),wenotethatthecorrelationaftertheCFOcorrection)isapproximatelythereal-valuedscalarP plusacomplex-valuednoise.Henceweproposetousethefollowingreal-valuedcorrelationsequenceforcoarsesymboltimingdetermination.Wecalculatethecorrelationsequenceinaniterativeformsimilartothecomplex-valuedapproachin[17 ]asfollows:P R k +1= P R k +Re N n = 1 z n k + N C z n k +2 N C z n k z n k + N C = P R k )+ N n = 1 z n k + N C z n k +2 N C z n k + z n k + N C z n k +2 N C z n k (17) wherebothRe )and denotetherealpartofacomplexentityand standsfortheimaginarypart.Westarttheiter-ationbyusingP R (0) = ReP S ).Notethatthereal-valuedcor-relationapproachgivenin(17 issuperiortotheabsolute-valuedonegivenin[17 ]sincetheformerusesfewercompu-tations,lowersthenoiselevel(variancereducedinhalfinthecorrelationsequence,anddecreasesclosertozerowhenthedatasamplesintheslidingdatablocksareduetothe

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AMIMOSystemwithBackwardCompatibilityforOFDM-BasedWLANs701 t 10 t 9 t 8 t 7 t 6 t 5 t 4 t 3 t 2 t 1 t 10 t 9 t 8 t 7 t 6 t 5 t 4 t 3 t 2 t 1 GI2 GI2 T 1 T 1 T 2 T 2 T 0 T F T C P R T P T C +3 N C T I N S h (t) Thresholds Figure4:Illustrationofsymboltimingdetermination.inputbeingGI2orthelongOFDMtrainingsymbolsfollow-ingtheshortOFDMtrainingsymbolsinthepreamble.WhensomeofthedatasamplesoftheslidingdatablocksaretakenfromthereceiveddataduetotheinputbeingGI2orthelongtrainingsymbolsfollowingtheshortOFDMtrain-ingsymbol,P R k willdropsince11 nolongerholds.Thispropertyisusedtoobtainthecoarsesymboltiming.LetT P ,asshowninFigure4,denotethersttimesamplewhenP R k dropstolessthanhalfofitspeakvalue.ThecoarsesymboltimingT C = T P + 3 2 N C + N C (18) isthecoarseestimateofthebeginningtimesampleduetotheinputbeingthelongOFDMtrainingsymbolT1 beforethesignaleld.Thesecondtermattheright-handsideoftheaboveequationisduetothefactthatP R k )willdroptoapproximatelyonehalfofitsmaximumvaluewhenthedatasamplesofthesecondhalfofthesecondofthetwoslidingblocksareduetotherstGI2inthepreambleasinput;thethirdtermisduetoonehalfofthelengthofGI2.Whenthechannelspreadingdelayt D = max { p l } isassumedtosatisfy t D N C ,onlythersthalfofGI2cansu erfromISI.HenceourgoalofcoarsetimingdeterminationistoplacethecoarsetimingestimatebetweenthetruetimingT 0 = 193 and T 0 N C = 177tomakeaccurateCFOestimationpossible.ThisexplainswhyweuseN C insteadof2N C forthethirdtermin18 ). 3.3.FineCFOestimationForeachreceiveantennaoutput,wecalculatethecorrelationbetweenthetwolongOFDMtrainingsymbolsbeforethesig-naleld.Wethensumthecorrelationsforallreceiveanten-nasasfollows:P L = N n = 1 N S 1 l = 0 z n l + T C z n l + T C + N S (19) ThentheCFOestimatecanbecomputedas F 1 2 N S P L (20) Wecanuse F inthesamewayas C tocorrecttheCFO.Weassumethatforthedataweusebelow, F hasbeenalreadycorrected.NotethattheaforementionedsimpleCFOestima-tionapproachmaynotbeoptimal.Forexample,theCFOestimationaccuracycouldbeimprovedbyusingthelongOFDMtrainingsymbolsafterthesignaleldaswell;how-ever,oursimpleCFOestimationapproachissu ciently accurateinthattheoverallsystemperformancecannolongerbeimprovedwithamoreaccurateCFOestimate,especiallywhenpilotsymbolsareexploited.Forthedatabitdetection,nomatterhowaccuratetheCFOestimateis,itcanneverbeperfectduetothepresenceofnoise.PilotsymbolsareusedtotracktheCFOresidualphaseforeachOFDMdatasymbolbeforedatabitdetection.Amaximumlikelihood(ML)CFOresidualtrackingschemeisgiveninAppendixC. 3.4.FinesymboltimingestimationWenowmoveontoobtainthesymboltimingbyusingthelongOFDMtrainingsymbolsbeforethesignaleld.ThesymboltimingisestimatedbyusingtheN datablocksoflengthN S ,startingfromthetimesampleT C +3 N C .Withthischoice,duetothefactthatT1 isidenticaltoT2 ,thedatablocksaremostlikelyduetotheinputbeingthesecondhalfofT1 andthersthalfofT2 ,evenwhenthecoarsesymboltiminghasalargeerror.Lety n denotetheN S -pointFFTofthedatablockfromthe n threceiveantennaandleth (t) n m betheequivalentFIRchannelinthetimedomainbetweenthem thtransmitan-tennaandthen threceiveantenna,n = 1,2,... N m = 1,2.Then,byneglectingtheexistenceoftheresidualCFO,y n can bewrittenas(cf.6 y n = X B W N S 2 m = 1 h (t) n m + W N S e n ,21)whereX B isadiagonalmatrixwiththe52knownBPSKsymbolsand12zeros,whichformtheT1 in Figure3,onthediagonal.SincetheMoore-PenrosepseudoinverseofX B is X B itselfandW N S /N 1 / 2 S isunitary,wegetanestimateofh (t) n = 2 m = 1 h (t) n m as

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702EURASIPJournalonAppliedSignalProcessing h (t) n = 1 N S W H N S X B y n (22) LetT I ,asshowninFigure4,denotetheindexoftherstele-mentof| h (t) |= N n = 1 | h (t) n | thatisabove1/ 3ofthemaximumvalueoftheelementsof N n = 1 | h (t) n | .Ourempiricalexperi-encesuggeststhatselectingthethresholdtobe1/ 3givesthebestresult.)ThenthesymboltimingT F isobtainedasT F = T C N C + T I 3 (23) Thesecondtermaboveisusedtocompensatefortheafore-mentioned3N C shiftduetothefactthatN S 3 N C = N C and thelasttermaboveischosentobe3toensurethatT F >T 0 occurswithverylowprobability.3.5.MIMOchannelresponseestimationAfterweobtainedT F ,wecannowestimatetheMIMOchan-nelresponse.Lety n ,1 denotetheN S -pointFFToftheaverageofthetwoconsecutiveblocks,eachofwhichisoflengthN S associatedwiththetwolongtrainingsymbolsbeforethesig-naleld,fromthen threceiveantenna.Lety n ,2 denotethecounterpartofy n ,1 afterthesignaleld.Then,forthen S th subcarrier,wehavey n ,1 x B h n ,1 + h n ,2 ,24)y n ,2 x B h n ,1 h n ,2 ,25)wherex B denotesthen S thdiagonalelementofX B y n i denotesthen S thelementofy n i i = 1,2,andwehavedroppedthedependenceonn S fornotationalsimplicity.Solving24 and25 yields h n ,1 = x B y n ,1 + y n ,2 2 ,26) h n ,2 = x B y n ,1 y n ,2 2 (27) WhenthereservedbitinthesignaleldindicatesaSISOtransmission,weonlyneedtoestimateh n ,1 n = 1,2,... N inawaysimilarto26 ). 4.ASIMPLEMIMOSOFTDETECTORWiththeCFO,symboltiming,andMIMOchannelresponsedeterminedandaccountedfor,wecanproceedtodetectthedatabitscontainedineachBLASTlayerandsubcarrieroftheOFDMdatasymbolsintheOFDMdataeld.Inthesequel,wepresentaverysimplesoftdetectorfortheMIMOsystem.NotethatthissoftdetectorcanbeusedinageneralsettingoftheBLASTsystemandhencewepresentitinageneralframe-workbasedonthedatamodelof10 ),whereH isassumedtobeN M and x tobeM 1.Weuse H toreplaceH in oursimulations.)ConsiderrsttheMLharddetectoroftheBLASTsystem.Forthedatamodelof10 ),theMLharddetectorisgivenby x = argminx C M 1 y Hx 2 ,28)where 2 denotestheEuclideannorm.Thecostfunctionin28 )canbewrittenas y Hx 2 = y H y + x H H H Hx y H Hx x H H H y = x H y H H H H H H x H y + y H y y H H H H H HH y (29) whereH = H H H 1 H H .Wenote,fromtheaboveequa-tion,thatbyignoringtheconstellationconstraintonx ,wecanobtainanunstructuredLSestimate x us of x ,whichisgivenby x us = H y = x + H e x + c (30) Notethat x us isthesoftdecisionstatisticthatweareinter-estedin.Werefertothissimpleschemeofobtainingasoftde-cisionstatisticastheMIMOsoftdetectionscheme.NotethatanecessaryconditionforH H H tobenonsingularisN M Alsonotethatc isstillGaussianwithzeromeanandcovari-ancematrixE cc H = 2 H H H = 2 H H H 1 (31) Duetotheuseoftheinterleaveranddeinterleaver,thedatabitscontainedinx areindependentofeachother.Byig-noringthedependenceamongtheelementsofc ,wecancon-sideronlythemarginalprobabilitydensityfunctionpdffortheelements x us m ), m = 1,2,... M ,of x us .Notethatanapproximationismadehere,whichcanleadtoperformancedegradation.However,thecomputationisgreatlysimpliedbytheapproximation.)LetH h T 1 h T 2 h T M C M N (32) Thenthem thelementofc m = 1,2,... M ,canbewrittenas c m = h T m e (33) Obviously,c m isstillGaussianwithzeromeanandvariance 2 m = E c m 2 = h m 2 2 (34) Theestimateoftheabovenoisevariance 2 canbeeasilyob-tainedviathedi erenceofthetwoconsecutiveblocksofthen threceiveantenna,fromwhichwegoty n ,1 (cf.24 )).Notethat 2 m alongwith x m us providethesoftinformationforthem th, m = 1,2,... M ,symbolin x us ,neededbytheViterbidecoder.Notealsothatthenoisescorrespondingtodi erentlayershavedi erentvarianceswhichmeansthatthesymbolscorrespondingtodi erentlayershavedi erentquality.ThisunbalancedlayerqualityisthereasonwhywehaveusedaspatialinterleaverbeforetheFDinterleaver.

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AMIMOSystemwithBackwardCompatibilityforOFDM-BasedWLANs703 NotethatforSISOsystemsweusuallyconsideranor-dinaryQAMsymbolastwoPAMsymbols(e.g.,a64-QAMsymbolcanbeconsideredastwo8-aryPAMsymbols)duetotheorthogonalitybetweentherealandimaginarypartsofaQAMsymbolaswellastheindependencebetweentherealandimaginarypartsoftheadditivecircularlysymmetricGaussianerror.AbitmetriccomputationschemeforPAMsymbolsispresentedinAppendixA.InAppendixB,weshowthattherealandimaginarypartsofc m areindependentofeachother.Hencewecansignicantlysimplifythebitmetriccomputationsbyexploitingtheseindependencies.TheminimummeansquareerrorMMSE)detectorisoftendeemedtobebetterthantheLS-basedone[12 ].Al-thoughthiscanbetruefortheconstantmodulusconstella-tions,suchasPSK,itisnotnecessarilytrueforQAMsym-bols,assuggestedbyoursimulationsduetothedi erentpowerlevelsoftheQAMsymbols.Hence,wedonotprovideanMMSEcounterpartoftheLS-basedsoftdetector.WhenthereservedbitinthesignaleldindicatesaSISOtransmission,theH in28 isinfactavector.Hencethe x us in 30 andtheE[cc H ]in(31 arescalars,andtheyarethesoftinformationusedasintheSISOsystemfordatabitdetection.5.NUMERICALEXAMPLESInthissection,weprovidenumericalexamplestodemon-stratethee ectivenessandperformanceofoursequentiales-timationmethodforCFO,symboltiming,andMIMOchan-nelresponsebasedonourMIMOpreambledesignaswellasthesimpleMIMOsoftdetector.IntheIEEE802.11astandard,themaximumtransmis-siondatarateis54Mbps;inthiscase,the64-QAMconstella-tionisusedandthechannelcodingrateisR = 3 / 4,whichcomesfrompuncturingtheR C = 1 / 2encodedsequencewiththepuncturingrateR P = 2 / 3.Weconsiderdoublingthemaximum54Mbpstransmissiondataratebyusingtwotransmitandtworeceiveantennas,thatis,M = N = 2.Inoursimulations,eachoftheMN = 4timedomainMIMOchannelsisgeneratedaccordingtotheChayatmodel;the4channelsareindependentofeachother.Duetothefactthat52outof64subcarriersareusedintheOFDM-basedWLANsystem,thesignal-to-noisera-tioSNR)fortheSISOsystemusedinthispaperisdenedas52 / (64 2 fortheconstellationswhoseaverageenergiesarenormalizedto1.WhereasfortheMIMOsystem,theSNRisdenedas52/ (128 2 i.e.,weusethesametotaltransmis-sionpowerfortheMIMOsystemasforitsSISOcounter-part).Werstprovideasimulationexampleforsymboltimingestimation.TwocurvesinFigure5showthe104 MonteCarlosimulationresultsofthecoarsesymboltimingestimatesfortheChayatchannelswitht r = 25andt r = 50nanoseconds,respectively,whenSNR= 10dB.Notethatthecoarsesymboltimingestimatesfallwithinthedesiredintervalwithahighprobability.Notealsothattheadversee ectofthecoarsesymboltimingestimatebeingsmallerthanT 0 N C = 177 isusuallynotsignicantsince,duetotheexponentiallyde-cayingpropertyofthechannels,theISIinthereceiveroutProbabilityDesiredintervalforcoarsetiming195 190 185 180 175 170 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 SymboltimingFinetiming(tr = 25ns)Finetiming(tr = 50ns)Coarsetiming(tr = 25ns)Coarsetiming(tr = 50ns) Figure5:Coarseandsymboltimingestimates.putduetotheinputbeingthelatterportionofthersthalfofGI2isminimal.Sincetheadversee ectofthecoarsesymboltimingestimatebeinglargerthanT 0 = 193isusuallytrou-blesomefortheCFOestimation,wepreferthecoarsetimingestimateT C tobewellaheadofT 0 .)TheothertwocurvesinFigure5showthe104 MonteCarlosimulationre-sultsofthesymboltimingestimatesfortheChayatchan-nelswitht r = 25andt r = 50nanoseconds,respectively,whenSNR = 10dB.Notethatoursimplesymboltimingap-proachgiveshighlyaccuratetimingestimates.Wethenprovideasimulationexampletoshowthee ectivenessoftheMIMOchannelestimatorandthePERper-formanceoftheMIMOsoftdetector.Onepacketconsistsof1000bytes.BasedontheIEEE802.11astandard,evenifonlyoneerroroccursinapacket,theentirepacketisdiscarded.)In Figure6,weshowthe104 MonteCarlosimulationresultsofthePERperformanceofoursoftdetectorasafunctionoftheSNRfortheMIMOsystem,witht r being50nanosec-ondsfortheChayatchannels,whenthetransmissiondatarateis108Mbps.Weconsidertwocases:thecaseofperfectchannelknowledgeandthecaseofestimatedchannelparam-eters.Fortheformercase,weassumetheexactknowledgeofCFO,symboltiming,andMIMOchannel,whereasforthelattercase,weusetheestimatesofalloftheaforementionedparametersobtainedwithoursequentialapproachfromtheMIMOpacketpreambleaswellastheCFOresidualphasetracking.Asareference,wealsogivethePERcurvesofthesoftdetectorfortheSISOsystem(withthedataratebeing54Mbps)asareference.Wenote,fromthePERcurves,thatboththeMIMOpreambledesignandthesequentialchannelparameterestimationalgorithmaree ectiveinthatthegapbetweenthePERcurvescorrespondingtotheperfectchannelknowledgecaseandtheestimatedchannelparametercasefortheMIMOsystemisnomorethanthatoftheSISOsystem.

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704EURASIPJournalonAppliedSignalProcessing PER30 29 28 27 26 25 24 23 22 21 20 10 1 10 0 SNRdB)Estimatedchannel2 2) Perfectknowledge(2 2) Estimatedchannel1 1) Perfectknowledge(1 1) Figure6:PERversusSNRatthe108MbpsdataratefortheChayatchannelswitht r = 50nanoseconds.NotealsothattheMIMOsoftdetectorise ectiveinthattheMIMOsystemneedsonly2to3dBextratotaltransmissionpowertokeepthesamePERwearemostlyinterestedinPERsbeing0.1,accordingtotheIEEE802.11astandardasitsSISOcounterpart,butwiththedataratedoubled.Finally,weshowtheperformancecomparisonsoftheSPDharddetector[8 ],theMIMOsoftdetector,aswellastheLSD-basedsoftdetector[11 ],anapproximationofSTBICM,theidealMIMOsoftdetector.Figure7givesPERcurvesob-tainedfrom104 MonteCarlosimulationsforthesedetec-torsasafunctionofSNRfortheMIMOsystem,withesti-matedchannelparameters.Thesimulationparametersarethesameasthoseinthepreviousexample.)NotethattheMIMOsoftdetectorismuchbetterthanSPDandisoutper-formedbyLSDintermsofPER.However,theMIMOsoftdetectorismuchmoree cientthanLSD.Wedidnotat-tempttooptimizeourMatlabsimulationcodes.Evenso,ourpreliminaryresultsindicatethattheLS-basedMIMOsoftdetectorrequiresonlyaboutonefthofthecomputationsneededbySPD.UnlikeSPD,whosesphereradiusshrinkswhenndingbettersolutions,LSDkeepsthesphereradiusconstant,whichmeansthatitiscomputationallymuchmoredemandingthanSPD,especiallywhenthesphereradiusislarge.Wedonothaveanexactpcomparison,yetwebe-lieveLSDps SPDops 5timesMIMOsoftdetectorps,whichmeansthattheLS-basedsoftdetectorcanbeor-dersofmagnitudemorecomputationallye cientthantheLSD-basedone.6.CONCLUDINGREMARKSWehaveproposedapreambledesignfortheMIMOsystemwithtwotransmitandtworeceiveantennas.ThisMIMO PER35 34 33 32 31 30 29 28 27 26 25 24 23 22 10 1 10 0 SNRdB)HardSPDSoftLSListSPD Figure7:PERversusSNRatthe108MbpsdataratefortheChayatchannelswitht r = 50nanoseconds.preambledesignisbackwardcompatiblewithitsSISOcoun-terpartasspeciedbytheIEEE802.11astandard.BasedonthisMIMOpreambledesign,wehavedevisedasequen-tialmethodfortheestimationofCFO,symboltiming,andMIMOchannelresponses.WehavealsoprovidedasimplesoftdetectorfortheMIMOsystembasedontheunstruc-turedLSapproachtoobtainthesoftinformationfortheViterbidecoder.Boththesequentialparameterestimationmethodandthesoftdetectorareverye cientandideallysuitedforreal-timeimplementations.Thee ectivenessofourmethodshasbeendemonstratedvianumericalexam-ples. APPENDICES A.BITMETRICCALCULATIONFORTHEQAMSYMBOLTomakethispaperself-contained,wedescribeinthisap-pendixbrieyourbitmetriccalculationmethodfortheQAMsymbol.NotethattherealandimaginarypartsofaQAMsymbolplustheadditivecircularlysymmetriccomplexGaussiannoiseareindependentofeachother.HencethesoftdecisionstatisticcorrespondingtoatransmittedQAMsym-bolcanbeeasilydividedintotherealandimaginaryparts,whichcorrespondtothesoftdecisionstatisticoftworealvaluedPAMsymbols.ThevarianceofthenoiseadditivetothePAMsymbolsishalvedascomparedtotheQAMsym-bols.Inviewofthis,weonlypresentthemethodforcalcu-latingthebitmetricforasymbolinthePAMconstellationR LetD i j ={ s : s R } denotethesetofallthepossiblePAMsymbolswiththei thbitv i = j i = 1,2,... ,log2 A/ 2, j = 0,1.TheformationofD i j dependsonthewaythePAMsymbolsarelabeled.Forexample,fortheGrayindexed8-ary

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AMIMOSystemwithBackwardCompatibilityforOFDM-BasedWLANs705 7 3 5 7 000011001100 Figure8:Illustrationof8-aryPAMsymbolswithGraylabelingandtheirpdfcurves.PAMconstellationshowninFigure8,wehaveD 1,0 ={7, 5, 3, 1 } ,(A.1wheretherstbitistheleftmostoneshowninFigure8. Then,foragivensoftinformationx ofthePAMsymbol,thebitmetricforv i isgivenby v i = log p v i = 1 x p v i = 0 x ,(A.2wherep v i = j x = p D i j x = s D i j p s | x = s D i j f x | s p s p x (A.3) withf x | s = 1 2 e x s 2 / 2 2 (A.4) beingthepdfgiventhesymbols andthevariance 2 ,asshowninFigure8.TheoccurrenceofeachsymbolinR is oftenassumedtobeequallylikely,thatis,p s = (1 / 2) B/ 2 foralls R .Inthiscase,wehavep v i = j | x = 1 2 B/ 2 p x s D i j f x | s ),(A.5)whichleadsto v i = log s D i ,1 e x s 2 / 2 2 log s D i ,0 e x s 2 / 2 2 (A.6) Tospeedupthebitmetriccalculationinpracticalap-plications,wecanmakeagridforx and toprecalculatealookuptableforthe v i s.Thebitmetriccalculationinoursimulationsisbasedonsuchatable.B.APROPERTYOFTHEc m IN 33 With x and x denotingtherealandimaginarypartsofx ,re-spectively,wegetfrom33 c = H + j H e + j e = H e H e + j H e + H e (B.1)Then c = H e H e c = H e + H e (B.2)Hence,wehaveE c c T = E H e H e e T H T + e T H T = E H e e T H T H e e T H T +E H e e T H T H e e T H T = 1 2 2 H H T H H T = 1 2 2 H H T H H T T (B.3)wherewehaveusedthefactthatE[ e e T ] = E[ e e T ] = 2 I / 2 andE[ e e T ] = E[ e e T ] = 0 .Equation(B.3impliesthatthediagonalelementsofE[ c c T ]arezeroandhenceE[ c m c m ] = 0, m = 1,2,... M C.PHASECORRECTIONUSINGPILOTSYMBOLSTheCFOcorrectionwillneverbeperfectinpracticeduetothepresenceofnoise.Hence,therewillbeaphaseerror for eachOFDMdatasymbolcausedbytheerrorintheCFOestimate F .Theerror increaseslinearlywithtime.Aswementionedearlier,eachOFDMdatasymbolcon-tainsfourknownpilotsymbols.Wedenotethesepilotsym-bolsbya4 1vectorp .Thepilotsymbolscanbeusedtocorrect foreachOFDMdatasymbolafterCFOcorrectionusing F .Lety (p) n bethevectorcontainingthecorrespond-ingfourelementsoftheFFToutputofanOFDMdatasym-bolintheOFDMdataeldreceivedfromthen thantenna,n = 1,2,... N .Let h (p) n m bethe4 1estimatedchannelvec-torfromtransmitantennam toreceiveantennan forthefourcorrespondingsubcarriers.LetP = diag { p } .Wehavey (p) n = e P 2 m = 1 h (p) n m + e (p) n n = 1,2,... N ,(C.1where{ e (p) n } N n = 1 arezero-meanwhitecircularlysymmetriccomplexGaussiannoisevectorsthatareindependentlyandidenticallydistributed.ThentheMLcriterionleadsto ML = argmin N n = 1 y (p) n e P 2 m = 1 h (p) n m 2 = N n = 1 2 m = 1 h (p) n m H P H y (p) n (C.2) ACKNOWLEDGMENTThisworkwassupportedinpartbytheNationalScienceFoundationGrantCCR-0097114andtheIntersilCorpora-tionContract2001056.

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706EURASIPJournalonAppliedSignalProcessing REFERENCES [1]R.vanNee,G.Awater,M.Morikura,H.Takanashi,M.Web-ster,andK.W.Halford,ewhigh-ratewirelessLANstan-dards,IEEECommunicationsMagazine,vol.37,no.12,pp.82,1999.[2]IEEE802.11a-1999,IEEEStandardforInformationtechnol-ogyTelecommunicationsandinformationexchangebe-tweensystemsLocalandmetropolitanareanetworkspecicrequirementsPart11:WirelessLANMediumAc-cessControlMAC)andPhysicalLayer(PHYspecica-tionsAmendment1:High-speedPhysicalLayerinthe5GHzband,1999.[3]A.F.Naguib,N.Seshadri,andA.R.Calderbank,ncreas-ingdatarateoverwirelesschannels,IEEESignalProcessingMagazine,vol.17,no.3,pp.76,2000.[4]G.J.Foschini,yeredspace-timearchitectureforwirelesscommunicationinafadingenvironmentwhenusingmulti-elementantennas,BellLabsTech.Journal,vol.1,no.2,pp.41,1996.[5]G.D.Golden,G.J.Foschini,R.A.Valenzuela,andP.W.Wol-niansky,DetectionalgorithmandinitiallaboratoryresultsusingtheV-BLASTspace-timecommunicationarchitecture,ElectronicsLetters,vol.35,no.1,pp.14,1999.[6]G.J.FoschiniandM.J.Gans,Onlimitsofwirelesscommu-nicationsinafadingenvironmentwhenusingmultiplean-tennas,WirelessPersonalCommunications,vol.6,no.3,pp.311,1998.[7]I.E.Telatar,Capacityofmulti-antennaGaussianchannels,EuropeanTransactionsonTelecommunications,vol.10,no.6,pp.585,1999.[8]O.Damen,A.Chkeif,andJ.-C.Belore,Latticecodedecoderforspace-timecodes,IEEECommunicationsLetters,vol.4,no.5,pp.161,2000.[9]X.Li,H.Huang,G.J.Foschini,andR.A.Valenzuela,Ef-fectsofiterativedetectionanddecodingontheperformanceofBLAST,inProc.IEEEGLOBECOM,vol.2,pp.10611066,SanFrancisco,Calif,USA,November2000.[10]A.M.Tonello,Space-timebit-interleavedcodedmodulationwithaniterativedecodingstrategy,inProc.IEEE52thVehic-ularTechnologyConference,vol.1,pp.473,Boston,Mass,USA,September2000.[11]B.M.HochwaldandS.tenBrink,Achievingnear-capacityonamultiple-antennachannel,IEEETransactionsonCom-munications,vol.51,no.3,pp.389,2003.[12]C.Z.W.HassellSweatman,J.S.Thompson,B.Mulgrew,andP.M.Grant,AcomparisonoftheMMSEdetectoranditsBLASTversionfortheMIMOchannel,inIEESeminaronMIMO:CommunicationsSystemsfromConcepttoImplemen-tations,pp.19/1/6,London,UK,December2001.[13]J.-J.vandeBeek,O.Edfors,M.Sandell,S.K.Wilson,andP.O.B orjesson,OnchannelestimationinOFDMsystems,inProc.IEEE45thVehicularTechnologyConference,vol.2,pp.815,Chicago,Ill,USA,July1995.[14]Z.WangandG.B.Giannakis,Wirelessmulticarriercom-munications,IEEESignalProcessingMagazine,vol.17,pp.29,May2000.[15]N.Chayat,Tentativecriteriaforcomparisonofmodulationmethods,IEEEP802.11-97/96,September1997.[16]E.G.LarssonandJ.Li,reambledesignformultiple-antennaOFDM-basedWLANswithnullsubcarriers,IEEE SignalProcessingLetters,vol.8,no.11,pp.285,2001.[17]T.M.SchmidlandD.C.Cox,obustfrequencyandtimingsynchronizationforOFDM,IEEETransactionsonCommu-nications,vol.45,no.12,pp.1613,1997.[18]J.Li,G.Liu,andG.B.Giannakis,Carrierfrequencyo set estimationforOFDM-basedWLANs,IEEESignalProcessingLetters,vol.8,no.3,pp.80,2001. JianhuaLiureceivedtheB.S.degreeinelec-tricalengineeringfromDalianMaritimeUniversity,Dalian,China,in1984,theM.S.degreeinelectricalengineeringfromtheUniversityofElectronicScienceandTech-nologyofChina,Chengdu,China,in1987,andthePh.D.degreeinelectronicengi-neeringfromTsinghuaUniversity,Beijing,China,in1998.FromMarch1987toFebru-ary1999,heworkedattheCommunica-tions,Telemetry,andTelecontrolResearchInstitute,Shijiazhuang,China,wherehewasanAssistantEngineer,Engineer,SeniorEn-gineer,andFellowEngineer.FromMarch1995toAugust1998,hewasalsoaResearchAssistantatTsinghuaUniversity.FromFebru-ary1999toJune2000,heworkedatNanyangTechnologicalUni-versity,Singapore,asaResearchFellow.SinceJune2000,hehasbeenaResearchAssistantintheDepartmentofElectricalandCom-puterEngineering,theUniversityofFlorida,Gainesville,workingtowardsaPh.D.degreemajoringinelectricalengineeringandmi-noringinStatistics.Hisresearchinterestsincludewirelesscommu-nications,statisticalsignalprocessing,andsensorarrayprocessing. JianLireceivedtheM.S.andPh.D.degreesinelectricalengineeringfromTheOhioStateUniversity,Columbus,in1987and1991,respectively.FromApril1991toJune1991,shewasanAdjunctAssistantProfes-sorwiththeDepartmentofElectricalEngi-neering,TheOhioStateUniversity,Colum-bus.FromJuly1991toJune1993,shewasanAssistantProfessorwiththeDepartmentofElectricalEngineering,UniversityofKen-tucky,Lexington.SinceAugust1993,shehasbeenwiththeDe-partmentofElectricalandComputerEngineering,UniversityofFlorida,Gainesville,wheresheiscurrentlyaProfessor.Hercurrentresearchinterestsincludespectralestimation,arraysignalprocess-ing,andtheirapplications.Dr.LiisamemberofSigmaXiandPhiKappaPhi.Shereceivedthe1994NationalScienceFoundationYoungInvestigatorAwardandthe1996O ceofNavalResearchYoungInvestigatorAward.ShewasanExecutiveCommitteeMem-berofthe2002InternationalConferenceonAcoustics,Speech,andSignalProcessing,Orlando,Florida,May2002.ShehasbeenanAs-sociateEditoroftheIEEETransactionsonSignalProcessingsince1999andanAssociateEditoroftheIEEESignalProcessingMag-azinesince2003.SheispresentlyamemberoftheSignalProcess-ingTheoryandMethods(SPTM)TechnicalCommitteeoftheIEEESignalProcessingSociety.