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System Design and Signal Processing for OFDM-Based Wireless Communication Systems

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Title:
System Design and Signal Processing for OFDM-Based Wireless Communication Systems
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LIU, JOANHUA ( Author, Primary )
Copyright Date:
2008

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Subjects / Keywords:
Antennas ( jstor )
Blasts ( jstor )
Decryption ( jstor )
Estimate reliability ( jstor )
Estimation methods ( jstor )
Signal processing ( jstor )
Signals ( jstor )
Simulations ( jstor )
Supernova remnants ( jstor )
Systems design ( jstor )

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University of Florida
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University of Florida
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Copyright Joanhua Liu. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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8/31/2005
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TABLEOFCONTENTS page ACKNOWLEDGMENTS.............................iv LISTOFTABLES.................................ix LISTOFFIGURES................................x ABSTRACT....................................xv CHAPTER 1INTRODUCTION..............................1 1.1OFDM.................................21.2 An OFDM-Based SISO System . . . . . . . . . . . . . . . . . . . 5 1.2.1OverviewoftheIEEE802.11aStandard...........5 1.2.2OFDMDataSymbolGeneration...............7 1.2.3DetectionoftheDataSymbols................9 1.3ScopeoftheWork...........................10 1.3.1SISOSystemsoverTime-InvariantChannels........10 1.3.2MIMOSystemsoverTime-InvariantChannels.......12 1.3.3SISOSystemsoverTime-VaryingChannels.........13 1.3.4MIMOSystemsoverTime-VaryingChannels........14 1.4LiteratureSurveywithWorkDescription..............14 1.4.1HistoryofOFDM.......................14 1.4.2SignalProcessingforSISOSystemsoverTime-InvariantChannels...............................16 1.4.3SystemDesignandSignalProcessingforMIMOSystems overTime-InvariantChannels................18 1.4.4SystemDesignandSignalProcessingforSISOSystemsover Time-VaryingChannels....................26 1.4.5SystemDesignandSignalProcessingforMIMOSystems overTime-VaryingChannels.................28 1.5OrganizationoftheDissertation...................30 2SISOSYSTEMSOVERTIME-INVARIANTCHANNELS........31 2.1ChannelandDataModel.......................33 2.1.1ChannelModel.........................34 2.1.2DataModel..........................38 2.2SequentialChannelParameterEstimation.............39 v

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2.2.1CoarseCFOEstimation....................40 2.2.2CoarseSymbolTimingEstimation..............41 2.2.3FineCFOEstimation.....................44 2.2.4FineSymbolTimingEstimation...............44 2.2.5ChannelResponseEstimation................46 2.2.6NumericalExamplesforSymbolTimingEstimation....47 2.3ErrorReduction............................49 2.3.1MLPhaseTracking......................49 2.3.2LSPhaseFitting........................50 2.3.3SamplingClockSynchronization...............51 2.3.4Semi-BlindChannelEstimation...............53 2.3.5NumericalExamples......................56 2.4ConcludingRemarks.........................62 2.5Appendixes..............................63 2.5.1DerivationsofEquations(2.8)and(2.9)...........63 2.5.2Equivalenceof(2.28)withtheMatchedFilterMethodof vanNeeandPrasad[120]...................65 2.5.3ASimpleSISOSoft-Detector.................66 2.5.4BitMetricCalculationfortheQAMSymbol........67 2.5.5MLPhaseFitting.......................68 2.5.6ANoteontheCFOEectSimulation............70 3MIMOSYSTEMSOVERTIME-INVARIANTCHANNELS.......71 3.1OverviewofSTBC..........................73 3.2ChannelCodingandDataModelforaGenericBLASTSystem..75 3.2.1ChannelCoding........................75 3.2.2DataModel..........................78 3.3SystemDesign.............................79 3.3.1PreambleDesignfortheMIMOSystems..........79 3.3.2OFDMDATAFieldDesignfortheSTBCSystem.....81 3.3.3OFDMDATAFieldDesignfortheMIMOBLASTSystem83 3.4ChannelParameterEstimation....................84 3.4.1ANoteonCFOandSymbolTimingEstimation......84 3.4.2MIMOChannelResponseEstimation............85 3.5ErrorReduction............................86 3.5.1MLPhaseTrackingfortheSTBCSystem..........86 3.5.2MLPhaseTrackingfortheBLASTSystem.........87 3.5.3Semi-BlindChannelResponseEstimationfortheSTBCSystem...............................88 3.5.4Semi-BlindChannelResponseEstimationfortheBLAST System.............................90 3.6DetectionfortheSTBCSystem...................92 3.6.1ASimpleSTBCSoft-Detector................92 3.6.2NumericalExamples......................94 vi

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3.7AnLS-BasedSimpleSoft-DetectorfortheBLASTSystem....97 3.7.1TheSoft-Detector.......................99 3.7.2NumericalExamples......................102 3.8AHybridSoft-DetectorfortheBLASTSystem..........105 3.8.1CommentsontheLSD-andLS-basedSoft-Detectors....105 3.8.2TheHybridSoft-Detector...................107 3.8.3SimulationResults.......................111 3.9AnIterativeSoft-DetectorfortheBLASTSystem.........113 3.9.1TheCLSD-BasedSoft-Detector...............114 3.9.2AlgorithmoftheTurboProcessing..............119 3.9.3SimulationResults.......................120 3.10Line-of-SightProblemwiththeBLASTSystems..........123 3.11ConcludingRemarks.........................127 3.12Appendixes..............................128 3.12.1APropertyofthe c m in(3.55)................128 3.12.2DerivationoftheSTBICM-BasedSoft-Detector.......129 4SISOSYSTEMSOVERTIME-VARYINGCHANNELS.........132 4.1PacketDesignandDataModel...................133 4.1.1E2213-02Standard......................134 4.1.2PacketDesign.........................134 4.1.3ChannelModel.........................135 4.1.4DataModel..........................137 4.2SignalProcessingAlgorithmsforChannelTrackingattheReceiver138 4.2.1DecisionFeedbackFDChannelResponseEstimation....139 4.2.2WeightedPolynomialFitting.................141 4.2.3WeightedPolynomialFittingwithSymbolTimingCompensation..............................142 4.2.4ChannelResponsePrediction.................143 4.3NumericalExamples.........................144 4.4ConcludingRemarks.........................150 5MIMOSYSTEMSOVERTIME-VARYINGCHANNELS.........151 5.1MIMOPreambleDesignandChannelEstimation.........152 5.1.1MIMOPreamblewithModiedClassITrainingSymbols.154 5.1.2ChannelEstimation......................155 5.1.3NumericalExamples......................158 5.2ChannelUpdating/TrackingfortheMIMOSystems........160 5.2.1PacketDesignfortheSTBCSystems............160 5.2.2ChannelUpdating/TrackingfortheSTBCSystems....161 5.2.3NumericalExamples......................162 5.3ConcludingRemarks.........................165 vii

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6SUMMARYANDFUTUREWORK....................166 6.1Summary...............................166 6.1.1SISOSystemsoverTime-InvariantChannels........166 6.1.2MIMOSystemsoverTime-InvariantChannels.......168 6.1.3SISOSystemsoverTime-VaryingChannels.........169 6.1.4MIMOSystemsoverTime-VaryingChannels........170 6.2FutureWork..............................171 REFERENCES...................................172 BIOGRAPHICALSKETCH............................183 viii

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LISTOFTABLES Table page 1{1Rate-dependentparametersspeciedbytheIEEE802.11astandard..8 2{1Comparisonoftheaveragenumbersofopsbetweenanexistingchannel parameterestimationmethodandtheproposedmethodforaSISO system....................................63 5{1Frequency-domaintrainingsequencesforthemodiedClassItraining symbolsfortheMIMOsystemsoverthetime-varyingchannels.....155 6{1Summaryofcompletedtasks.......................167 ix

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LISTOFFIGURES Figure page 1{1BasiccomponentsofthetransmitterofanOFDM-basedsystem....3 1{2IllustrationofsubchannelsofanOFDM-basedsystem:(a)onesubchannel;(b)multiplesubchannelsofaatfadingchannel;(c)multiple subchannelsofafrequency-selectivefadingchannel...........4 1{3ThepacketstructuredescribedbytheIEEE802.11astandard.....6 1{4Diagramofthetransmitter(upperpart)andreceiver(lowerpart)of anIEEE802.11aconformableSISOsystem...............7 2{1ThepacketstructurespeciedbytheIEEE802.11astandard.....32 2{2IllustrationsoftheanalogwaveformsforOFDMdatasymbolsandthe outputofamultipathchannel.......................34 2{3Illustrationoftheamplitudeoftwotypesofequivalentdiscretechannel responseforamultipathchanneldescribedby h ( t )= ( t 2 : 5 t S ) j 0 : 5 ( t 4 : 8 t S ):(a)TypeA;(b)TypeB.................36 2{4Illustrationofthediscontinuitiesintheamplitudeofthefrequencydomainchannelresponseforamultipathchanneldescribedby h ( t )= ( t 2 : 5 t S ) j 0 : 5 ( t 4 : 8 t S ).......................37 2{5Illustrationofdenitionofvarioussymboltimingsandthedeterminationofthesesymboltimings........................42 2{6Correlationvaluecomparisonforthereal-valuedcalculationalgorithm withtheabsolute-valuedcalculationalgorithmforatime-invariant channelwith t r =50nsforaSISOsystem................43 2{7Probabilitydistribution(orhistogram)ofcoarseandnesymboltimingestimateswhenSNR=10dBfortime-invariantchannelsgenerated accordingtotwochannelmodelswithvarious t r 'sforaSISOsystem: (a)themodiedexponentialchannelmodel;(b)theexponentialchannelmodel..................................47 2{8Probabilitydistributionofcoarseandnesymboltimingestimatesfor time-invariantchannelswith t r =100nsforaSISOsystematvarious SNRs:(a)coarsesymboltiming;(b)nesymboltiming........48 x

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2{9PERversusSNRcomparisonforcoarsesymboltimingandnesymbol timingfortime-invariantchannelswithvarious t r 'sforaSISOsystem atvariousdatarates:(a)datarate=12Mbps;(b)datarate=24 Mbps;(c)datarate=54Mbps......................57 2{10PERversusSNRfortime-invariantchannelswith t r =50nsforaSISO systematvariousdatarates:(a)datarate=12Mbps;(b)datarate =24Mbps;(c)datarate=54Mbps...................58 2{11PERversusSNRinthepresenceofsamplingclockerrorfortimeinvariantchannelswith t r =50nsforaSISOsystematvariousdata rates:(a)datarate=12Mbps;(b)datarate=24Mbps;(c)datarate =54Mbps.................................60 2{12PERversusSNRfortime-invariantchannelswith t r =50nsforrealtimeprocessinginthepresenceofsamplingclockerrorforaSISO systematvariousdatarates:(a)datarate=12Mbps;(b)datarate =24Mbps;(c)datarate=54Mbps...................62 2{13Illustrationof8-aryPAMsymbolswithGraylabelingandtheirpdf curves....................................69 3{1DiagramofaMIMOsystem........................72 3{2DiagramofaBLASTtransmitteremployingconvolutionalchannel coding....................................77 3{3DiagramofaBLASTreceiveremployingViterbialgorithmforconvolutionalchannelcoding...........................77 3{4PacketpreamblesfortheWLANsystems:(a)thestandardizedSISO preamble;(b)the(proposed)MIMOpreamble..............81 3{5OFDMDATAeldsfortheSISOandSTBCsystems:(a)SISO;(b) STBC....................................82 3{6DiagramofanOFDM-basedSTBCtransmitterbasedonasimple OFDMdatasymbolgeneratingscheme..................83 3{7OFDMDATAeldsfortheSISOandBLASTsystems:(a)SISO;(b) BLAST...................................84 3{8PERversusSNRforaSTBCsystemwithtwotransmitantennasand onereceiveantennafortime-invariantchannelswith t r =50nsat variousdatarates:(a)datarate=12Mbps;(b)datarate=24Mbps; (c)datarate=54Mbps..........................95 xi

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3{9PERversusSNRforaSTBCsystemwithtwotransmitantennasand tworeceiveantennasfortime-invariantchannelswith t r =50nsatthe 24Mbpsdatarate.............................97 3{10PERversusSNRforaBLASTsystemwithtwotransmitantennas andtworeceiveantennasusingtheLS-basedsoft-detectorfortimeinvariantchannelswith t r =50nsatvariousdatarates:(a)datarate =24Mbps;(b)datarate=48Mbps;(c)datarate=108Mbps....103 3{11PERcomparisonoftheLS-basedsoft-detectorwithtwootherdetectors foraBLASTsystemwithtwotransmitantennasand2receiveantennas asafunctionofSNRforthetime-invariantchannelswith t r =50ns atthe108Mbpsdatarate(withestimatedchannelparameters)....105 3{12PERcomparisonofthehybridsoft-detectorwithothersoft-detectors foraBLASTsystemwithtwotransmitantennasand2receiveantennas asafunctionofSNRforthetime-invariantchannelswith t r =50ns atthe108Mbpsdatarate(withperfectchannelknowledge)......112 3{13ProbabilitiesofusingLSDinthehybridandCN-hybridsoft-detectors foraBLASTsystemwithtwotransmitantennasandtworeceiveantennasasafunctionofSNRfortime-variantchannelswith t r =50ns atthe108Mbpsdatarate.........................113 3{14Diagramofthereceiverwithiterativedetection/decodingforanOFDMbasedBLASTsystem...........................115 3{15DiagramofthereceiverwithturboprocessingforanOFDM-based BLASTsystem...............................119 3{16PERversusSNRforaBLASTsystemwithtwotransmitantennasand tworeceiveantennasusingtheiterativeLSD-basedsoft-detectorwith variousiterationsfortime-invariantchannelswith t r =50nsatthe 108Mbpsdatarate(withperfectchannelknowledge)..........121 3{17PERversusSNRforaBLASTsystemwithtwotransmitantennas andtworeceiveantennasusingtheiterativeCLSD-basedsoft-detector withvariousiterationsfortime-invariantchannelswith t r =50nsat the108Mbpsdatarate(withperfectchannelknowledge).......122 3{18PERcomparisonoftheiterativeCLSD-andLSD-basedsoft-detectors with4iterationsaBLASTsystemwithtwotransmitantennasandtwo receiveantennasasafunctionofSNRfortime-invariantchannelswith t r =50nsatthe108Mbpsdatarate(withperfectchannelknowledge).123 xii

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3{19PERversusSNRforaBLASTsystemwithtwotransmitantennasand tworeceiveantennasusingthenon-iterativeLSD-basedsoft-detector withvariousiterationsofchannelupdatingfortime-invariantchannels with t r =50nsatthe108Mbpsdatarate................124 3{20PERversusSNRforaBLASTsystemwithtwotransmitantennasand tworeceiveantennasusingtheturboprocessingwithvariousiterations fortime-invariantchannelswith t r =50nsatthe108Mbpsdatarate.125 3{21PERversusSNRcomparisonofaBLASTsystemwithtwotransmit antennasandtworeceiveantennastoaSISOsystematrespective maximumdataratesusingvariousprocessingalgorithmsandunder variousconditionsfortime-invariantchannelswith t r =50ns.....126 3{22PERversustransmitantennaangularspacingforaBLASTsystem withtwotransmitantennasandtworeceiveantennasusingthenoniterativesoft-detectorsfortime-invariantRicianfadingchannelswith t r =50nsand K =20(withestimatedchannelparameters)atthe30 dBSNRandthe108Mbpsdatarate...................127 4{1ThepacketstructureforaSISOsystemasspeciedbytheE2213-02 standard...................................134 4{2SubpacketstructureoftheOFDMDATAelddesignedfortheSISO systemsovertiming-varyingchannels...................135 4{3Real-timechannelresponseupdatingbyusingthedetected/decoded databitsinasubpacket..........................139 4{4MSEsofchannelestimationforaSISODSRCsystemusingPFand MPFasfunctionsof P fortime-invariantchannelswith t r =100nsat variousSNRsatthe27Mbpsdatarate..................145 4{5MSEsofchannelestimationforaSISODSRCsystemusingDFFE,PF, MPF,andMPF/PasfunctionsofSNRfortime-invariantchannelswith t r =100nsatthe27Mbpsdatarate...................146 4{6MSEsofchannelestimationforaSISODSRCsystemusingMPFand MPF/Pasfunctionsofthemovingspeedforquasi-statictime-varying channelswith t r =100nsatvariousSNRsatthe27Mbpsdatarate.147 4{7PERversusSNRforaSISODSRCsystemconformingtotheE221302 standardforvariousmovingspeedsforquasi-statictime-varyingfrequencyselectivefadingchannelswith t r =100nsatthe27Mbpsdatarate..148 4{8PERversusSNRforaSISODSRCsystemusingthesegmentedpacket designandchannelupdating/trackingbyDFFE,PF,MPF,andMPF/P fortime-invariantchannelswith t r =100nsatthe27Mbpsdatarate.149 xiii

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4{9PERversusSNRforaSISODSRCsystemusingthesegmentedpacket designandchannelupdating/trackingbyMPF/Pforvariousquasistatictime-varyingchannelswith t r =100nsandvariousmoving speedsatthe27Mbpsdatarate.....................150 5{1AshortMIMOpreamblewithonlyonepairoflongOFDMtraining symbolsfortheOFDM-basedMIMODSRCsystems..........155 5{2MSEsofchannelestimationforaMIMODSRCsystemusingPF/I andML-FIRasfunctionsof P withashortMIMOpreambledesign fortime-invariantrealisticOFDMchannelswith t r =100nsatvarious SNRs(withperfectsymboltiming)....................159 5{3MSEsofchannelestimationforaMIMODSRCsystemusingPF/I (with P =5)andML-FIR(with P =19)asfunctionsofSNRwitha shortMIMOpreambledesignfortime-invariantrealisticOFDMchannelswith t r =100ns(withperfectsymboltiming)...........160 5{4PERversusSNRforanSTBCDSRCsystemwithtwotransmitantennasandonereceiveantennausingvariouscombinationsofoursystem designsandsignalprocessingalgorithmsfortime-invariantchannels with t r =100nsatthe27Mbpsdatarate................164 5{5PERversusSNRforanSTBCDSRCsystemwithtwotransmitantennasandonereceiveantennausingacombinationofoursystemdesigns andsignalprocessingalgorithmsforvariousquasi-statictime-varying channelswith t r =100nsandvariousmovingspeedsatthe27Mbps datarate..................................165 xiv

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Orthogonalfrequency-divisionmultiplexing(OFDM)isanupandcomingmod-ulationtechniqueforcommunications.Herein,weconsidervarioussystemdesignandsignalprocessingissuesforOFDM-andpacket-basedSISO(single-input/single-output)andMIMO(multiple-input/multiple-output)wirelesscommunicationsys-temsovertime-invariantandtiming-varyingfrequency-selectivefadingchannels.Inparticular,weconsider|andmakecontributionsin|thefollowingrelatedparts. P1.SISOsystemsovertime-invariantchannels.Wedeviseorselectsignalpro-cessingalgorithmsthatcanprovidebenettotheoverallsystemperformanceandcanbeecientlyimplementedinreal-time.Specically,rst,weprovideasequen-tialmethodfortheestimationofcarrierfrequencyoset(CFO),symboltiming,andchannelresponsebyexploitingthestructureofthepacketpreamble.Second,weaccountforvariouserrors:theresidueCFOinducedphaseerror,thesamplingclockinducedtimedelayerror,andthechannelestimationerror. P2.MIMOsystemsovertime-invariantchannels.WedesignpacketstructuresanddevisesignalprocessingalgorithmsbasedonthesedesignsforMIMOsystemsincludingtheSTBC(space-timeblockcoding)systemandtheBLAST(Bell-labs'xv

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P3.SISOsystemsovertime-varyingchannels.Wedesignpacketstructureanddevisesignalprocessingalgorithmstoaddresstheproblemofhigh-ratetransmissionovertime-varyingchannels.Forthepacketdesign,wesegmentanentirepacketintomultiplesubpacketstotrackthechannelvariationovertime.Eachsubpacketcontainsazerotailsequencetoresetthechannelencodersothatthedetection/decodingresultofeachsubpacketcanbeusedtoupdatethechannelresponseforthissubpacket.Atthereceiver,weuseweightedpolynomialttingtoimprovethechannelupdatingaccuracyandthechannelpredictioneectivenesstoimprovethetrackingaccuracyforthechannelresponseforeachsubpacket. P4.MIMOsystemsovertime-varyingchannels.Wedesignpacketstructuresanddevisesignalprocessingalgorithmsbasedonthesedesignstoaddresstheproblemofhigh-robustness/high-rateMIMOtransmissionoverthetime-varyingchannels.ThesedesignsandsignalprocessingalgorithmsarebasedonourpacketdesignandsignalprocessingalgorithmsofP1,P2,andP3.xvi

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Orthogonalfrequency-divisionmultiplexing(OFDM)isanupandcomingmod-ulationtechniqueforcommunications.Itisespeciallyusefulforhigh-speedwire-lesscommunicationswhichusewidebandwidth,wherethechannelscanexperiencefrequency-selectivefading.Itsusefulnessismainlyduetoitssuperiorabilityofmit-igatingoreliminatingtheimpactsoffrequency-selectivefadingchannelscausedbymultipath.ThisusefulnesscanbeseenfromtheemploymentofOFDMinagoodspectrumofapplications.First,OFDMhasbeenusedinimplementingdigitalaudiobroadcasting(DAB)systems[24]anddigitalvideobroadcasting(DVB)systems[99].Second,OFDMhasbeenselectedasthebasisfortheairinterfaceforseveralnewhigh-speedwirelesslocalareanetwork(WLAN)|alsoknownasWi-Fi|standards[119]includingIEEE802.11a[49],IEEE802.11g[50],andHIPERLAN/2[30].Third,OFDMhasbeenselectedasthebasisfortheairinterfaceforthenewbroadbandwire-lessaccess(WiMax)standardIEEE802.16[51].Fourth,OFDMhasbeenselectedasthebasisfortheairinterfaceforthededicatedshortrangecommunication(DSRC)standardE2213-02[4].Fifth,OFDMisastrongcandidateforanumberoffuturestandards,suchasthemobilebroadbandwirelessaccess(MBWA)standard[56]. ThisdissertationmainlyconcernstheOFDM-basedwirelesscommunicationsys-temsthataddressthecurrentandfutureapplications,includingtheaforementionedWLAN,MiMax,DSRC,andMBWA.Weconsidervarioussystemdesignandsignalprocessingissuesforthesesystems,whicharesingle-input/single-output(SISO)sys-tems,aswellastheirmultipleantennacounterparts,i.e.,themultiple-input/multiple-output(MIMO)systems.Specically,fortheOFDM-basedWLANandMiMaxap-plications,wherethechannelsareconsideredfrequency-selective(i.e.,multipath)1

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Beforemovingontointroducethescopeoftheworkofthisdissertation,letusgiveabriefintroductionofOFDM.1.1OFDM

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Figure1{1:BasiccomponentsofthetransmitterofanOFDM-basedsystem. ForanOFDM-basedSISOcommunicationsystem,thesingledatastreamisde-multiplexed,byserialtoparallel(S/P)conversion,intomultiplesubstreamsofdata,eachofwhichistransmittedonasubchannelcenteredatasubcarrier.Toavoidtheintercarrierinterference(ICI),thesesubchannelsshouldbeorthogonaltoeachother,hence,thetermOFDM|orthogonalfrequency-divisionmultiplexing.OFDMcanbeecientlyimplementedbyusingthefastFouriertransform(FFT)[22,94]andtheinverseFFT(IFFT).Theorthogonalityandeciencycanbesimultaneouslyguar-anteedbyutilizingIFFT(atthetransmitter)andFFT(atthereceiver).Figure1{1showsthebasiccomponentsofthetransmitterofanOFDM-basedsystem.Figures1{2(a),(b),and(c)showasubchannel,multiplesubchannelsofaatfadingchannel,andmultiplesubchannelsofafrequency-selectivefadingchannel,respectively.Itcanbeseenthateveniftheentirechannelexperiencesfrequency-selectivefading,thesubchannelscanstillbeseenasexperiencingatfading. ByusingOFDM,asingledatasymbol,whichoccupiesthewholebandwidthandthusisshortinduration,isreplacedbymultipleparalleltransmitteddatasymbols,whichhavelongerduration.ThisincreaseddurationiseectiveinmitigatingtheISIeectofmultipathchannelsincetheintervalofISI,whichcouldbeconsiderablecomparedtothesymboldurationforthesinglecarriercase,canbeonlyasmallfractionofthesymboldurationforOFDM.Byaddingacyclicprex(CP),aswillbedetailedinthefollowingsection,theeectofISIcanbecompletelyremoved,providedthattheCPislongerthanthespreadtime-delayofthemultipathchannelandacorrectsymboltimingisavailable. ThemainadvantagesofOFDMareasfollows.

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Figure1{2:IllustrationofsubchannelsofanOFDM-basedsystem:(a)onesubchan-nel;(b)multiplesubchannelsofaatfadingchannel;(c)multiplesubchannelsofafrequency-selectivefadingchannel. Yet,OFDMhasthefollowingdisadvantages.

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Inthenextsection,wegiveamoredetaileddescriptionofanOFMD-basedsys-tem.NotethatIEEE802.11g,HIPERLAN/2,andIEEE802.16areverysimilartoIEEE802.11aintermsofOFDMdatasymbolgenerationanddatabitsdetec-tion/decoding.NotealsothatE2213-02isdesignedbasedonIEEE802.11a.Hence,weuseIEEE802.11atoexemplifyourpresentation.1.2AnOFDM-BasedSISOSystem

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Figure1{3:ThepacketstructuredescribedbytheIEEE802.11astandard. TheinformationcarryingdataareencodedintheOFDMDATAeld.Theen-codingprocess,asdemonstratedintheupperpartofFigure1{4,isasfollows.AblockofsourcedatabitsisrstscrambledandthenconvolutionallyencodedbyanindustrialstandardconstraintlengthK=7,rate1/2encoder,whichhasgenerationpolynomialsg0=(133)8andg1=(171)8.Theencodedoutputisthenpuncturedac-cordingtothetransmissiondatarate(seeTable1{1)andissegmentedintosubblocksofdatabitsoflengthNCBPS(thenumberofcodedbitsperOFDMdatasymbol,whichisdeterminedbythetransmissiondatarate),eachofwhichcorrespondstoanOFDMdatasymbol.Thedataineachsubblockarerstinterleavedamongthesubcarriersandthenmapped(ingroupsofNBPSCbits)intoA-QAM(quadratureamplitudemodulation,NBPSC=log2A)orPSK(phase-shiftkeying)symbols,whichareusedtomodulatethedierentdatacarryingsubcarriers.EachOFDMdatasymbolintheOFDMDATAeldemploysNS=64subcarriers,48ofwhichareusedfordatasym-bolsand4forpilottones.Therearealso12nullsubcarrierswithoneinthecenterandtheother11onthetwoendsofthefrequencyband.TheOFDMdatasymbols,eachofwhichconsistsofNS=64samples,areobtainedviatakingIFFTofthedatasymbols,pilottones,andnullsontheseNSsubcarriers.Toeliminatetheintersymbolinterference(ISI),eachOFDMdatasymbolisprecededbyaCPorguardinterval(GI),whichcontainsthelastNCsamplesoftheOFDMdatasymbol. TheSIGNALeldcontainstheinformationincludingthetransmissiondatarateanddatalengthofthepacket.Theinformationisencodedin16databits.Thereisalsoareservedbit(forfutureuse)andaparitycheckbit.These18bits,paddedwith6zeros,arethenconvolutionallyencoded(bythesameencoderasfortheOFDM

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Figure1{4:Diagramofthetransmitter(upperpart)andreceiver(lowerpart)ofanIEEE802.11aconformableSISOsystem.DATAeld)toobtaina48bitencodedsequence.Theencodedsequenceistheninterleavedandusedtomodulatethe48datacarryingsubcarriersviaBPSK(binaryphase-shiftkeying).TheSIGNALeldconsistsof64samplesandisobtainedviatakingIFFTofthese48BPSKsymbols,4pilottones,and12nulls.Also,thereisaCPoflengthNCtoseparatethepreamblefromtheSIGNALeld.1.2.2OFDMDataSymbolGeneration beavectorofNSsymbols,wherexk;nS,nS=1;2;:::;NS,isthesymbolmodu-latingthenSthsubcarrier,whichisequalto0forthenullsubcarriers,1or1

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Datarate(Mbps) Constellation Codingrate BPSK 1/2 1 48 24 9 BPSK 3/4 1 48 26 12 QPSK 1/2 2 96 48 18 QPSK 3/4 2 96 72 24 16-QAM 1/2 4 192 96 36 16-QAM 3/4 4 192 144 48 64-QAM 2/3 6 288 192 54 64-QAM 3/4 6 288 216 where()Hdenotestheconjugatetranspose.ToeliminateISI,skisprecededbyaCPorguardinterval(GI)sk;CformedusingthelastNCelementsofsk. TheshortandlongOFDMtrainingsymbolsinthepacketpreambleandtheOFDMsymbolintheSIGNALledcanbegeneratedsimilarly. BystackingthepacketpreambleandtheOFDMsymbols(togetherwiththeircorrespondingCPs)intheSIGNALandOFDMDATAelds,weobtaintheentirepackets2C(5+K)(NS+NC)1,asshowninFigure1{3.

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whereHC=266666666664h0hNS1h2h1h1h0h3h2...............hNS2hNS3h0hNS1hNS1hNS2h1h0377777777775(1.4) istheNSNScirculantmatrixformedfromhLF.(Notethatfornotationalconve-nience,wehaveaugmentedhLFbylettinghl=0forl=LF;LF+1;;NS1in(1.4)togethNS.)Thenthereceivedsignalvectorzcanbewrittenasz=zne+w;(1.5)

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wherewN(0;(2=NS)INS)istheadditivezero-meanwhitecircularlysymmetriccomplexGaussiannoisewithvariance2.TheFFToutputofthereceiveddatavectorzcanbewrittenasy=WNSz=1 wheree=WNSwandH=diagfhgwithh=WNShNS=[h0h1hNS1]TbeingtheFFT,withzero-paddingtolengthNS,ofthechannelimpulseresponsehLF. WithknownH(canbeestimatedbyusingthetwolongtrainingsymbols),thedetecteddatasymbolscanbeexpressedinthevectorformas^x=H1r:(1.7) Tofacilitatetheabovedetection,weneedtoobtainCFO,symboltiming,aswellasthechannelresponse.Thischannelparameterestimationtaskisoneoftheissuesconsistingofthescopeofthisdissertation.1.3ScopeoftheWork

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First,weconsiderthechannelmodelingfortheOFDMsignaling.Thetime-invariantfrequency-selectivefadingchannelsfortheOFDMsignalingisoftenmodeledasanFIR(niteimpulseresponse)lter[16,124].Weclarifythefact,whichwasrstrevealedin[118]butlargelyoverlookedeversince,thattheFIRchannelmodelisonlyanapproximatemodelfortheOFDMsignaling.WeconsiderachannelmodelusefulforcharacterizingtherealisticchannelsfortheOFDMsignaling.ThischannelmodelisespeciallyusefulforassessingtheperformanceofthechannelparameterestimationmethodsdevisedforOFDM-basedwirelesscommunicationsystems. Second,weconsiderthechannelparameterestimationproblem.Wegivehereinasequentialparameterestimationmethodthatcanbeusedtoestimatealloftheaforementionedchannelparametersfortherealisticchannels.ThesequentialmethodfullyexploitsthestructureofthepacketpreambleasspeciedbytheIEEE802.11astandard. Finally,weconsidertheerrorreductionproblem.First,wegiveamaximum-likelihood(ML)phasetrackingapproachusingpilottonestoestimate(andcorrect)theresidueCFOinducedphaseerror(CPE)foreachreceivedOFDMdatasymbol.Second,wesupplyaleast-squares(LS)phasettingapproachusingtheestimatedCPEsfromeachoftheOFDMdatasymbolstoimprovetheaccuracyofthephaseerrorestimates.Third,weprovideasamplingclocksynchronizationapproachtomitigatetheunsynchronizedsamplingclockinducedtimedelayerror.Thissynchronizationapproachisespeciallyusefulforthepacket-basedtransmissionsandcaneliminatetheneedofusingtheAFC(automaticfrequencycontrol)clockrecoverycircuit.Fourth,wepresentasemi-blindmethod,whichisamodiedversionoftheonein[59],toimprovethechannelresponseestimationaccuracyusingthedetected/decodeddatabitsiteratively.

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AsacommonpartforbothoftheOFDM-basedBLASTandSTBCsystems,weproposeaMIMOpreambledesignfortheMIMOsystems.ThisMIMOpreambleisbackwardcompatiblewithitsSISOcounterpartasspeciedbytheIEEE802.11astandard.Basedonthispreambledesign,wecanusethesequentialparameteresti-mationmethodproposedfortheSISOsystemstoestimateCFOandsymboltimingfortheMIMOsystems.WecanalsoestimatetheMIMOchannelresponses. FortheOFDM-basedSTBCsystem,weproposeanOFDMDATAelddesignwhichincludesthepairingoftheOFDMdatasymbolsaswellasthepilottones.WeprovideasimpleOFDMdatasymbolgenerationschemefortheSTBCtransmitter.TheSTBCtransmitterbasedonthisschemehasacomplexityapproximatelythesameasitsSISOcounterpart.WealsoprovideerrorreductionalgorithmswhicharesimilartothosefortheSISOsystems.Furthermore,wepresentasimplesoft-detectortoobtainthesoft-informationusedforthechanneldecoder,suchastheViterbialgorithm. FortheOFDM-basedBLASTsystem,weproposeanOFDMDATAelddesignwhichincludesthepairingofthepilottonesusedforerrorreduction.Wealsopro-posethreeBLASTsoft-detectors,whichcandeliversoft-informationtothechanneldecoder,withdierentperformance/complexitytrade-ostofacilitatethereal-time

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Weproposeanewschemeforchanneltrackingthatcansignicantlyoutper-formalltheexistingapproachesforhighratetransmissions.Ournewschemecom-binespacketdesignwithsignalprocessing(atthereceiver)toaddressthechanneltrackingproblem.Forthepacketdesign,wesegmentanentirepacketintomulti-plesubpackets|witheachsubpackethavingzerotailbitstoresettheconvolutionalchannelencoder|sothatthedetection/decodingresultofasubpacketcanbeusedtoupdatethechannelestimationforthesubpacket.Thisway,weprovideamechanism

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ThetermOFDMwasrstusedinaUSpatentissuedin1970[117],andtheOFDMtechniqueinthemodernsensewasproposedin1971[125],wherethedis-creteFouriertransform(DFT)wasusedinthemodulationanddemodulationoftheparalleltransmitteddata.OFDMisaspecialformofMC;OFDMemphasizestheorthogonalityamongthesubcarriers,i.e.,thesubchannels.TheorthogonalityofthesubchannelsisautomaticallyguaranteedbyusingDFT,andtheintensiveana-loghardwareimplementationscanbereplacedbydigitalhardware(plussoftware)implementations,suchasfastFouriertransform(FFT)[22]chips. Inthe1980s,OFDMwasmainlystudiedforhigh-speedmodems[43,44],digitalmobilewirelesscommunications[19],anddigitalaudiobroadcasting(DAB)[62]. Inthe1990sandearly2000s,theresearchonOFDMgainedsignicantlyin-creasedattention[7],duetotheconvergenceofthefollowingfactors:(a)high-speedapplications,suchasDSL(digitalsubscriberlines)[17,18],DABandDVB,aswellasWLANs;(b)FFThardwareimprovements[42,88,126];and(c)theoret-icallyprovenoptimalperformanceofOFDM[53,100].Inthemeantime,OFDMhasbeencombinedwithmanyemergingtechniques,suchastheMIMOschemes[8,9,47,64,70,95,103,111],toimprovetheperformance. MoredetailedinformationabouttheearlyhistoryofOFDMcanbefoundin[7,120,130].

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WeconsiderthefollowingsignalprocessingissuesfortheOFDM-basedSISOsystems:channelparameterestimationanderrorreduction.Wealsoconsiderthechannelmodelingissuewhichisespeciallyimportantforassessingthechannelpa-rameterestimationmethodsdevisedfortheOFDM-basedsystems.Anliteraturereviewandworkdescriptionaregiveninthefollowingthreeareas.1.4.2.1Channelmodel

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Wealsoreducethechannelresponseestimationerror.Wepresentasemi-blindmethod,whichisamodiedversionoftheonein[60],toimprovethechannelresponseestimationaccuracyusingthedetected/decodeddatabitsiteratively.Thismethodismoresuitablethanthecommonlyuseddecisionfeedbackmethod[98],since,thankstothepoweroferrorcorrectionduetodecoding,theformercanexploitthedecodedresultwhichismuchbetterthanthedetectionresultfromasimplehard-decisionofthedatasymbolsofthelatter.1.4.3SystemDesignandSignalProcessingforMIMOSystemsoverTime-InvariantChannels Aswehavementionedearlier,weconsidervarioussystemdesignandsignalprocessingissuesfortwokindsofOFDM-basedMIMOsystems:STBCandBLAST.NotethatourdesignistargetingataSTBCsystemwithtwotransmitantennasandonereceiveantennaaswellasaBLASTsystemwithtwotransmitantennasandtworeceiveantennas.WhilesomeofthesignalprocessingalgorithmsareconnedtotheMIMOsystemswithtwotransmitantennas,otherscanbeusedinmoregeneralcases,aswillbeclearinthesequel.WeexemplifyourpresentationoftheOFDM-basedMIMOsystemswiththeWLANapplications.LiteraturereviewandworkdescriptionaboutsystemdesignandsignalprocessingissuesforthesetwoMIMOsystemsaregivenbelow.1.4.3.1TheSTBCandBLASTschemesSTBC.

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Toimprovetheup-linktransmissionrobustness(i.e.,thetransmissionfromthemobileterminal[MT]totheaccesspoint[AP])wecanusemaximal-ratioreceivercombining(MRRC)[52,71]byinstallingmultiplereceiveantennasattheAP.Forthedown-linktransmission(fromtheAPtotheMT),weshouldkeeptheMTdeviceassimpleaspossible.AsimplefeasiblewayistodeploymultipletransmitantennasattheAPandexploitthetransmitdiversityoeredbyspace-timecoding(STC)(see,forexample,[85]andthereferencestherein)overtheMIMOorMISO(multiple-input/single-output)channels.ThebenetsofusingSTCaretwofold.First,wecanincreasethetransmissionrobustnessandreducetherequestforretransmission.(FortheSISOcase,ifthefadingchannelisreallypoor,retransmissiondoesnotreallyhelpunlessthechannelchanges.)Second,lesstotaltransmissionpowerisrequired.AlthoughreducingthetotaltransmissionpowerattheAPisnotcriticalsinceitisalreadyverylowcomparedtoothersystemssuchasbasestationsforcellularphones,itcantranslatetoanotheradvantage:reducedSNRrequirementwhichenablesustoinstallmoreAPsinacertainareatoaccommodatemoreusers. AmongthevariouspopularSTCschemesfortheatfadingchannels,theorthog-onalspace-timeblockcoding(STBC)approach(see,forexample,[2,33,35,34,113],andthereferencestherein)isparticularlyattractivesinceitcanentailasimplere-ceiver;asimplelinearprocessingcandecoupletheMISOchannelintoanumberofindependentSISOchannels.Inaddition,whenconcatenatedwithconvolutionalcodes,soft-informationneededfortheViterbidecodercanbeobtainedeasilyfromtheSTBCreceiver.ManySTBCmethodshavebeenproposedforthefrequency-selectivefadingchannels(see,forexample,[61,63,72,84,110,122,129]andthereferencestherein).WeemployasimpleSTBCschemehereinthatusesSTBCon

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OurfocushereinistoincreasethetransmissionrobustnessbyusingaSTBCWLANsystemwithtwotransmitantennasandonereceiveantenna.BLAST. OurfocushereinisondoublingthedatarateoftheSISOsystemasspeciedbytheIEEE802.11astandardbyusingaBLASTsystemwithtwotransmitantennasandtworeceiveantennas.STBCandBLASTcomparison.

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ThesameasfortheSISOsystem,theMIMOpreambleisusedtoestimatechannelparameterssuchasCFO,symboltiming,and(MIMO)channelresponse.Earlierpreambledesigns(see,e.g.,[57]andthereferencestherein)aremainlybasedontheFIRchannelmodel,andareproblematicfortherealisticchannels.WeproposeaMIMOpreambledesignfortheMIMOsystems,whichisbackwardcompatiblewithitsSISOcounterpartasspeciedbytheIEEE802.11astandard[78].Thatis,aSISOreceivercanperformCFO,symboltiming,andchannelresponseestimationbasedontheproposedMIMOpreambledesignanddetectuptotheSIGNALeld.TheSISOreceiveristheninformed,byusing,e.g.,thereservedbitintheSIGNALeld,thatatransmissionisaSISOornot.OurMIMOpreambledesigncanbeusedwithtwotransmitandanynumberofreceiveantennas.OFDMDATAelddesignfortheSTBCsystems.

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TostayasclosetoIEEE802.11aaspossible,weusethesamescrambler,convo-lutionalencoder,puncturer,interleaver,symbolmapper,andCPasspeciedintheIEEE802.11astandardforourSTBCsystem.OFDMDATAelddesignfortheBLASTsystems.

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ParalleltotheerrorreductionalgorithmsdevisedfortheSISOsysteminthepreviouschapter,wehavethecorrespondingerrorreductionalgorithmsfortheMIMOsystems.NotethattheLSphasettingandthesamplingclocksynchronizationmethodsdevisedfortheSISOsystemscanbeusedintheMIMOsystemsdirectly.Hence,weonlyneedtoaddressspecicallythealgorithmsforMLphasetacking[78]andsemi-blindchannelresponseestimationfortheMIMOsystems.ThesealgorithmsaredependentonthetransmittingschemesandwillbepresentedseparatelyintheSTBCandBLASTsettings.1.4.3.4DetectionfortheMIMOsystems

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ThechannelresponseoftheMIMOsystemcanbeestimatedviausingtheMIMOpreamble[78].Duetothelimiteddurationofthepreamble,thedetection/decodingperformancewiththeestimatedchannelresponseisabout2dBfromthatwiththeperfectchannelknowledge.Thedetection/decodingperformancecanbeimprovedifmoreaccuratechannelestimatecanbeobtained.Duringiterativeprocessing,wecanupdate/improvethechannelresponseestimatebyusingthedetected/decodeddatabitsviatheaforementionedsemi-blindtechnique.Thecombinationoftheiterativedetection/decodingutilizingtheCLSD-basedsoft-detectorandthechannelupdating,referredtoastheturboprocessing,cansignicantlyimprovethesystemperformance.1.4.4SystemDesignandSignalProcessingforSISOSystemsoverTime-VaryingChannels LiketheSISOWLANsystems,theE2213-02conformableSISODSRCsystemsemploypacket-basedtransmission.Theyusethepacketpreambletoobtainthechan-nelparametersandusecoherentdetectiontodetectthedatabitscontainedinthepayload.Tocontrolthepacketerrorrate,convolutionalchannelcoding/decodingisusedintheSISODSRCsystems.Thepacket-basedtransmissioniswellsuitedfortime-invariantchannelsbutwillsuerfromsevereperformancedegradationsfortime-varyingchannels,whichcanoccur,forexample,duetousermobility.Thelackofmechanismfortrackingthetime-varyingchannelsforcoherentdetectionisthemaincausefortheperformancedegradations.Asaresult,theSISODSRCsystems

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Therearemainlytwoclassesofapproachestodealwiththetime-varyingchannelsifwearenotlimitedtoconformingtothestandard.Theapproachesinoneclassusedierentialmodulation/detectionschemeswhichobviatetheneedofchanneltracking[27,96,107].Forhighdataratetransmissions,dierentialamplitude/phaseshift-keying(DAPSK)canbeusedtoimprovetheperformance[29,74].Theapproachesintheotherclassusechanneltrackingschemestotrackthechannelvariationtosupportthecoherentdetection.Theapproachesinthelatterclasscanbefurtherdividedintotwomaingroups.Theapproachesinonegroup,suchasthosein[69,89],usedecisionfeedbacktoupdatethechannelresponses.Theapproachesintheothergroup,suchasthosein[28,93],usepilottonestoupdatethechannelresponses.Unfortunately,theapproachesintheformergroupworkwellonlyforPSKconstellations,suchasQPSK,andwillsuerfromsignicantperformancedegradationsforQAMconstellations,whichareoftenusedforhighdataratetransmissions.TheproblemwiththeQAMsymbolsisthatthosewithsmallamplitudescancauselargeerrorsinchannelresponseupdating.TheapproachesinthelattergroupassumeandexploitthepropertiesofFIRchannelmodelsfortheOFDMsignaling.However,FIRmodelshavebeenshowntobeonlyroughapproximationsoftherealisticchannelsforOFDMsignaling[118,77].Asaresult,inpracticalapplications,theseapproacheswillperformmuchworsethanthedierentialapproachesforhighratetransmission. Weproposeanewschemeforchanneltrackingthatcansignicantlyoutperformalltheexistingapproachesincludingthedierentialonesforhighratetransmission.Ournewschemecombinespacketdesignwithsignalprocessing(atthereceiver)toaddressthechanneltrackingproblem.Forthepacketdesign,wesegmentanentirepacketintomultiplesubpackets|witheachsubpackethavingzerotailbitstoresettheconvolutionalchannelencoder|sothatthedetection/decodingresultofasubpacket

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OurnewschemecanbeeasilyusedtomodifytheE2213-02standardsothattheDSRCsystemsconformingtothemodiedstandardcanbeusedfortime-varyingchannels.ThesamemodicationcanbeusedintheWiMaxstandardaswellastheWi-FistandardsincludingIEEE802.11a,IEEE802.11g,andHIPER/LAN2,whichareallpacket-andOFDM-based,toaccommodatemobility.Moreover,thenewschemecanbereadilyadoptedintotheMBWAstandardswhichareunderdevelop-ment.1.4.5SystemDesignandSignalProcessingforMIMOSystemsoverTime-VaryingChannels AsfortheSISOsystems,therearemainlytwoclassesofapproachestodealwiththetime-varyingchannelproblemfortheSTBCsystems|dierentialmodula-tion/detectionorchannelupdating/tracking.FortheBLASTsystems,however,thereisonlyoneclassofapproachestodealwiththisproblem|channelupdating/tracking.

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Letusconsiderthedierentialmodulation/detectionproblemfortheSTBCsys-temsrst.Forthetime-varyingchannels,dierentialspace-timemodulation(DSTM)schemes[45,48,82,75,112]wererecentlyintroducedasextensionsofthetraditionaldierentialphaseshiftkeying(DPSK)schemesfortheSISOsystems.LikeDPSKschemesfortheSISOsystems,theDSTMschemescanobviatetheneedforchannelestimationatthereceiver.Inaddition,DSTMcanmaintainthedesiredpropertiesofspace-timecodingtechniques,suchasthetransmitdiversity.Morerecently,STBC-baseddierentialspace-timeblockcode(DSTBC)modulationschemewasproposed[36,81].DSTBChasthesamediversitygainas|butoershighercodinggainthan|theDSTMschemes.ThisisnotsurprisinginviewofthefactthatSTBCisoptimalintermsofSNR[109].Moreover,thedecodingoftheDSTBCmodulationschemeismoreecientthanfortheDSTMschemes,especiallyforlargeconstellations,sincetheformerallowsforadecoupledlineardetection,withwhichitistodetecteachinformationsymbolseparately.However,bothoftheDSTMandDSTBCschemescanonlybeusedwiththePSKsymbolsduetotheneedofkeepingthesameampli-tudeintheprocessofdierentialmodulation.Thiswillleadtoaproblem|severeperformancedegradationsforhightransmissiondataratecasesduetotheproblemofsmallminimumconstellation-distancewithlargePSKconstellations.Forexample,forthehighesttransmissiondatarateforDSRC,64-PSK,whichhasamuchsmallerminimumconstellation-distancethan64-QAM,needstobeused.Asaconsequence,neitherDSTMnorDSTBCisapplicableforthecaseofhightransmissiondatarates. Duetotheaboveproblems,weusechannelupdating/trackingfortheMIMOsystemsoverthetime-varyingchannels.WeconsidervarioussystemdesignandsignalprocessingissuesfortheMIMOsystems.TheMIMOpreambledesignwe

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Aswehavementionedearlier,theOFDM-basedSISOwirelesscommunicationsystemsconsideredinthischapterincludetheSISOWLANsystemsandtheSISOWiMaxsystemswhichworkovertime-invariantfrequency-selectivefadingchannels.Duetothesimilarityofthesesystems,wewillexemplifyourpresentationofthischapterusinganIEEE802.11a[49]conformableSISOWLANsystem.TheconsideredSISOsystemusespacket-basedtransmission.Eachpacket,asshowninFigure2{1,consistsofanOFDMpacketpreamble,aSIGNALeld,andanOFDMDATAeld.Thepacketpreambleisusedtodeterminethechannelparameters,includingthecarrierfrequencyoset(CFO),symboltiming,aswellasthechannelresponsebetweenthetransmitantennaandthereceiveantenna.Theseparametersareneededforthedetection/decodingofthedatabitscontainedintheSIGNALeldandtheOFDMDATAeld. Inthischapter,weconsiderthefollowingsignalprocessingissuesfortheIEEE802.11aconformableSISOWLANsystem:channelparameterestimationanderrorreductionatthereceiver.Wedeviseorselectsignalprocessingalgorithmsthatcanhelpimprovetheoverallsystemperformanceandcanbeecientlyimplementedinreal-time.Theimportantissuesthatweaddressarebriefedasfollows. First,weconsiderthechannelmodelingfortheOFDMsignaling.Thetime-invariantfrequency-selectivefadingchannelsfortheOFDMsignalingisoftenmodeledasanFIR(niteimpulseresponse)lter[16,124].Weclarifythefact,whichwasrstrevealedin[118]butlargelyoverlookedeversince,thattheFIRchannelmodelisonlyanapproximatemodelfortheOFDMsignaling.WeconsiderachannelmodelusefulforcharacterizingtherealisticchannelsfortheOFDMsignaling.Thischannelmodel31

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Figure2{1:ThepacketstructurespeciedbytheIEEE802.11astandard.isespeciallyusefulforassessingtheperformanceofthechannelparameterestimationmethodsdevisedforOFDM-basedwirelesscommunicationsystems. Second,weconsiderthechannelparameterestimationproblem.Wegivehereinasequentialparameterestimationmethodthatcanbeusedtoestimatealloftheaforementionedchannelparametersfortherealisticchannels.ThesequentialmethodfullyexploitsthestructureofthepacketpreambleasspeciedbytheIEEE802.11astandard. Finally,weconsidertheerrorreductionproblem.First,wegiveamaximum-likelihood(ML)phasetrackingapproachusingpilottonestoestimate(andcorrect)theresidueCFOinducedphaseerror(CPE)foreachreceivedOFDMdatasymbol.Second,wesupplyaleast-squares(LS)phasettingapproachusingtheestimatedCPEsfromeachoftheOFDMdatasymbolstoimprovetheaccuracyofthephaseerrorestimates.Third,weprovideasamplingclocksynchronizationapproachtomitigatetheunsynchronizedsamplingclockinducedtimedelayerror.Thissynchronizationapproachisespeciallyusefulforthepacket-basedtransmissionsandcaneliminatetheneedofusingtheAFC(automaticfrequencycontrol)clockrecoverycircuit.Fourth,wepresentasemi-blindmethod,whichisamodiedversionoftheonein[59],toimprovethechannelresponseestimationaccuracyusingthedetected/decodeddatabitsiteratively. Notethattheabovechannelparameterestimationanderrorreductionalgorithmscanalsobeused(withappropriatemodicationorextension)intheOFDM-basedMIMOwirelesscommunicationsystemswhichwillbeconsideredinthelaterchap-ters.TofacilitatetheapplicationofthesealgorithmsintheMIMOsystems,we

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Theeectivenessofouralgorithmsandtheoverallsystemperformanceofusingouralgorithmsaredemonstratedvianumericalexamplesfollowingthedescriptionofthealgorithms. Theremainderofthischapterisorganizedasfollows.Section2.1givesachannelmodelwhichcanbeusedtocharacterizetherealisticchannelsfortheOFDMsignalinganddescribesthedatamodelfortheOFDM-basedWLANs.Section2.2presentsthesequentialparameterestimationmethod.TheerrorreductionalgorithmsareprovidedinSection2.3.Finally,weconcludethischapterinSection2.4.2.1ChannelandDataModel beavectorofNS=64symbols,wherexk;nS,nS=1;2;:::;NS,isthesymbolmodulatingthenSthsubcarrierandisequalto0forthenullsubcarriers,1or1forthepilottones,oramemberinaconstellationCfordatacarryingsubcarriers.HereKisthenumberofOFDMdatasymbolsinapacketandCisaniteconstellation,suchasBPSK,QPSK,16-QAM,or64-QAM.LetWNS2CNSNSbetheFFTmatrix.ThenthekthOFDMdatasymbolskcorrespondingtoxkisobtainedbytakingIFFTofxk.Thatis,sk=WHNSxk=NS;(2.2)

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Figure2{2:IllustrationsoftheanalogwaveformsforOFDMdatasymbolsandtheoutputofamultipathchannel. where()Hdenotestheconjugatetranspose.ToeliminateISI,skisprecededbyacyclicprex(CP)orguardinterval(GI)sk;CformedusingthelastNC=16elementsofsk. BystackingthepacketpreambleandtheOFDMsymbols(togetherwiththeircorrespondingCPs)intheSIGNALandOFDMDATAeld,weobtaintheentirepackets2C(5+K)(NS+NC)1.(Here5isduetothefactthatthelengthofthepacketpreambleis4OFDMdatasymbollongandthatthelengthoftheSIGNALeldis1OFDMdatasymbollong.)PassingsthroughapairofD/Aconverters(forboththerealandimaginarypartsofs)andthecorrespondinglower-passlters,weobtainthe(complex)analogwaveforms(t)whichisshowninFigure2{2.2.1.1ChannelModel denotethe(baseband)time-domainanalogchannelimpulseresponseofatime-invariantfrequency-selective(multipath)fadingchannel(calledtherealisticchannelforshort),

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Leth(t)=[h(t)0h(t)1:::h(t)NS1]T(2.4) bethecorrespondingtime-domaindiscretechannelresponse[referredtoastheequiv-alentdiscretechannelresponseofh(t)]forthesampleddatablocks.Notethatbyequivalencewemeanthediscretefrequency-domainresponseof(2.4)isthesameasthatof(2.3)ontheinterestedsubcarriers;i.e.,ifh=WNSh(t)4=[h1h2:::hNS]T(2.5) isthefrequency-domainchannelresponsecorrespondingtothesampledsignals,thenforaninterestedsubcarriernS,wehave:hnS=XppejptS!!=2[nS1]NS NStS;(2.6) wheretSisthesamplingintervaland[nS1]NS=8><>:nS1;nSNS=2;nS1NS;nS>NS=2:(2.7) Wehavetwotypesofequivalentdiscretechannelresponsesdependingonthechoicesoftheinterestedsubcarriers,whichareusedtotransmitdata.IfwewereinterestedinalloftheNS=64subcarriers,thenthelth,l=0;1;:::;NS1,elementofh(t),whichisreferredtoastheTypeAequivalentdiscretechannelresponse,couldbewrittenash(t)l=Xppej(pl) sin((pl)=NS):(2.8)

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(a)(b) Figure2{3:Illustrationoftheamplitudeoftwotypesofequivalentdiscretechannelresponseforamultipathchanneldescribedbyh(t)=(t2:5tS)j0:5(t4:8tS):(a)TypeA;(b)TypeB. Ontheotherhand,ifweareinterestedinonlytheNSC=52non-zerosubcarriers(withthe12null-subcarriersbeingsettozero),then,thelthelementofh(t),whichisreferredtoastheTypeBequivalentdiscretechannelresponse,canbewrittenash(t)l=Xppej(pl) sin((pl)=NS)1:(2.9) SeeAppendix2.5.1forthederivationof(2.8)and(2.9). WeremarkthatbothofthetwoequivalentdiscretechannelsusuallyareoflengthNS(thisistrueevenwhenwehaveonlyonepathbutt0isnotamultipleoftS),asshowninFigure2{3.Thereasonisthat,asshowninFigure2{4,thetransactionfromoneperiodofhtoanotherisnotsmoothduetofactthatp,p=0;1,isnotaninteger. NotethatthelengthoftheequivalentchannelsismuchlongerthanNC,thelengthofCP.However,thiswillnotcausetheISIproblemaslongasPM0NC1andwehavethecorrectsymboltiming.Thereasonisthatthe(sampled)receiveroutputduetothemultipathchannelisnotobtainedfromtheconvolutionoftheequivalentdiscretechannelresponsewiths;rather,itisobtainedfromsampling

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Figure2{4:Illustrationofthediscontinuitiesintheamplitudeofthefrequency-domainchannelresponseforamultipathchanneldescribedbyh(t)=(t2:5tS)j0:5(t4:8tS).z(t),theconvolutionofh(t)ands(t),asshowninFigure2{2,withtheportioncontaminatedbyISIdiscarded. Theequivalentdiscretechannelresponseh(t)isoftenapproximatedbyanexpo-nentiallydecayingFIRlterwithlengthLF[124],denotedashe(t)=LF1XlF=0h(t)lF(tlFtS);(2.10) whereh(t)lFN0;1e1=tnelF=tn;(2.11) withnh(t)lFoLF1lF=0beingindependentofeachother,tn=tr=tS,trbeingtheroot-mean-square(rms)delayspreadofthefrequency-selectivefadingchannel,LF=d10tne+1,anddxedenotingthesmallestintegernotlessthanx.Thischannelmodelissometimesreferredtoastheexponentialchannelmodel[16]. Theexponentialchannelmodelisecientfornumericalsimulationsforsystemperformance;however,weshouldnotdeviseparameterestimationmethodsbasedonitslimitedlengthproperty,since,aswasshownin(2.8)or(2.9),theFIRlterisonlyanapproximationoftherealisticchannelfortheOFDMsignaling.

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Thedirectsimulationof(2.3)isdicultduetoapotentiallylargePMandthedistributionsofp.Instead,wemodify(2.10)slightlytobettercharacterizetherealisticchannelsasfollows:hm(t)=LF1XlF=0h(t)lF(tlFtStlF);(2.12) whereh(t)lFisasin(2.11)andtlF,lF=0;1;:::;LF1,isuniformlydistributedover[0;tS].Wealsoassumethatnh(t)lFoLF1lF=0andftlFgLF1lF=0areindependentofeachother.Thisnewchannelmodel,referredtoasthemodiedexponentialchannelmodel,ismorerealisticthantheonein(2.10)duetothetimedelaysftlFgLF1lF=0introducedin(2.12).2.1.2DataModel Leth=[h1h2:::hNS]T(2.13) bethefrequency-domainchannelresponse(ontheNSsubcarriers)correspondingtotheemployedsymboltiming.Thereceiveddatavectorisdenotedaszk=znek+wk;(2.14) wherewkN(0;(2=NS)INS)istheadditivezero-meanwhitecircularlysymmetriccomplexGaussiannoisewithvariance2=NS.TheFFToutputofzkcanbewrittenas[124]yk=WNSzk

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wherediagfhgdenotesthediagonalmatrixformedfromhandekN(0;2INS).Thedatamodelin(2.15)canalsorepresenttheOFDMsymbolsintheSIGNALeldandthepacketpreamble. Equation(2.15)canalsobewrittenasyk=diagfxkgh+ek:(2.16) Notethat(2.15)isusefulforthedatasymbol(ordatabit)detectionwhereas(2.16)canbeusedforthechannelresponseestimation(withxkbeingtheknowntrainingdatasymbolvector,denotedasxBinthesequel,forthecaseofchannelresponseestimation).2.2SequentialChannelParameterEstimation

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Aswehavementionedearlier,wepresentthesequentialchannelparameteres-timationalgorithmintheSIMOcontext,whereweassumethatwehaveNreceiveantennas.2.2.1CoarseCFOEstimation whereisthenormalizedCFO(withrespecttothesamplingfrequency),whichwestillrefertoitasCFOforconvenience.(TheabsoluteCFO,i.e.,thedierencebetweentheoscillatorsofthereceiverandtransmitter,is=tS,wheretSisthesamplingperiod.)Foreachreceiveantennaoutput,considerthecorrelationbetweentwoconsecutivenoise-freereceiveddatablocks,eachofwhichisoflengthNC.ThenthesumofthecorrelationsforallreceiveantennascanbewrittenasNXn=1v+NC1Xl=vznen(l)(znen(l+NC))=ej2NCNXn=1NC1Xl=0jznen(l)j24=Pej2NC;(2.18) where()denotesthecomplexconjugateandvisanynon-negativeintegersuchthatznen(v+2NC1)isasampleofthenthreceiveantennaoutputduetotheinput(transmitantennaoutput)beingasampleoftheshortOFDMtrainingsymbolsof

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whereePisduetothepresenceofthenoise.WecalculatethecoarseCFOas[65]^C=1 2NC\PS;(2.20) where\xdenotestakingtheargumentofx. WenextcorrecttheCFOusing^Ctogetthedatasamplesz(C)n(l),n=1;2;;N,asfollows:z(C)n(l)=zn(l)ej2l^C:(2.21) Correspondingly,wehaveP(C)S=PSej2NC^C:(2.22) Inthesequel,weonlyconsidertheCFOcorrecteddatagivenabove.Fornotationalconvenience,wedropthesuperscriptofz(C)n(l),n=1;2;;N.2.2.2CoarseSymbolTimingEstimation Nowwecanuseacorrelationmethod,modiedbasedontheapproachpresentedin[105],toestimatethecoarsesymboltiming.

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Figure2{5:Illustrationofdenitionofvarioussymboltimingsandthedeterminationofthesesymboltimings. From(2.19)and(2.22),wenotethatthecorrelation(aftertheCFOcorrection)isapproximatelythereal-valuedscalarPplusacomplex-valuednoise.Henceweproposetousethefollowingreal-valuedcorrelationsequenceforcoarsesymboltimingdetermination.Wecalculatethecorrelationsequenceinaniterativeformsimilartothecomplex-valuedapproachin[105]asfollows:PR(v+1)=PR(v)+Re(NXn=1[zn(v+NC)zn(v+2NC)zn(v)zn(v+NC)])=PR(v)+NXn=1fzn(v+NC)[zn(v+2NC)zn(v)]+~zn(v+NC)[~zn(v+2NC)~zn(v)]g;(2.23) wherebothRe()and()denotetherealpartofacomplexentityand~()standsfortheimaginarypart.WestarttheiterationbyusingPR(0)=Re(PS).Notethatthereal-valuedcorrelationapproachgivenin(2.23)issuperiortothecomplex-andabsolute-valuedonegivenin[105]sincetheformer(a)usesfewercomputations,(b)lowersthenoiselevelinthecorrelationsequence(noisevariancereducedinhalf),and(c)dropsclosertozeroastheslidingdatablocksstartingtoslideoutoftheregionduetotheinputbeingtheshortOFDMtrainingsymbolsinthepreamble.Theadvantagesof(a)and(b)canbeseenreadilyin(2.19)and(2.23)andtheadvantage

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Figure2{6:Correlationvaluecomparisonforthereal-valuedcalculationalgorithmwiththeabsolute-valuedcalculationalgorithmforatime-invariantchannelwithtr=50nsforaSISOsystem.of(c)canbeobservedfromasimulationexampleshowninFigure2{6,wherePA(v)denotestheabsolute-valued(complex-valued)correlationsequenceof[105]. WhensomeofthedatasamplesoftheslidingdatablocksaretakenfromthereceiveddataduetotheinputbeingGI2orthelongOFDMtrainingsymbolsfollowingtheshortOFDMtrainingsymbols,thevalueofPR(v)willdropsince(2.17)nolongerholds.Thispropertycanbeusedtoobtainthecoarsesymboltimingestimate.LetTP,asshowninFigure2{5,denotethersttimesamplewhenPR(v)dropstobelowhalfofitspeakvalue.Then,wecanhavethecoarsesymboltimingestimateas:TC=TP+3 2NC+NC:(2.24) NotethatthesecondtermattherighthandsideoftheaboveequationisduetothefactthatthevalueofPR(v)willdroptoapproximatelyonehalfofitsmaximumwhenthedatasamplesofthesecondhalfofthesecondofthetwoslidingblocksareduetotheGI2inthepreamble;thethirdtermisduetoonehalfofthelengthofGI2|2NC.WhenPM0NC1,onlythersthalfofGI2cansuerfromISI.Henceourgoalofcoarsetimingdeterminationistoplacethecoarsetimingestimatebetween

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Wehavenoticedthatthecoarseorstandardsymboltimingisnotsucientlyaccurate,especiallyforfrequency-selectivefadingchannelswithlargetr,aswillbeshownviasimulationexamples.Henceanesymboltimingestimateisneeded.Beforewepresentournetimingestimationmethod,letusconsidertheneCFOestimationrst,sinceabettercorrectedCFOcanleadtoabetternesymboltimingestimation.2.2.3FineCFOEstimation (NotethatthedatasamplesusedherestartfromTC.)ThentheneCFOestimatecanbecomputedas^F=1 2NS\PL:(2.26) Wecanuse^Finthesamewayas^CtocorrecttheresidueoftheCFO.Weassumethatthedataweusebelowhave^Fcorrectedalready.2.2.4FineSymbolTimingEstimation ThenesymboltimingisestimatedbyusingadatablockoflengthNS,startingfromthetimesampleTC+3NC.Withthischoice,duetothefactthatT1isidenticaltoT2,thedatablockismostlikelyduetotheinputbeingthesecondhalfofT1andthersthalfofT2,evenwhenthecoarsesymboltiminghasalargeerror.

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LetyndenotetheoutputoftheNS-pointFFTofthedatablockfromthenthre-ceiveantennaandh(t)nbetheequivalentdiscretechannelresponseinthetime-domainbetweenthetransmitantennaandthenthreceiveantenna.Then,byneglectingtheexistenceoftheresidueCFO,yncanbewrittenas[cf.(2.16)]yn=XBWNSh(t)n+en;(2.27) whereXBisadiagonalmatrixformedfromxB,whichisthetrainingdatasymbolvectorconsistingofthe52knownBPSKsymbolsand12zeros,usedtoformtheT1inFigure2{1.(herethesubscript\B"denotetheBPSKsymbols).SincetheMoore-Penrosepseudo-inverseofXBisXBitselfandWNS=N1=2Sisunitary,wegetanestimateofh(t)nas^h(t)n=1 whichisactuallythenoisecontaminatedTypeBequivalentdiscretechannelresponse[cf.Figure2{3(b)]. LetTI,asshowninFigure2{5,denotetherstindexoftheelementsofNXn=1j^h(t)nj(2.29) thatareabove1=3ofthemaximumvalueoftheelementsofj^h(t)j.(Ourempiricalexperiencesuggeststhatselectingthethresholdtobe1=3givesveryrobustresult.)ThenthenesymboltimingTFisobtainedasTF=TCNC+TI3:(2.30) Thesecondtermaboveisusedtocompensatefortheaforementioned3NCshiftduetothefactthatNS3NC=NCandthelasttermaboveischosentobe3toensurethatTF>T0withnegligibleprobability. Notethatwecouldhavedevisedmethodsmoresophisticatedthan(2.30)todeterminethenesymboltimingbasedontheelementvaluesgivenin(2.29).Yet,the

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wherexBdenotesthenSthdiagonalelementofXB,yn;LdenotesthenSthelementofyn;L,anden;LN(0;2=2).(Herethevariance2=2isduetoaveragingthetwoconsecutivedatablocks.)Wecanreadilyhave,from(2.31),that:^hn=xByn;L:(2.32) Wecanreadilyseethatthevarianceoftheestimationerrorforhnis2=2.Moreover,^2,theestimateof2,canbeeasilyobtainedfromthedierencezn;4=zn;T1zn;T2,n=1;2;:::;N.Letyn;denotetheFFTofzn;.Thenyn;

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(a)(b) Figure2{7:Probabilitydistribution(orhistogram)ofcoarseandnesymboltimingestimateswhenSNR=10dBfortime-invariantchannelsgeneratedaccordingtotwochannelmodelswithvarioustr'sforaSISOsystem:(a)themodiedexponentialchannelmodel;(b)theexponentialchannelmodel.N(0;22INS).Let~yn;beasub-vectorofyn;containingtheelementsontheNSC=52non-zerosubcarriers.Wecompute^2as^2=1 2NSCNNXn=1k~yn;k2;(2.33) wherekkdenotestheEuclideannorm.This^2isneededlaterbythesoft-detector.2.2.6NumericalExamplesforSymbolTimingEstimation

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(a)(b) Figure2{8:Probabilitydistributionofcoarseandnesymboltimingestimatesfortime-invariantchannelswithtr=100nsforaSISOsystematvariousSNRs:(a)coarsesymboltiming;(b)nesymboltiming.frequency-selectivefadingchannelsgeneratedaccordingtothemodiedexponentialmodelandtheexponentialmodelareshowninFigures2{7(a)and2{7(b),respec-tively.Thetwodashedcurvesintheguresshowtheprobabilitydistributionofthecoarsesymboltimingestimatesforchannelswithtr=50and150nsatalowSNR=10dB.Thetwosolidcurvesintheguresshowtheprobabilitydistributionofthenesymboltimingestimates.Wecanseethatthesimplenesymboltimingapproachgiveshighlyaccuratetimingestimates,comparedtothecoarseone,evenatthelowSNRof10dB.Wecanalsoseethatthenesymboltimingestimationmethod,likethecoarsesymboltimingone,worksequallywellforthechannelsgeneratedaccordingtobothofthechannelmodels.Example2.

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wherePk=diagfpkgande(p)n;kdenotesthecorrespondingzero-meanwhitecircularlysymmetriccomplexGaussiannoisevector.TheMLestimateofkisthencomputed

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Withthecomputed^k,wecancompensateout^keitherinthetimedomainviazn;kej^korinthefrequencydomainviayn;kej^k,n=1;2;:::;N.2.3.2LSPhaseFitting First,weconsiderthecasewhereallKOFDMdatasymbolsintheOFDMDATAeldareusedforttingthephaseerrorsintoaline.ThephaseerrorduetotheresidueCFOcanbemodeledask=1+k2;k=0;1;:::;K;(2.36) wherek=0correspondstotheOFDMsymbolintheSIGNALeld.WecouldhaveusedanMLapproachtoobtain1and2.However,theMLapproachiscomputationallyexpensiveanddoesnotprovideappreciabledetectionimprovementoverthefollowingsimpleLSphaseerrorttingapproach,whichwepreferovertheformer.(Tobeself-contained,weprovidethederivationoftheMLphasetting

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wheref^kgKk=0arerstobtainedwiththeMLphasetrackingapproachin(2.35)andthenphaseun-wrappedbeforeusedin(2.37).TheLSestimateofisthengivenby:^=(DTD)1DT^:(2.38) Onceweobtain^,weremovethelinearphaseerror^F=D^fromthedatasequenceintheOFDMDATAeld. Second,weconsiderthecasewherethekthOFDMdatasymbolandKS
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ThesamplingclocksynchronizationisdierentfromthesymboltiminginthatthelatterisconcernedwithdeterminingthecorrectpositiontodiscardtheGIwhiletheformerfocusesonaligningthereceiversamplingclocktothetransmittersymbolclock.ThesamplingclocksynchronizationisimportantsincetheWLANperformancewilldegradeseverelywithoutit. TheIEEE802.11astandardsuggeststhatatthetransmitter,bothofthetransmitcenterfrequencyandthesymbolclockfrequencybederivedfromthesamereferenceoscillator.Wesuggestthesameatthereceiver,i.e.,thesamplingclockfrequencyandthereceivecenterfrequencybederivedfromthesamereferenceoscillator.TheneachsampleofthereceivedsignalwillsuerfromadelayoffS whereisthenormalizedCFOaddressedearlier,fS=1=tSisthesamplingfrequency,andfRisthereceivecenterfrequencyorradiofrequency(RF).BothfSandfRareknowntothereceiver.Althoughthisdelayistoosmalltocausetheintercarrierinterference(ICI),itwillaccumulateovertime. LetNT=NS+NC=80.TheaccumulatedtimedelaysforthesamplesofthekthOFDMdatasymbolaredierentbutareapproximatelyequaltoTk=kNTfS ThetimedelayssueredbythesamplesofanOFDMdatasymbolwillresultinanapproximatelylinearphasechangeacrossthesubcarriers,whichcandrastically

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Toaddresstheaboveproblem,weconsiderthefollowingstrategy.Let^TkdenotetheestimateofTk.(Toinitialize,weset^T0=0.)ForthekthOFDMdatasymbol,werstcompensateoutthetimedelayusing^Tk1toachievesamplingclocksynchro-nization.ThisapproachshouldworkwellsinceTk1andTkshouldbequiteclosetoeachother.Next,wecompute^Tkasfollows:rst,calculatethephaseerror^kusingthepilottonesusingthemethodsuggestedin(2.35),second,un-wrapthephaseerror^k,third,obtainanupdatedCFOestimateas^I+^k where^IdenotestheinitialCFOestimate(whichincludesboththecoarseandneCFOestimatesobtainedfromthepacketpreamble),andnally,replacethein(2.41)usingtheupdatedCFOestimatetoget^Tk=fS Notethatwhenweusethesemi-blindmethod,whichwillbepresentedinthesequel,toimprovethechannelestimationaccuracy,wemustrstaccountforthephasechangecausedbyTkforthekthOFDMdatasymbol,k=1;2;:::;K,beforedoingtheaveragein(2.49).2.3.4Semi-BlindChannelEstimation First,wereconstructthedatasymbolsbyusingthesameproceduresofscram-bling,convolutionalencoding,interleaving,andsymbolmappingasfortheoriginal

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Second,wedeterminewhetherornotthe48reconstructeddatasymbolsassoci-atedwithsk,k=1;2;:::;K,willbeusedforthesemi-blindchannelestimation;i.e.,wedoaselectionontheOFDMdatasymbollevel.WeaddthisselectionstepontheOFDMdatasymbollevelfortworeasons:(a)thedecodingerrorfortheconvolutionalcodestendstooccurinburstsand(b)theinterleavingisperformedwithinanOFDMdatasymbol.Letxk,k=1;2;:::;K,denotethedatasymbolonthenSthsubcarrier,whichisusedtoformtheOFDMdatasymbolskintheOFDMDATAeld.(Fornotationalconvenience,weagaindropthedependenceofournotationonnS.)Letxkdenotethereconstructeddatasymbolcorrespondingtoxk.Wecalculatethedistancedk=NXn=1jyn;k^hnxkj2(2.44) forthenSthsubcarrier,whereyn;kisthenSthelementofyn;k[cf.(2.15)]fromthenthreceiveantennaand^hnisthechannelestimatefromthetransmitantennatothenthreceiveantennaonthenSthsubcarrier.Weaddupsuchdistancesforallofthe48datacarryingsubcarriersandcomparethesumwithathreshold,whichwechoosetobe96N^2(i.e.,twicethenoisevariance,48N^2)herein.Ifthesumisbelowthethreshold,weusetheestimateddatasymbolxkassociatedwithskforthesemi-blindchannelestimation;otherwise,wedropitfromfurtherconsiderations. Finally,weestimatethechannelresponseforeachsubcarrierinasemi-blindway.ConsiderthenSthsubcarrier.Letyn;R(herethesubscribeRstandsfor"recon-struction")bethecolumnvectorformedbystackingalltheyn;k'sassociatedwiththesk'sthatsurvivetheaforementionedselectionprocess.LetxRbethevectorformedfromthecorrespondingxk'ssimilarlytoyn;R.Wecanalsoformvectorsyn;SandxS

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fromwhichwecanobtainthesemi-blindestimateofhnviatheLSapproach:^hn;SB=xHSBxSB1xHSByn;SB:(2.46) (HerethesubscriptsSBstandsfor"semi-blind.")Duetothestructureofthepream-ble,itiseasytoverifythatxHSxS=2:(2.47) LetR4=xHRxR:(2.48) Thenwehave^hn;SB=1 2+RxHSyn;S+xHByn;B=2 2+R^hn+R where^hnistheoneestimatedin(2.32). Sinceitismorecomplicatedtodeterminetheaccuracyof^hn;SBduetothestatisticaldependenceofxRandyn;R,wesimplyignoretheerrorin^hn;SBwhenusingtheSISOsoft-detector[cf(2.61)ofAppendix2.5.3]anduse^2=PNn=1j^hnj2insteadinthesoft-detector.Thisapproximationwillnothavenoticeableeectonthedecodingperformancesincewhatisimportantforthesoft-informationisnottheabsoluteSNR,^2=PNn=1j^hnj2,oneachsubcarrier,buttherelativeSNRamongthesubcarriers.Wealsoremarkthatthissemi-blindapproachcanbeusediteratively.However,ournumericalexamplesindicatethattheiterationisnotneededduetonegligibleperformanceimprovement.

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NotethatwedidnotusetheSIGNALeldtoimprovetheaccuracyofthesemi-blindchannelestimationsincetheimprovementofusingtheSIGNALeldisnegligibleforlargeK.However,ifonedoesnotwishtoexperiencethelatencycausedbyusingtheOFDMDATAeld,onecanusetheSIGNALeldinstead.Thisisreferredtoasthereal-timesemi-blindchannelestimationmethod.Forthisapproach,thedataselectionstepisnotneededduetothehighdetectionaccuracyasaresultofthesimpleBPSKusedintheSIGNALeld,aswellasthehighdecodingaccuracyduetotheun-punctured(lowrate)channelcoding.2.3.5NumericalExamples

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(a)(b) (c) Figure2{9:PERversusSNRcomparisonforcoarsesymboltimingandnesymboltimingfortime-invariantchannelswithvarioustr'sforaSISOsystematvariousdatarates:(a)datarate=12Mbps;(b)datarate=24Mbps;(c)datarate=54Mbps.showstheadvantageofthenesymboltimingoverthecoarsesymboltiming.Weconsiderfourcaseshere:CaseA: estimatedchannelparameters(CFO,symboltiming,andchannelareallestimatedwiththesymboltimingbeingtheshiftedversionofthecoarsesymboltimingestimatewhichisobtainedbyTS=TC+NC=2[cf.(2.24)])togetherwithMLphasetrackingusingpilottones;tr=50ns;CaseB: estimatedchannelparameters(CFO,symboltiming,andchannelareallestimatedwiththesymboltimingbeingthenesymboltiming)togetherwithMLphasetrackingusingpilottones;tr=50ns;CaseC: thesameasCaseAexceptthattr=100ns;

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(a)(b) (c) Figure2{10:PERversusSNRfortime-invariantchannelswithtr=50nsforaSISOsystematvariousdatarates:(a)datarate=12Mbps;(b)datarate=24Mbps;(c)datarate=54Mbps.CaseD: thesameasCaseBexceptthattr=100ns. Figures2{9(a),2{9(b),and2{9(c)showthePERcurvesfortheabovefourcasesasafunctionoftheSNRwhenthetransmissiondataratesare12,24,and54Mbps,respectively.WecanseethatalthoughthenesymboltimingdoesnotprovideappreciablePERperformanceimprovementoverthecoarsesymboltimingforchannelswithtr=50ns(whichmeansthattheshifttothecoarsesymboltimingisreasonable),itdoessupportmuchbetterPERperformancethanthelatterforchannelswithtr=100ns,especiallyinthehighdataratecases.

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estimatedchannelparameters(CFO,symboltiming,andchannelresponseareallestimatedwiththesymboltimingbeingthenesymboltiming)togetherwithMLphasetrackingusingpilottones;Case2: estimatedchannelparameters(thesameasCase1)withLSphasettingusingallOFDMdatasymbolsintheOFDMDATAeld;Case3: estimatedchannelparameters(thesameasCase1)withLSphasetting(thesameasCase2)plusoneiterationofthesemi-blindchannelestimationmethod;Case4: estimatedchannelparameters(thesameasCase1)withLSphaset-ting(thesameasCase2)plustwoiterationsofthesemi-blindchannelestimationmethod;Case5: perfectchannelknowledge(withCFO,symboltiming,andchannelre-sponseallassumedknown).(SeeAppendix2.5.6foranoteontheCFOeectsimulationfortheperfectchannelknowledgecase.) InFigures2{10(a),2{10(b),and2{10(c),weshowthePERperformanceasafunctionoftheSNRforchannelswithtr=50nswhenthetransmissiondataratesare12,24,and54Mbps,respectively.WeobservefromthePERcurvesthat:

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(a)(b) (c) Figure2{11:PERversusSNRinthepresenceofsamplingclockerrorfortime-invariantchannelswithtr=50nsforaSISOsystematvariousdatarates:(a)datarate=12Mbps;(b)datarate=24Mbps;(c)datarate=54Mbps.forthepresenceofthesamplingclockerror;(b)fR=5:25GHz;(c)osetsofthereferenceoscillatorsatthetransmitterandthereceiverareidentically,independently,anduniformlydistributedover[20;20]ppm(partspermillion).(Themaximumreferenceoscillatortolerancesuggestedbythestandardis20ppmforboththetransmitterandthereceiver.)Weconsiderthreecaseshere:Casei: standardprocessing|theprocessingbasedonestimatedchannelparame-tersandtrackedCPE,asinCase1ofExample4;Caseii: standardprocessingplussamplingclocksynchronization;

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withoutsamplingclockerror,i.e.,thesameasCase1ofExample4,in-cludedasareference. Figures2{11(a),2{11(b),and2{11(c)showthePERcurvesfortheabovethreecasesasafunctionoftheSNRforchannelswithtr=50nswhenthetransmissiondataratesare12,24,and54Mbps,respectively.Thefollowingobservationsareimmediate: thesameasCaseiiofExample5;Caseb: processingofCaseaplusreal-timeLSphasettingwithKS=9;Casec: processingofCasebplusreal-timesemi-blindchannelresponseestimation;Cased: thesameasCase5ofExample4,includedasareference. Figures2{12(a),2{12(b),and2{12(c)showthePERcurvesfortheabovefourcasesasafunctionoftheSNRforchannelswithtr=50nswhenthetransmissiondataratesare12,24,and54Mbps,respectively.SignicantPERimprovementscanbeobservedfromthesegures,especiallyforthecasesoflowerdatarates.ThelargergapsbetweenCasecandCasedinFigure2{12ascomparedtothegapsbetweenCase3andCase5inFigure2{10areduetothepoorerperformanceofthereal-timesemi-blindchannelresponseestimationmethod,whichusestheSIGNALeldonly.Example7.

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(a)(b) (c) Figure2{12:PERversusSNRfortime-invariantchannelswithtr=50nsforreal-timeprocessinginthepresenceofsamplingclockerrorforaSISOsystematvariousdatarates:(a)datarate=12Mbps;(b)datarate=24Mbps;(c)datarate=54Mbps.thecomputationaleciencyofournewmethod.Numericalresults,asshowninTable2{1,manifestthatournewsequentialmethodisabout70timesmoreecientthantheexistingone.2.4ConcludingRemarks

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Table2{1:ComparisonoftheaveragenumbersofopsbetweenanexistingchannelparameterestimationmethodandtheproposedmethodforaSISOsystem. Existingmethod Proposedmethod Coarsetiming 4,247 1,171 Finetiming 712,694 5,783 Channelestimation 9,443 3,278 Overall 726,384 10,232 Inparticular,rst,wehavegivenachannelmodelwhichcanbeusedtocharacterizetherealisticchannelsfortheOFDMsignaling.Second,byexploitingthestructureofthepacketpreamblespeciedbytheIEEE802.11astandard,wehaveprovidedasequentialmethodfortheestimationofCFO,symboltiming,andchannelresponse.Unlikesomeoftheexistingchannelparameterestimationmethods,ourmethodworkswellfortherealisticchannelsandisabout70timesmoreecientthananexistingmethod.Finally,wehaveconsideredtheerrorreductionproblem.TocorrecttheCPEusingthepilottones,wehaveconsideredMLphasetrackingandLSphasettingapproaches.Theproperphasettingcanleadtoagainofupto1.2dBintermsofPER.Toimprovethechannelestimationaccuracy,wehavepresentedasemi-blindchannelresponseestimationmethodwhichcanresultinanadditionalgainofupto1.5dB.Tocorrectthesamplingclockerror,wehaveprovidedasamplingclocksynchronizationapproachthatavoidstheuseofanAFCcircuit.Thealgorithmsgivenhereinhavebeenextendedtothecaseofmultiplereceiveantennas.2.5Appendixes2.5.1DerivationsofEquations(2.8)and(2.9) NS;nS=1;2;:::;NS;(2.50)

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where[nS1]NS=8><>:nS1;nSNS=2;nS1NS;nS>NS=2:(2.51) second,withtheaboveexpression,wegivethederivationof(2.8).Thelth,l=0;1;:::;NS1,elementoftheTypeAequivalentdiscretechannelresponseofh(t)canbewritten,duetoDFT,as:h(t)l=NSXnS=1Xppej2p[nS1]NS NS!ej2l(nS1) NS!ej2ln NS=XppNS=21Xn=NS=2ej2(pl)n NS=Xppej(pl)NS1Xn=0ej2(pl)n NS=Xppej(pl)1ej2(pl) sin((pl)=NS):(2.53) Notethatwithoutadequatelyaddressingthe[]NSproblemin(2.52),i.e.,mis-takenlyusingnS1insteadof[nS1]NS,whichputthediscontinuedpointsoftheperiodicfrequency-domainresponseat:::;64;0;64;:::ratherthan:::;32;32;:::,(cf.Figure2{4,and),[118]givesanincorrectresultas:h(t)l=Xppej(l+(NS1)p)=NSsin(p) sin((pl)=NS):(2.54) Third,wegivethederivationof(2.9).Usingthesameapproachasbefore,thelthelementoftheTypeBequivalentdiscretechannelresponseofh(t)canbewritten

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NS11A=Xppej(pl)NSC NSNSC1Xn=0ej2(pl)n NS1!=Xpp0@ej(pl)NSC NS1ej2(pl)NSC NS sin((pl)=NS)1:(2.55)2.5.2Equivalenceof(2.28)withtheMatchedFilterMethodofvanNeeandPrasad[120] Considerablockofr(t)(v)withlengthNS.ThenthenSthsubcarrier,nS=1;2;:::;NS,ofitsFFTcanbewrittenas:r(nS)=NS1Xv=0r(t)(v)ej2v(nS1)=NS=NS1Xl=0sB(l)ej2l(nS1)=NSNS1Xv=0z(l+v)ej2(l+v)(nS1)=NS=xBy;(2.57)

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wherexBandyare,respectively,thenSthelementofxBandyTintroducedaround(2.27).(Herewedropthesubscriptnforconvenience.)TakingtheFFTof(2.28),weobtainexactly(2.57)forthenSthsubcarrier.Thisconcludestheproofoftheequivalence. NotethatournesymboltimingestimationmethodusesareceiveddatablockoflengthNS,whilethematchedltermethodusesareceiveddatablockoflength2NS.Hence,inthepresenceofnoise,theequivalenceonlyholdsapproximatelyduetodierentnoises.Nevertheless,thereisnonoticeabledierenceinthesymboltimingestimationperformancebetweenthesetwomethods.2.5.3ASimpleSISOSoft-Detector Let^h4=hh:(2.59) Sinceonlytheestimatedchannelresponseisavailabletothedetector,wehave^xk=1 NotethattheerrortermnkincludesboththeerrorduetotheestimatedchannelandtheperturbationduetotheadditivewhiteGaussiannoise.Byassumingstatistical

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Keepingthevaluesof^xkandusingittogetherwith3^2=(2j^hj2)(see(2.33)onobtaining^2)canprovidethesoft-detectionoutputforthedatasymbolxk.Thisoutputcanbeconvertedtothesoft-informationforeachdatabitinxk,usingthemethodshowninthenextAppendix,whichparallelsthemethodin[13].Thissoft-informationisneededbytheViterbialgorithm.2.5.4BitMetricCalculationfortheQAMSymbol LetDi;j=fs:s2RgdenotethesetofallthepossiblePAMsymbolswiththeithbitbi=j,i=1;2;;B=2,j=1;1.TheformationofDi;jdependsonthewaythatthePAMsymbolsarelabeled.Forexample,fortheGrayindexed8-aryPAMconstellationshowninFigure2{13,wehaveD1;1=f7;5;3;1g;(2.62)

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wheretherstbitistheleftmostoneshowninFigure2{13(0inthegurecorrespondsto1inthecontext).Then,foragivensoft-information(x;)ofthePAMsymbol,thebitmetricforbiisgivenbyli=logp(bi=1jx) wherep(bi=jjx)=p(Di;jjx)(2.64)=Xs2Di;jp(sjx)=Xs2Di;jf(xjs)p(s) 22(2.65) beingtheprobabilitydensityfunction(pdf)giventhesymbolsandthevariance2,asshowninFigure2{13.TheoccurrenceofeachsymbolinRisoftenassumedtobeequally-likely,i.e.,p(s)=(1=2)B=2;8s2R.Inthiscase,wehavep(bi=jjx)=1 2B=2p(x)Xs2Di;jf(xjs);(2.66) whichleadstoli=log8<:Xs2Di;1e(xs)2 229=;log8<:Xs2Di;1e(xs)2 229=;:(2.67) Tospeedupthebitmetriccalculationinpracticalapplications,wecanmakeagridforxandtopre-calculatealook-uptableforthe^vi's.ThebitmetriccalculationinoursimulationsforPERisbasedonsuchatable.2.5.5MLPhaseFitting

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Figure2{13:Illustrationof8-aryPAMsymbolswithGraylabelingandtheirpdfcurves.givenin(2.34)asy(p)k=ej(1+k2)bkPh(p)+e(p)k;k=0;1;:::;K;(2.68) wherePisaknown44diagonalmatrix,bkisaknownBPSKsymbol,andh(p)isthetruebutunknownchannelresponseonthepilottones.(Notethatwemodelthechannelresponseasunknownratherthanusingitsestimate^h(p)heresincetheformerwaycanresultinbetterestimateofh(p)aswellas1and2usingtheexactMLcriteria.)Thelog-likelihoodfunctionisf=KXk=0y(p)kej(1+k2)bkPh(p)2;(2.69) which,bylettingY=hy(p)0y(p)1:::y(p)Ki;(2.70)u=boej1b1ej(1+2):::bKej(1+K2)T;(2.71) andh(p)=Ph(p);(2.72) canberewrittenasf=trhYh(p)uTYh(p)uTHi=trYYHYuhH(p)h(p)uTYH+h(p)uTuhH(p)=trhh(p)Yu(uTu)1uTuh(p)Yu(uTu)1H+YYHYu(uTu)1uTY:(2.73)

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Byletting^h(p)=Yu(uTu)1(2.74) andconsideringthefactthat(uTu)1=1=(K+1);(2.75) wehave[^1;^2]=arg[1;2]mintrYYH1 whichcanbeimplementedbyatwo-dimensionalsearch.Tospeedupthesearch,wecanusetheLSttingresultasaninitialvalue.2.5.6ANoteontheCFOEectSimulation whereh(t)ej2fStistheactualchannelresponseforthiscase.

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TheOFDM-basedMIMOwirelesscommunicationsystemsconsideredinthischapterinclude:(A)theSTBC(space-timeblockcoding)systemthatfocusesonincreasingthetransmissionrobustnessand(B)theBLAST(Bell-labs'layeredspace-time)systemthatfocusesonincreasingthetransmissiondatarate.AgenericMIMOsystemisshowninFigure3{1,whereMandNdenotethenumberoftransmitandreceiveantennas,respectively. Weconsidervarioussystemdesignandsignalprocessingissuesfortheafore-mentionedtwokindsofMIMOsystems.NotethatourdesignistargetingataSTBCsystemwithtwotransmitantennasandonereceiveantennaaswellasaBLASTsystemwithtwotransmitantennasandtworeceiveantennas.WhilesomeofthesignalprocessingalgorithmsareconnedtotheMIMOsystemswithtwotransmitantennas,otherscanbeusedinmoregeneralcases,aswillbeclearinthesequel.WeexemplifyourpresentationoftheOFDM-basedMIMOsystemswiththeWLANapplications. AsacommonpartforbothoftheOFDM-basedBLASTandSTBCsystems,weproposeaMIMOpreambledesignfortheMIMOsystems.ThisMIMOpreambleisbackwardcompatiblewithitsSISOcounterpartasspeciedbytheIEEE802.11astandard.Basedonthispreambledesign,wecanusethesequentialparameterestima-tionmethodpresentedinthepreviouschaptertoestimateCFOandsymboltiming.WecanalsoestimatetheMIMOchannelresponses. FortheOFDM-basedSTBCsystem,weproposeanOFDMDATAelddesignwhichincludesthepairingoftheOFDMdatasymbolsaswellasthepilottones.WeprovideasimpleOFDMdatasymbolgenerationschemefortheSTBCtransmitter.71

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Figure3{1:DiagramofaMIMOsystem.TheSTBCtransmitterbasedonthisschemehasacomplexityapproximatelythesameasitsSISOcounterpart.WealsoprovideerrorreductionalgorithmswhicharesimilartothosefortheSISOsystemsprovidedinthepreviouschapter.Furthermore,wepresentasimplesoft-detectortoobtainthesoft-informationusedforthechanneldecoder,suchastheViterbialgorithm. FortheOFDM-basedBLASTsystem,weproposeanOFDMDATAelddesignwhichincludesthepairingofthepilottonesusedforerrorreduction.Wealsopro-posethreeBLASTsoft-detectors,whichcandeliversoft-informationtothechanneldecoder,withdierentperformance/complexitytrade-ostofacilitatethereal-timehardwareimplementationofthedetector.Therstoneisasimplesoft-detectorbasedontheunstructuredLSapproach.ThisLS-basedsoft-detectorisverysimplesinceitdecouplesthemulti-dimensionalQAMsymboldetectionintomultipleone-dimensionalQAMsymbol|andfurtherPAMsymbol|detections.Thesecondoneisahybridsoft-detectorthatcombinesthemeritsoftheLS-basedsoft-detectorandalistspheredecoder(LSD)-basedsoft-detectorfordatabitdetection.Thishybridsoft-detectorhasagoodperformance/complexitytrade-o.Thethirdoneisanitera-tivesoft-detectorwhichusesconstrainedvalueoftheaprioriinformationtoimprovethedetection/decodingperformance.Thisiterativesoft-detectorhasthebesteverknownperformance.

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Inthischapter,weprovideseveralsimulationexamplestoshowtheoverallsystemperformanceoftheMIMOsystemsthatuseoursystemdesignandvarioussignalprocessingalgorithms. Theremainderofthischapterisorganizedasfollows.Section3.1givesanoverviewofSTBC.Section3.2introducesthechannelcodinganddatamodelforagenericBLASTsystemwhichareusefulfordevelopingsoft-detectors.Section3.3describesthedesignoftheMIMOsystems,whichincludesthenewMIMOpreambledesignandtheOFDMDATAelddesignforthetwokindsofMIMOsystems.Sec-tion3.4presentsoursequentialmethodforCFO,symboltiming,andMIMOchannelresponseestimation.Section3.5provideserrorreductionapproachesfortheMIMOsystems.Section3.6describesthesimplesoft-detectorfortheSTBCsystem.Sec-tions3.7{3.9deliverthethreesoft-detectorsfortheBLASTsystem:theLS-basedsoft-detectorinSection3.7,thehybridsoft-detectorinSection3.8,andtheiterativesoft-detectorinSection3.9.Section3.10considertheproblemofcorrelatedchannelsandtheimpactonthedesignandperformancefortheBLASTsystems.Finally,weendthischapterwithcommentsandconclusionsinSection3.11.3.1OverviewofSTBC wherehm,m=1;2,denotesthechannelgainfromthemthtransmitantenna(Tx)tothereceiveantenna. Letx1;x22Cbethetwodatasymbolstobetransmittedintwoconsecutivetimeintervals.FortheAlamouticode[2](aspecialcaseofSTBC[33,35,34,113]

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wheree1;e2N(0;2)aretheadditivezero-meanwhitecircularlysymmetriccom-plexGaussiannoiseswithvariance2.Notethatwewillabsorbthecoecient1=p FortheOFDM-basedSTBCsystem,theaboveSTBCcodingschemeanddatamodelarevalidforeachdata-carryingsubcarrier,andhencetheabovey1,y2,h,x1,x2,e1,ande2areallsubcarrierdependent.Letx1andx2bethetwovectorsofNSdatasymbolsintendedtobetransmittedontheNSsubcarriers,ands1=WHNSx1=NSands2=WHNSx2=NSbethetwoSISOOFDMdatasymbolscorrespondingtox1andx2,respectively.Then,thetwotransmitantennasoftheOFDM-basedSTBCsystemsimultaneouslytransmits1;1=s1;(3.3)s2;1=s2(3.4) attime1,andthens1;2=WHNSx2=NS;(3.5)s2;2=WHNSx1=NS(3.6) attime2.(Ofcourse,CPsareattached.)Hence,thetransmissionoftheOFDMdatasymbolsarepairedfor2consecutivetimeintervals.

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Adirectcalculationoftheabovetwoequationsindicatesthat[122]s1;2=Js2;(3.7)s2;2=Js1;(3.8) whereJ=26666666666410000001...............00100100377777777775(3.9) isapermutationmatrixwhichperformstheoperationoftime-invert(withone-shift).Equations(3.7)and(3.8)willleadtoasimpleOFDMdatasymbolgenerationscheme,asdetailedlater.3.2ChannelCodingandDataModelforaGenericBLASTSystem

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Atthetransmitter,asshowninFigure3{2,theconvolutionalchannelencodertakesablockofdatabitsd=fd1;d2;:::;dKg2f1;+1g1Kasitsinputandgivesalargerblockofdatabitsu=C(d)=fu1;u2;:::;uKg2f1;+1g1Kasitsoutput,where1and+1denotethelogic0and1,respectively.Thecodingrateisthende-nedasRC=K=K.Insomecaseswepuncturetheencoderoutputblockutoobtainasmallerblockofdatabitsv=fv1;v2;:::;v~Kg2f1;+1g1~K(~K
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Figure3{2:DiagramofaBLASTtransmitteremployingconvolutionalchannelcod-ing. Figure3{3:DiagramofaBLASTreceiveremployingViterbialgorithmforconvolu-tionalchannelcoding. Atthereceiver,asshowninFigure3{3,thesoft-detectorrstgeneratesthebitmetricsl(k)=[l(k)1l(k)2:::l(k)BM]T,withl(k)ibeingthebitmetriccorrespond-ingtob(k)i,i=1;2;:::;BM,attimek=1;2;:::;K.Thesoft-detectorthenrearrangesthebitmetricstoobtainf^v([k1]B+1)m;^v([k1]B+2)m;:::;^v(kB)mgforthedatabitsfv([k1]B+1)m;v([k1]B+2)m;:::;v(kB)mg,whichweremappedtothesymbolx(k)m.Let^vm=f^v(1)m;^v(2)m;:::;^v(K0)mg,m=1;2;:::;M,denotethebitmetricsequencecorre-spondingtothemthtransmittedlayer.TheMbitmetricsequencesarecombinedintoonelongerbitmetricsequence^v=f^v(1);^v(2);:::;^v(~K)gbytheM:1MUX.Passingtheabovebitmetricsequence^vthroughthedeinterleaver,weobtainthedeinterleavedbitmetricsequence^v=f^v1;^v2;:::;^v~Kg.Forthepuncturedbitse-quence,weneedthebitmetricforeachpuncturedbitaswellbeforeusingtheViterbialgorithm.Thiscanbedoneeasilybyusingzeroasthebitmetricforeachpunc-turedbit.Oncewegetthebitmetricsequence^u=f^u1;^u2;:::;^uKgcorrespondingtotheencoderoutputu,wecanusetheViterbialgorithmtoobtaintheestimate^d=f^d1;^d2;:::;^dKgofthesourcebinarybitsequenced.

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Herein,wefocusonthecalculationofthebitmetricsforthedatabitsintheQAMsymbol,duetoourWLANapplication.3.2.2DataModel whereh(k)n;misthe(complex)channelgainfromthemthtransmitantennatothenthreceiveantennaattimek,whichisassumedtobeknown.Withx(k)=[x(k)1x(k)2:::x(k)M]TdenotingtheM1QAMsymbolvectorbeingsentattimek,thereceivedsignalcanbewrittenasy(k)=H(k)x(k)+e(k)2CN1;k=1;2;:::;K;(3.11) wheree(k)N(0;2kIN)istheadditivezero-meanwhitecircularlysymmetriccom-plexGaussiannoise. Withanappropriatepairofinterleaveranddeinterleaver,theMIMOchannelcanbeassumedtobeblockat-fading[38,86],i.e.,H(k)isconstantattimekforthetransmissionofx(k)butchangesindependentlyfromonetimeindextoanother.Inthelatersections,wefocusonobtainingthesoft-informationgiventhatweknowthechannelmatrixH(k),thenoisevariance2k,andthereceiveddatavectory(k).Fornotationalconvenience,wewilldropthesuperscriptkinEquation(3.11)togetthedatamodely=Hx+e2CN1;(3.12) ory=Hx(b)+e;(3.13)

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ifthedependenceonbneedstobestressed.3.3SystemDesign TheMIMOsystemsconsideredhereinhavetwotransmitandoneortworeceiveantennas.Twopacketsaretransmittedsimultaneouslyfromthetwotransmitan-tennas.Wedesigntwopreambles(collectivelycalledtheMIMOpreamble),oneforeachtransmitantenna.WeassumethatthereceiveantennaoutputssuerfromthesameCFOandhavethesamesymboltiming.TobebackwardcompatiblewiththeSISOpreamblethatisshowninFigure3{4(a),weusethesameshortOFDMtrainingsymbolsasintheSISOpreambleforbothofthetwotransmitantennas,asshowninFigure3{4(b).Hence,theMIMOpreambledesignismainlyaboutthedesignofthelongOFDMtrainingsymbols,whichareusedfortheMIMOchannelresponseestimation.

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ChannelresponseestimationfortheMIMOsystemshasattractedmuchresearchinterestlately.Orthogonaltrainingsequencestendtogivethebestperformance(see,forexample,[57]andthereferencestherein).Weadoptthisideaoforthogonaltrainingsequencesinourpreambledesign.Intheinterestofbackwardcompatibility,weusethesameT1andT2(aswellasGI2)asfortheSISOsystemforbothofthetwotransmitantennasbeforetheSIGNALeld,asshowninFigure3{4.AftertheSIGNALeld,weuseT1andT2(andGI2)foronetransmitantennaandT1andT2(andGI2)fortheother.Thisway,whenthesimultaneouslytransmittedpacketsarereceivedbyaSISOreceiver,theSISOreceivercansuccessfullydetectuptotheSIGNALeld,whichisdesignedtobethesameforbothtransmitantennas.ThereservedbitintheSIGNALeldcantelltheSISOreceivertostopitsoperationwheneveraMIMOtransmissionfollowsorotherwisetocontinueitsoperation.ThelongOFDMtrainingsymbolsbeforeandaftertheSIGNALeldareusedintheMIMOreceiversforchannelresponseestimation.Althoughtheemploymentofanadditionalpairoflongtrainingsymbolswillincreasetheoverhead,thecorrespondinglossofeciencyisnotsignicantforlargerpackets.ThereservedbitintheSIGNALeldcanalsoinformtheMIMOreceiversthatatransmissionisaSISOone.Whenthisoccurs,theMIMOreceiverscanmodifytheirchannelestimationanddatabitdetectionstepsslightly. OtherMIMOpreambledesignoptionswithbackwardcompatibilityarepossible.Forexample,byexploitingthetransmit/receivediversitiesviausingsubcarrierlevelorthogonalityoftheshortOFDMtrainingsymbols,wemaygetimprovedsymboltim-ingand/orCFOcorrection.However,theseimprovementsdonotnecessarilyresultinimprovedpacketerrorrate(PER)performance.HencewepreferthestraightforwardMIMOpreambledesignshowninFigure3{4(b).

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(a) (b) Figure3{4:PacketpreamblesfortheWLANsystems:(a)thestandardizedSISOpreamble;(b)the(proposed)MIMOpreamble. InthecaseofMIMOtransmissions,anadditionalbitisneededtodierentiatetheSTBCtransmissionfromtheBLASTone.Thisbit,alongwithotherbitsconveyingotherinformation,canbesentinanadditionalSIGNALeld.3.3.2OFDMDATAFieldDesignfortheSTBCSystem TostayasclosetoIEEE802.11aaspossible,weusethesamescrambler,convo-lutionalencoder,puncturer,interleaver,symbolmapper,andCPasspeciedintheIEEE802.11astandardforourSTBCsystem.(However,zerosmaybepaddedtotheendofthedatabitsequencetomakethenumberofOFDMdatasymbolseven.)TofacilitatetrackingtheresidueCFOinducedphaseerrorusingthepilottones,wealsopairthepilotsymbols,i.e.,weusethesamepilotsymbolsfors2k1ands2k,k=1;2;;K,fortheSTBCsystem;thisisdierentfromtheSISOsystem.

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(a) (b) Figure3{5:OFDMDATAeldsfortheSISOandSTBCsystems:(a)SISO;(b)STBC. BasedontheSTBCOFDMDATAeldshowninFigure3{5(b),asimpletrans-mitterwithasimpleOFDMdatasymbolgenerationschemeisshowninFigure3{6,whereFIFO(rst-inrst-out)andStack(rst-inlast-out)1andStack2havethesamewidth(forexample,12bits)forboththerealandimaginarypartsofthe(com-plex)datasamples.ThecomponentswithinthedashedframeareidenticaltothoseforanIEEE802.11aconformableSISOWLANtransmitter. ThisSTBCtransmitterworksinthefollowingwaytogeneratetheOFDMdatasymbolsintheOFDMDATAeld.Attime2k2(thetimeassociatedwithOFDMdatasymbols2k2fortheSISOpacket[cf.Figure3{5(a)]withk=1standingforthelastlongtrainingsymbol),s2kisgeneratedandthewaveformisputintotheFIFO(usedasadelaydevice)andStack1(usedasatime-reverseandshiftdevicetorealizetheoperationofJ).Attime2k1,s2k1isgeneratedandputintotheFIFOandStack1;theswitchesareconnectedtotheupperswitchpolesands2k1ands2k(s2kisobtainedbyusingtheFIFO)aretransmittedsimultaneouslyfromTx1andTx2,respectively;theoutputoftheFIFOisalsostoredintoStack2.Attime2k,s2(k+1)isgeneratedandputintotheFIFOandStack1(thesearethepreparationforthenextOFDMdatasymbol,inthesamewayasfors2kattime2k2);theswitchesareconnectedtothelowerswitchpolesandtime-reversedwaveformsJs2kandJs2k1

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Figure3{6:DiagramofanOFDM-basedSTBCtransmitterbasedonasimpleOFDMdatasymbolgeneratingscheme.(obtainedbyusingtheStack2and1,respectively)aretransmittedsimultaneouslyfromTx1andTx2,respectively.3.3.3OFDMDATAFieldDesignfortheMIMOBLASTSystem Figures3{7(a)and3{7(b)showtheOFDMDATAeldsfortheSISOsystemandtheBLASTsystem,respectively.SimilartotheSISOandSTBCsystems,weneedtousethepilottonestotracktheresidueCFOinducedphaseerror.Tofacilitatethistracking,thetwoOFDMdatasymbolsthataresentfromthetwotransmitantennassimultaneouslyarepairedtohaveidenticalpilotsymbols,asfortheSTBCpilottones.

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(a) (b) Figure3{7:OFDMDATAeldsfortheSISOandBLASTsystems:(a)SISO;(b)BLAST. Notethatwithoutconsideringthespatialinterleaver,theOFDMDATAeldfortheBLASTsystemcanbeseenasobtainedfromthatoftheSTBCsystembydiscardingtheOFDMdatasymbolsattheeventimeintervals,i.e.,theOFDMdatasymbolsgeneratedbyusingtheoperationofJ.3.4ChannelParameterEstimation AsfortheMIMOchannelresponseestimation,itisonlyasimpleextensionfromthatoftheSISOcase.3.4.1ANoteonCFOandSymbolTimingEstimation

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Similarly,thenesymboltimingestimationcouldalsobeimprovedbyusingthelongOFDMtrainingsymbolsaftertheSIGNALeld.Yet,duetothesucientaccuracyofthenesymboltiming,asdemonstratedinthepreviouschapter,wedonotfurtherconsidertheimprovedmethod.3.4.2MIMOChannelResponseEstimation wherexBdenotesthenSthelementofxB(thevectorofthetrainingsymbols),yn;idenotesthenSthelementofyn;i,i=1;2[cf.(2.15)],anden;iN(0;2=2),i=1;2,wherethevariance2=2isduetoaveragingthetwoconsecutiveblocks.(Herein,wehavedroppedthedependenceonnSfornotationalsimplicity.)Solving(3.14)and(3.15)yields^hn;1=(yn;1+yn;2)=(2xB);(3.16)^hn;2=(yn;1yn;2)=(2xB):(3.17)

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Fromtheaboveequations,weseethatthevarianceoftheestimationerrorfor^hn;m,m=1;2,is2=2,thesameasfortheSISOcase,sinceEfjxBj2g=1=2fortheMIMOsystems. WhenthereservedbitintheSIGNALeldindicatesaSISOtransmission,weonlyneedtoestimatehn;1,n=1;2;;N,inawaysimilarto(3.16).3.5ErrorReduction Let^h(p)n;mbethe41estimatedchannelresponsevectorcorrespondingtothe4pilottonesfromthemthtransmitantennatothenthreceiveantenna,m=1;2,n=1;2;:::;N.Fork=1;2;;K,letgn;2k1=^h(p)n;1+^h(p)n;2(3.19)

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andgn;2k=^h(p)n;1+^h(p)n;2:(3.20) Lety(p)n;k,k=1;2;;2K,denotethe41datavectorcorrespondingtothepilottonesinyn;kobtainedbyperformingtheFFTonthekthreceiveddatavector,fromthenthreceiveantenna,duetotheinputbeingthesimultaneouslytransmittedOFDMdatasymbolsattimekintheOFDMDATAeld.Then,fork=1;2;:::;2K,wehave,similarto(2.34)ofthepreviouschapter:y(p)n;k=ejkPkgn;k+e(p)n;k;(3.21) wherekistheresidueCFOinducedphaseerror,Pk=diagfpkg,ande(p)n;kdenotesthecorrespondingzero-meanwhitecircularlysymmetriccomplexGaussiannoisevector.TheMLestimateofkisthencomputedas^k=argminkNXn=1y(p)n;kejkPkgn;k2=\(NXn=1gHn;kPHky(p)n;k):(3.22)3.5.2MLPhaseTrackingfortheBLASTSystem

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First,wereconstructthedatasymbolsusingthedetected/decodeddatabitsviafollowingthesameproceduresofscrambling,convolutionalencoding,interleaving,andsymbolmappingasfortheoriginaldatabitsattheSTBCtransmitter. Second,weselectthereconstructeddatasymbolswhichcanbeusedinthesemi-blindchannelestimation.HerewedotheselectioninthelevelofOFDMdatasymbolpairs;i.e.,wedetermineifthe248=96restructureddatasymbolsassociatedwiths2k1ands2kwillbeusedforthesemi-blindchannelestimation.Letxk,k=1;2;;2K,denotethedatasymbolonthenSthsubcarrier,whichisusedtoformtheOFDMdatasymbolskintheOFDMDATAeld.(Fornotationalconvenience,weagaindropthedependenceonnSbelow.)Letxkdenotethereconstructeddatasymbolcorrespondingtoxk.Then,fork=1;2;;K,weformthereconstructeddatamatrixas:Xk4=264x2k1x2kx2kx2k1375;(3.24) andletYk=266666664y1;2k1y1;2ky2;2k1y2;2k......yN;2k1yN;2k377777775(3.25)

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bethereceiveddatamatrix,whereyn;2k1andyn;2karethereceiveddataonthenSthsubcarrierattimes(2k1)and2k,respectively,fromthenthreceiveantenna.Let^H=266666664^h1;1^h1;2^h2;1^h2;2......^hN;1^hN;2377777775(3.26) bethechannelmatrix,where^hn;mistheestimatedchannelresponseonthenSthsubcarrierfromthemthtransmitantennatothenthreceiveantenna.Then,wecalculatethedistancedk=Yk^HXk2(3.27) forthenSthsubcarrier.Wethenaddupsuchdistancesforall48datacarryingsubcarriersandcomparethesumwithathreshold,whichwechoosetobe144N^2herein,i.e.,3Ntimesthenoisevariance,48^2,fordk'sonallofthe48datacarryingsubcarriers.Ifthesumisbelowthethreshold,weusetheestimateddatasymbolsassociatedwiths2k1ands2kforthesemi-blindchannelestimation;otherwise,wedropthemfromfurtherconsiderations. Finally,weestimatetheMIMOchannelresponseforeachsubcarrierinasemi-blindway.ConsiderthenSthsubcarrier.LetYRbetheN-rowmatrixformedbyconcatenatingalltheYk's[cf.(3.25)]associatedwiththes2k1ands2k'sthatsurvivetheaforementionedselectionforthesemi-blindchannelestimation.LetXRbethematrixformedfromthecorrespondingXk's[cf.(3.24)]similarlytoYR.WecanalsoformavectorYSandamatrixXSsimilarlytoYRandXRforthelongOFDMtrainingsymbolsbeforeandaftertheSIGNALeld.ThenYSB4=YSYRtHXSXR4=HXSB;(3.28)

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fromwhichwecanobtainthesemi-blindestimateofHviatheLSapproach:^HSB=YSBXHSBXSBXHSB1:(3.29) DuetotheorthogonalstructureofourpreambledesignaswellasXk,itiseasytoverifythatXSXHS=2I2(3.30) andXRXHR=RI2;(3.31) whereRisthesumofthesquaresofalltheestimateddatasymbolsfx2k1;x2kgusedtoformXR.Hence,wehave^HSB=1 2+RYSXHS+YRXHR=2 2+R^H+R wheretheelementsof^Harethoseestimatedin(3.16)or(3.17).3.5.4Semi-BlindChannelResponseEstimationfortheBLASTSystem Therststepisthestillthereconstructionofthedatasymbolsfromthede-tected/decodeddatabits,asfortheSISOandSTBCsystems. ThesecondstepperformstheselectionofOFDMdatasymbolsbasedononetimeintervalonlysince,aswehavementionedearlier,theOFDMDATAeldfortheBLASTsystemcanbeseenasobtainedfromthatoftheSTBCsystembydiscardingtheOFDMdatasymbolsfortheeventimeintervals.Fork=1;2;;K,weformthereconstructeddatavectorattimekas:xk4=264x2k1x2k375;(3.33)

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wherex2k1andx2k,k=1;2;:::;K,denotethedatasymbolonthenSthsubcarrier,whichareusedtoformtheOFDMdatasymbolss2k1ands2kintheOFDMDATAeld,andletyk=266666664y1;ky2;k...yN;k377777775(3.34) bethereceiveddatamatrixattimek,whereyn;kisthereceiveddataonthenSthsubcarrier,fromthenthreceiveantenna.LetHbeas(3.26).Wecalculatethedistancedk=yk^Hxk2(3.35) forthenSthsubcarrier.Weaddupsuchdistancesforallofthe48datacarryingsubcarriersandcomparethesumwithathreshold,whichwechoosetobe96N^2herein,i.e.,twicethenoisevariance,248N^2.Ifthesumisbelowthethreshold,weusetheestimateddatasymbolxkassociatedwiths2k1ands2kforthesemi-blindchannelresponseestimation;otherwise,wedropitfromfurtherconsiderations. Thethirdstep,likethatfortheSISOandSTBCsystems,estimatestheMIMOchannelresponseforeachsubcarrierinasemi-blindway.ConsiderthenSthsubcar-rier.LetYRbetheNrowmatrixformedbycollectingalltheyk'sassociatedwiththepairofs2k1ands2k'sthatsurvivetheaforementionedselectionprocess.LetXRbethematrixformedfromthecorrespondingxk'ssimilarlytoYR.WecanalsoformmatricesYSandXSsimilarlytoYRandXRforthelongOFDMtrainingsymbolsbeforeandaftertheSIGNALeld.ThenYSB4=[YSYR]tHXSXR4=HXSB;(3.36) fromwhichwecanobtainthesemi-blindestimateofHviatheLSapproach:^HSB=YSBXHSBXSBXHSB1:(3.37)

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ConsiderthenSthsubcarrieraftertakingtheFFTsoftworeceiveddatablocksattimes(2k1)and(2k).(Fornotationalconvenience,weagaindropthedependenceonnS.)ThereceiveddatamatrixonthenSthsubcarriercanbewrittenas266666664y1;2k1y1;2ky2;2k1y2;2k......yN;2k1yN;2k377777775=266666664h1;1h1;2h2;1h2;2......hN;1hN;2377777775264x2k1x2kx2kx2k1375+266666664e1;2k1e1;2ke2;2k1e2;2k......eN;2k1eN;2k377777775:(3.38) Theaboveequationcanberearrangedas2666666666666664y1;2k1...yN;2k1y1;2k...yN;2k3777777777777775| {z }y {z }H {z }~xk+2666666666666664e1;2k1...eN;2k1e1;2k...eN;2k3777777777777775| {z }e

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Let^H Itiseasytoverifythat^H where2=NXn=1j^hn;1j2+j^hn;2j2(3.42) andI2istheidentitymatrixofdimension2.Sinceonlytheestimatedchannelre-sponsesareavailabletothedetector,wehave^~xk4=264^x2k1^x2k375=1 Notethattheerrorterm~nkincludesboththeerrorsduetotheestimatedchannelandtheperturbationduetotheadditivewhiteGaussiannoise.Byassumingthestatisticalindependenceamong~xk,H 2I2,E[e

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Rounding^x2k1and^x2k,k=1;2;;K,tothenearestsymbolinCwillrenderaSTBChard-detector.Keepingthevaluesof^x2k1and^x2k,k=1;2;;K,andusingthemtogetherwiththenoisevariancewillleadtotheSTBCsoft-detector.SeeAppendix2.5.4ofthepreviouschapterforthebitmetric(soft-information)calcula-tion. Forthecaseofsemi-blindchannelresponseestimation,sinceitismorecompli-catedtodeterminetheaccuracyof^HSBduetothestatisticaldependenceofXRandyR,wesimplyignoretheerrorin^HSBwhenusingthesoft-detector[cf(3.44)]anduse^2=2insteadinthesoft-detectorfordatabitdetection.3.6.2NumericalExamples SimilartotheSISOsystems,weconsiderthreetransmissiondataratesfortheSTBCsystemsinsomeofoursimulationexamples:12Mbps,24Mbps,and54Mbps. Asstatedearlier,fortheSISOsystems,theSNRisdenedasNSC=(NS2)fortheconstellationswhoseaverageenergiesarenormalizedto1.FortheMIMOsystems,includingboththeSTBCandBLASTsystems,theSNRisdenedasNSC=(NS2)forconstellationswhoseaverageenergiesarenormalizedto1/2(i.e.,weusethesametotaltransmissionpowerfortheMIMOsystemsasfortheirSISOcounterpart).Allsimulationresultsareobtainedwith104Monte-Carlotrials.TheMNtime-invariant

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(a)(b) (c) Figure3{8:PERversusSNRforaSTBCsystemwithtwotransmitantennasandonereceiveantennafortime-invariantchannelswithtr=50nsatvariousdatarates:(a)datarate=12Mbps;(b)datarate=24Mbps;(c)datarate=54Mbps.frequency-selectivefadingchannelsbetweentheM=2transmitantennasandtheNreceiveantennasaregeneratedindependentlyviatheexponentialchannelmodelpresentedinthepreviouschapter,changingfromonetrialtoanother.Example1. estimatedchannelparameters(CFO,symboltiming,andMIMOchannelresponseareallestimatedwiththesymboltimingbeingthenesymboltiming)togetherwithMLphasetrackingusingthepilottones;

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estimatedchannelparameters(thesameasCase1)withLSphasettingusingallOFDMdatasymbolsintheOFDMDATAeld;Case3: estimatedchannelparameterswithLSphasetting(thesameasCase2)plusoneiterationofthesemi-blindMIMOchannelresponseestimationmethod;Case4: perfectchannelknowledge(withCFO,symboltiming,andMIMOchannelresponseallassumedknown). InFigures3{8(a),3{8(b),and3{8(c),weshowthePERperformanceofaSTBCsystemwithonereceiveantennaasafunctionoftheSNRforchannelswithtr=50nswhenthetransmissiondataratesare12,24,and54Mbps,respectively.Asareference,wealsogiveineachgureaPERcurvefortheSISOsystemwithperfectchannelknowledgeandatthesamedatarate.WeobservefromthePERcurvesthat: NotethatthegapbetweentheLSphasettingmethodandtheMLphasetrackingmethodfortheSTBCsystemislargerthanthatfortheSISOsystem.ThereasonisthatfortheSTBCdetection,thechannelresponsefortime2k1and2k,k=1;2;:::;K,shouldbethesame.Yet,duetotheusageofMLphasetracking,thechannelresponseforthesetimesarenotthesameanymore.Forthereal-timing

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Figure3{9:PERversusSNRforaSTBCsystemwithtwotransmitantennasandtworeceiveantennasfortime-invariantchannelswithtr=50nsatthe24Mbpsdatarate.processingcase,wecanusethereal-timeLSphasettingapproach,whichispresentedinthepreviouschapter,toimprovetheperformance.Example2.

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ConsiderthedatabitdetectionforagenericBLASTsystemwithadatamodelgivenby(3.13),whichisreproducedhereforconvenience:y=Hx(b)+e:(3.45) Thepurposeofdatabitdetectionistodeterminethebitmetricforeachdatabitcontainedinb.Thebitmetric(alsoknownastheL-value)fortheithbit,i=1;2;:::;BM,isdenedasl(i)=logP(bi=+1jy;H) Thesoft-detectorsthatcalculatetheabovebitmetriccanbedividedintotwoclasses,dependingonwhethertheaprioriinformationofthedatabitsisemployedornot.Thesoft-detectorsinClassonedonotusetheaprioriinformation,andthesedetectorscanbenon-iterativeaswellasiterative.Thesoft-detectorsinClasstwomakeuseoftheaprioriinformation,andthesedetectorsareiterativeinnature. FortheClassonesoft-detectors,undertheassumptionofequalprobabilityforeachdatabitsandwiththeusageoftheBayes'theorem,thebitmetriccanbewrittenasl(i)=logPb2Bi;+1P(yjb;H) whereBi;+1andBi;1arethesetsof2BM1bitvectorsbwithbibeing+1and1,respectively. Undertheassumptionofadditivezero-meanwhitecircularlysymmetriccomplexGaussiannoiseforthereceiveddata,theaboveequationcanbewrittenasl(i)=logPb2Bi;+1e1

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which,byusingthemax-logapproximation[39],canbewrittenas(seeAppendix3.12.2foraderivation):litmaxb2Bi;+11 Thisisinfacttheoptimalbutextremelyinecientnon-iterativeSTBICM-basedsoft-detector. Inthissectionandthenext,weconsidertwoClassonenon-iterativesoft-detectors:(a)thesimpleLS-basedsoft-detectorand(b)thehybridsoft-detector.Inthere-mainderofthissection,wepresentthesimpleLS-basedsoft-detectorfortheBLASTsystem.Thehybridsoft-detectorwillbepresentedinthenextsection.3.7.1TheSoft-Detector wherekkdenotestheEuclideannorm.Thecostfunctionin(3.50)canbewrittenaskyHxk2=yHy+xHHHHxyHHxxHHHy=xHyH(Hy)HHHHxHyy+yHyyH(Hy)HHHHHyy;(3.51) whereHy=(HHH)1HH.Wenote,fromtheaboveequation,thatbyignoringtheconstellationconstraintonxwecanobtainanunstructuredLSestimatex(LS)ofx,whichisgivenbyx(LS)=Hyy=x+Hye4=x+c:(3.52)

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Notethatx(LS)isthesoft-decisionstatisticthatweareinterestedin.Werefertothissimpleschemeofobtainingthesoft-decisionstatisticastheLS-basedsoft-detector.NotethatanecessaryconditionforHHHtobenonsingularisNM.AlsonotethatcisstillGaussianwithzero-meanandcovariancematrixE[ccH]=2Hy(Hy)H=2(HHH)1:(3.53) Duetotheuseoftheinterleaveranddeinterleaver,thedatabitscontainedinxareindependentofeachother.Byignoringthedependenceamongtheelementsofc,wecanconsideronlythemarginalprobabilitydensityfunction(pdf)fortheelementsx(LS)m,m=1;2;;M,ofx(LS).(Notethatanapproximationismadehere,whichcanleadtoperformancedegradation.However,thecomputationisgreatlysimpliedbytheapproximation.)LetHy4=266666664hT1hT2...hTM3777777752CMN:(3.54) Thenthemthelementofc,m=1;2;;M,canbewrittenascm=hTme:(3.55) Obviously,cmisstillGaussianwithzero-meanandvariance2m=E[jcmj2]=khmk22:(3.56) Theestimateoftheabovenoisevariance2canbeeasilyobtainedviathedierenceofthetwoconsecutiveblocksofthenthreceiveantenna,fromwhichwegotyn;1[cf.(3.14)];see(2.33).Notethat2malongwithx(LS)mcanbeusedtocalculatethebitmetricsforthedatabitscontainedinthemth,m=1;2;;M,symbolinx(LS),whichareneededbytheVA.Notealsothatthenoisescorrespondingto

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TheLS-basedsoft-detectorisimplementedinthefollowingtwosteps:StepLS1: ignoretheconstraintofconstellationforxtoobtainanunstructuredLSsymbolestimatex(LS)ofxasx(LS)=(HHH)1HHy;(3.57)StepLS2: forj=1;2;:::;Bandm=1;2;:::;M,obtainthebitmetricforeachbitusingtheBICMscheme[13](similarto(3.49),butfortheSISOcase)l(LS)m(j)=1 whereBm;j;+1andBm;j;1arethesetsof2B1bitvectorsbm2f1;+1gB1withthejthbitbeing+1and1,respectively. WeremarkthatfortheSISOsystemsweusuallyconsideranordinaryQAMsymbolastwoPAMsymbols(e.g.,a64-QAMsymbolcanbeconsideredastwo8-aryPAMsymbols)duetotheorthogonalitybetweentherealandimaginarypartsofaQAMsymbolaswellastheindependencebetweentherealandimaginarypartsoftheadditivecircularlysymmetricGaussianerror.InAppendix3.12.1,weshowthattherealandimaginarypartsofcmareindependentofeachother.HencewecansignicantlysimplifythebitmetriccomputationsinStepLS2byexploitingtheseindependencies;seeAppendix2.5.4ofthepreviouschapter. Byroundingx(LS)m,m=1;2;:::;M,totheclosestpointintheconstellationC,weobtainthehard-detectionoutput^x(LS)mofx(LS)m.Further,weobtaintheoutputoftheLS-basedhard-detector:^x(LS)=[^x(LS)1^x(LS)2:::^x(LS)M]T;(3.59) whichwillbeusedlater.

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Notethattheminimummean-squarederror(MMSE)-basedsoft-detectorisof-tendeemedtobebetterthantheLS-basedone[41].Althoughthiscanbetruefortheconstant-modulusconstellations,suchasPSK,itisnotnecessarilytrueforQAMsymbols,assuggestedbyoursimulationresults(notprovidedhere)duetothedif-ferentpowerlevelsoftheQAMsymbols.HencetheLS-basedsoft-detectorismorepreferablethantheMMSE-basedonesincetheformerisslightlymorecomputation-allyecientthanthelatter.3.7.2NumericalExamples WeconsiderdoublingthetransmissiondatausingaBLASTsystemwithM=2transmitantennasandN=2receiveantennas.Insomeofoursimulationexamples,weconsiderdoublingthetransmissiondataatthreedatarates:12Mbps,24Mbps,and54Mbps.Allsimulationresultsareobtainedwith104Monte-Carlotrials.TheMN=4frequency-selectivefadingchannelsbetweenthetransmitandthereceiveantennasaregeneratedindependentlyviatheexponentialchannelmodelpresentedinthepreviouschapter,changingfromonetrialtoanother.Example3.

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(a)(b) (c) Figure3{10:PERversusSNRforaBLASTsystemwithtwotransmitantennasandtworeceiveantennasusingtheLS-basedsoft-detectorfortime-invariantchannelswithtr=50nsatvariousdatarates:(a)datarate=24Mbps;(b)datarate=48Mbps;(c)datarate=108Mbps.gureaPERcurvefortheSISOsystemwithperfectchannelknowledgeandatonehalfofthecorrespondingBLASTdatarate.WeobservefromthePERcurvesthat:

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ComparingFigure3{10(b)andFigure3{9wehavethefollowingremark:(A)fordoublingthedatarateatlowdatarates,suchasfrom12Mbpsto24Mbps,aBLASTsystemwithtwotransmitantennasandtworeceiveantennasmaynotbeasgoodasaSTBCsystemwiththesamenumberofantennas;(B)fordoublingthedatarateathighdatarates,suchasfrom54Mbpsto108Mbps,theaforementionedBLASTsystemcertainlyoutperformsitSTBCcounterpart|infact,doublingthedataratefrom54Mbpsto108MbpsusingaSTBCsystemneedstouse4096-QAM,whichishardlypossibleinpracticalapplications.Assuch,weconcentrateondoublingthedatarateathighdatarates,say,from54Mbpsto108Mbps,usingtheBLASTsystem.Example4.

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Figure3{11:PERcomparisonoftheLS-basedsoft-detectorwithtwootherdetectorsforaBLASTsystemwithtwotransmitantennasand2receiveantennasasafunctionofSNRforthetime-invariantchannelswithtr=50nsatthe108Mbpsdatarate(withestimatedchannelparameters).thantheLSD-basedone.Infact,thesearethereasonthatweproposethehybridsoft-detector,whichwillbedetailedinthenextsection.3.8AHybridSoft-DetectorfortheBLASTSystem

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106 StepSD1: obtainaset ~ B ofbitvectors b 2f 1 ; +1 g BM 1 ,referredtoasthe candidatepool,whichsatises 8 > < > : k y Hx ( b ) k 2 d l ; 8 b 2 ~ B ; k y Hx ( b ) k 2 >d l ; 8 b = 2 ~ B ; (3.60) byusingthemodiedSPDalgorithmthathasaxedsphereradius d l ,determinedbythenumberofantennaaswellasthenoisevariance [46]; StepSD2: foreach i =1 ; 2 ;:::;BM ,calculatethesets: ~ B i; +1 = B i; +1 \ ~ B and ~ B i; 1 = B i; 1 \ ~ B ;(3.61) StepSD3: foreach i =1 ; 2 ;:::;BM ,obtainthebitmetricby l (SD) ( i )= 1 2 min b 2 ~ B i; 1 k y Hx ( b ) k 2 min b 2 ~ B i; +1 k y Hx ( b ) k 2 : (3.62) Atthecostofsomeperformancedegradation,theLSD-basedsoft-detectorimprovesthecomputationaleciencyoftheSTBICM-basedsoft-detectorsignicantly duetolimitingthesearchoverthemuchsmallersets.(Wedonotknowtheexact degradationforourWLANapplicationsincetheSTBICM-basedsoft-detectoristoo slowtomakeareasonablecomparison.)However,theLSD-basedsoft-detectorisnot asecientastheSPD-basedhard-detectorduetothefollowingreasons:(a)LSD inStepSD1usesxedsphereradiuswhereasSPDuseschangingsphereradiusthat shrinkswiththendingofanewpointinthespherewithashorterdistanceand(b) thebitmetriccalculationinStepSD2needsadditionalcomputations. TheLS-basedsoft-detector,ontheotherhand,focusesmainlyonthecomputationalcomplexitysideoftheperformance/complexitytrade-o.WhiletheLSD-based soft-detectorimprovestheeciencyof(3.49)bylimitingthesearchonsmallersets,

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TheLS-basedsoft-detectorisordersofmagnitudemoreecientthantheLSD-basedsoft-detectorduetothedecoupling,aswillbeanalyzedlater.However,theperformanceoftheformerisworsethanthelatter(morethan2dBfortheM=N=2caseforourWLANapplication,asshowninFigure3{11).3.8.2TheHybridSoft-Detector Now,letusexaminewhathinderstheperformanceoftheLS-basedsoft-detector.WecanreadilyseethatwhenHHHisclosetoascaledidentitymatrix,thebitmetricsfromtheLS-basedsoft-detectorwillnotbeworsethanthosefromtheLSD-basedone.However,whenHHHbecomesill-conditioned,thebitmetricsfromtheformerwillbemuchworsethanthosefromthelatter,becauseofthefollowingreasons:(a)someelementsofthenoisevectorcin(3.52)aremagnieddrasticallyduetothepoorchannelsand(b)usefulinformationislostduetothedecoupling.Hence,these(bad)bitmetricscorrespondingtotheill-conditionedchannelscanbeseenasthebottle-neckfortheperformanceoftheLS-basedsoft-detector.IfwecanidentifythesebadbitmetricsandreplacethembythosefromtheLSD-basedsoft-detector,wecanimprovethedetectionperformancesignicantly.

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WeidentifythebadbitmetricsbycomparingtheLS-basedhard-detectoroutput^x(LS)andtheSPD-basedhard-detectoroutput^x(SPD).Iftheyarenotthesame,^x(LS)ismorelikelytohaveerror(s)since^x(SPD)isanMLestimate,whichisbetterthantheformertheoretically.Inthiscase,thecorrespondingbitmetricsfromtheLS-basedsoft-detectorareconsideredbad;otherwise,thesebitmetricscanbeconsideredreliable. Inviewoftheabove,wehavethefollowingstepsforthehybridsoft-detector:StepHY1: obtaintheunstructuredLSsymbolestimatex(LS)byusing(3.57)ofStepLS1;StepHY2: determinetheLShard-detectionresult^x(LS);StepHY3: calculatetheSPDhard-detectionresult^x(SPD);StepHY4: checkthehard-detectionresults|if^x(LS)=^x(SPD),thengotoStepHY5;otherwise,gotoStepHY6;StepHY5: obtainbitmetricsby(3.58)ofStepLS2basedonx(LS)fromStepHY1andthenstop;StepHY6: obtainbitmetricsbyperformingStepsSD1,SD2andSD3andthenstop. Thecomputationalcomplexityofthehybridsoft-detectorisdominatedbySPDandtheLSD-basedsoft-detector,i.e.,StepsHY3andHY6.Tospeedupthecalcula-tionofSPDinStepHY3,weneedtoconsiderthedeterminationofitsinitialradius,whichisacrucialissueforSPD.Iftheinitialradiusistoosmall,therewillbenopoint(x)inthesphere|SPDcannotndtheMLsolution.Ontheotherhand,iftheinitialradiusistoolarge,SPDwillbeveryslowduetotheunnecessaryadditionalsearches.Thenumberoftheadditionalsearchescanbereducedbyusingamodiedsearchingapproachgivenin[14].However,itcomplicatesthealgorithmsitself.Here,wegiveatightsphereradius,basedontheLS-basedhard-detectoroutput^x(LS),byusingdr=kyH^x(LS)k+d;(3.63)

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whered>0isaverysmallvalue.Notethatthespherewiththisradiuswillcontainatleastonepoint|theoutputoftheLS-basedhard-detector.Notealsothat,formostcases(98outof100fortheSNRsofinterestinourWLANapplication,aswillbeshownbythesimulationresultsinnextsection),thespherewiththisradiuscontainsonlyonepoint.Byusingthistightradius,ourpreliminarysimulationresultsshowthatSPDcanbeasecientastheinterferencecancellationandnullingalgorithm[37]andusesonly5timesasmanyopsastheLS-basedsoft-detector. Thecomputationalcomplexity,intermsofops,foreachstepoftheLSD-basedsoft-detectorcanbeestimatedasfollows.(WeassumeM=Nforconvenience.)StepHY1: Negligible.StepHY3: Negligible.StepHY5: Negligiblebytable-checkingforthePAMsymbols;seeAppendix2.5.4ofthepreviouschapter.StepHY6: (a)O(M3)toO(M6)forLSD(StepSD1),which,asshownbysimula-tionresults,usestypically2to10timesasmanyopsasSPDinStepHY3,i.e.,10to50timesasmanyopsasLSinStepHY1.(Weusetheaverage25inthesequel.)(b)O(N2CBM)forbitmetriccalculation(StepSD3),whereNCisthenumberofcandidatesinthelistforLSDandtheoperationofndingtheminimumisperformedbyusingtheconventionalbubblingalgorithm;forexample,forM=2,B=6,andNC=120(whichistypicalforagoodperformance),thisamountstoabout43200ops(assumingjBi;+1j=jBi;1j=60,i=1;2;:::;12,forconvenience),whichisabout95timesasmanyopsasLSinStepHY1.

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Aswillbeseenfromthesimulationresultsinthenextsection,lessthan2%caseshavedierentSPDandLShard-detectionresults.Hence,wecanseethatthehybridsoft-detectorisabout1|{z}LS+5|{z}SPD+0:02(25|{z}LSD+95)=8:4 timesasslowastheLS-basedsoft-detector,whichindicatesthatthehybridsoft-detectorisabout10timesslowerthantheLS-basedsoft-detector.WecanalsoseethattheLSD-basedsoft-detectorneeds120timesasmanyopsastheLS-basedone,whichmeansthatthehybridsoft-detectorisabout10timesfasterthantheLSD-basedone.Notethatthenewhybridsoft-detectorismoreecientforhighSNRsthanforlowSNRssinceathighSNRstheprobabilitiesof^x(LS)=^x(SPD)arehighandthechancesofusingthecomputationallyexpensiveLSD-basedsoft-detectorarelow.Notealsothattheaboveanalysisofthecomplexityisonlyintendedtogiveafeelingabouttheeciencyofthehybridsoft-detectorandisbynomeansveryaccurate.Moreaccurateanalysisofthecomplexities,includingthoseforSPDandLSD,isstillanopentopic. Weremarkthatthebadbitmetricscanalsobeidentiedbyusingthecon-ditionnumber(CN)ofHHH,andtheresultingsoft-detectorcanbereferredtoastheCN-hybridsoft-detector.However,theCN-hybridsoft-detectorisinferiortoitshybridcounterpartduetothefollowingreasons.First,itishardtodetermineathresholdfortheCN.Ifthethresholdistoohigh,manybadbitmetricsfromtheLS-basedsoft-detectorwillbeusedinthehybridsoft-detector,whichwillleadtodegradedperformance.Ontheotherhand,ifthethresholdistoolow,thecomputa-tionallyexpensiveLSD-basedsoft-detectorwillbeusedtoooften,whichwillresultinincreasedcomputationalcomplexity.Second,alargeCNdoesnotnecessarilyre-sultindetectiondierencesbetweentheSPD-andLS-basedhard-detectors.Neither

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Figure3{12:PERcomparisonofthehybridsoft-detectorwithothersoft-detectorsforaBLASTsystemwithtwotransmitantennasand2receiveantennasasafunctionofSNRforthetime-invariantchannelswithtr=50nsatthe108Mbpsdatarate(withperfectchannelknowledge). Notethatevenwiththeneedofthis1.5dBextraSNR,i.e.,1.5dBmoreto-taltransmissionpower,thePERperformanceoftheOFDM-basedBLASTsystemwiththehybridsoft-detectorisstillimpressivesinceevenifwewishtodoublethetransmissiondatarateusingtwoseparateSISOsystemsovertwodierentphysicalchannelsbydoublingthebandwidth,westillneed3dBextraSNRortotaltrans-missionpower.Ifweconsiderthecaseof1%PER,wecandoublethedataratewithabout0.5dBlesstotaltransmissionpower.Example6.

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Figure3{13:ProbabilitiesofusingLSDinthehybridandCN-hybridsoft-detectorsforaBLASTsystemwithtwotransmitantennasandtworeceiveantennasasafunctionofSNRfortime-variantchannelswithtr=50nsatthe108Mbpsdatarate.3.9AnIterativeSoft-DetectorfortheBLASTSystem Ithasbeenshownin[46]thatiterativedetection/decodingwithaniterativeLSD-basedsoft-detectorcanbeusedtoachievenear-capacity.Yet,wendthatduetousingtheuncensoredaprioriinformation,whichiscausedbythelimitedLSDcandidatepoolduringeachiteration,theLSD-basedsoft-detectorcanperformworsewithincreasediterationnumber.Wemitigatethisproblembyconstrainingthevalueoftheaprioriinformationtoimprovethedetection/decodingperformanceofthesystem.TheresultingmethodisreferredtoastheiterativeconstrainedLSD(CLSD)-basedsoft-detector.Wealsoimprovethedetection/decodingperformancebyincorporatingtheupdatedchannelresponseobtainedusingtheaforementionedsemi-blindMIMOchannelresponseestimationmethodfortheBLASTsystem.The

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Inthesequel,werstpresenttheiterativeCLSD-basedsoft-detectorandthengivetheturboprocessingalgorithm.3.9.1TheCLSD-BasedSoft-Detector Withtheapplicationoftheaprioriinformation,thebitmetricfortheithbit,i=1;2;:::;of(3.46)canbewritten,byusingtheBayes'theorem,aslD(i)4=lnP(bi=+1jy;H) Here,lA(i)andlE(i)arereferredtoastheaprioriandextrinsicinformation,re-spectively.Theaprioriinformationcanbeobtainedfromthesoft-outputoftheBCJRdecoder[5],andtheextrinsicinformationcanbecalculatedbytheiterativeSTBICM-basedsoft-detector[12,46,116]:lE(i)t1 2maxb2Bi;+11 2maxb2Bi;11 whereBi;+1andBi;1arethesetof21bitvectorswithbibeing+1and1,re-spectively,andlA=[lA(1)lA(2):::lA()]T.(SeeAppendix3.12.2foraderivationof(3.65).)

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Figure3{14:Diagramofthereceiverwithiterativedetection/decodingforanOFDM-basedBLASTsystem. Intheiterativedetection/decodingprocess,asshowninFigure3{14,theaprioriinformationlA(i)fortheBLASTdetectors,whichcanbeseenastheinnerdecoder(s),isobtainedfromthesoft-outputoftheBCJRdecoder,whichcanbeseenastheouterdecoder.Specically,itisfromtheincrementalsoft-information(denotedasLEinthegure)duetotheBCJRdecoder.Similarly,theinputfortheBCJRdecoder(denotedasLAinthegure)isthedeinterleavedextrinsicinformation,lE,theincrementalsoft-informationfromtheBLASTdetectors.TheiterativedecodingstartsfromtheBLASTdetectionwiththeaprioriinformationbeingzero. TheiterativeSTBICM-basedsoft-detectorof(3.65)isoptimal;infact,itisthemaximumaposteriori(MAP)BLASTsoft-detector.However,itisextremelyine-cient.TheiterativeSTBICM-basedsoft-detectorcanalsobesimpliedbyusingLSD[46],similartothenon-iterativecase.Asforthenon-iterativecase,theiterativeLSD-basedsoft-detectorkeepstheSTBICMframe-workwhileimprovingtheeciencyof(3.65)bysearchinginmuchsmallersubsets~Bi;+1Bi;+1and~Bi;1Bi;1withj~Bi;+1j<<21andj~Bi;1j<<21.Assuch,theiterativeLSD-basedsoft-detectorisimplementedsimilarlytothenon-iterativeLSD-basedsoft-detectors|StepsSD1

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2maxb2~Bi;+11 2maxb2~Bi;11 whichisdoneiteratively. TheiterativeLSD-basedsoft-detectorisordersofmagnitudemoreecientcom-putationallythantheiterativeSTBICM-basedsoft-detector.However,thiseciencyisobtainedatthecostofsomeperformancedegradationduetothelimitedcandidatepoolusedbyLSD. Duetothelimitedcandidatepool,theaprioriinformationfromtheouterde-codercanbequitepoor.Whenthepooraprioriinformationisusedblindlywithoutbeingcensored,theperformanceoftheiterativeprocessingcandegradewiththein-creaseiniterationnumber.Fortunately,thisproblemcanbemitigatedbyusingasimpleapproachtocensortheaprioriinformationfromtheouterdecoder. Beforepresentingthemitigationmethod,letusrstexaminethereasonthatleadstothepooraprioriinformation.Aswementionedearlier,theaprioriinfor-mationfortheiterativeBLASTsoft-detectorsisobtainedfromtheBCJRdecoder.Henceitcanbeseenastheoutputofafeedbacksystem.WhenusingtheiterativeSTBICM-basedsoft-detector,thesystemcanbestable.Thereasonisthatforanyi2f1;2;:::;gandb12Bi;+1,theremustbeab22Bi;1suchthatbT1lAlA(i)=bT2lA+lA(i):(3.67) Thereforenomatterhowlargetheaprioriinformationis,theoutputoftheiterativeSTBICM-basedsoft-detectorisboundedby1 22maxkyHx(b)k2minkyHx(b)k2:(3.68)

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Moreover,thisboundeddetectoroutputleadstoboundedaprioriinformation,re-sultinginconvergence.WhenusingtheiterativeLSD-basedsoft-detector,ontheotherhand,thecorrespondingsystemcanbeunstable.Thereasonisthatforsomei2f1;2;:::;gandb12~Bi;+1,theremaynotexistb22~Bi;1suchthat(3.67)holds.Hence,iftheaprioriinformationislarge,bT1lAlA(i)maybethedominantfactorindeterminingthesoft-outputin(3.66),whichisnotboundedby(3.68).Speci-cally,thelimitedcandidatepoolcancausethefollowingtwoproblemsforiterativedecoding:Problem(a): TheoutputoftheiterativeLSD-basedsoft-detectorof(3.66)canbeverylargeandgotoinnityquicklyduringiterativedetection/decodingasaresultoftheuncensoredaprioriinformationfromtheBCJRde-coder.Problem(b): TheoutputoftheiterativeLSD-basedsoft-detectorof(3.66)canbeunreliable,whichcanleadtodegradeddecodingperformance. Problem(a)wasaddressedin[46]bysimplyclippingthesoft-outputto8.However,duetoProblem(b),thedetection/decodingperformancecanstillbeworsewiththeincreaseiniterationnumber,especiallyatlowtomoderateSNRs(see,forexample,Example7below). WebelievethatProblem(b)isthemainreasonoftheperformancedegradationwiththeincreaseiniterationnumber.Henceweproposetoimprovethedetec-tion/decodingperformanceoftheiterativeLSD-basedsoft-detectorbyadequatelyaddressingProblem(b),i.e.,byconstrainingtheaprioriinformationfromtheouterdecoder.Wefollowthefollowingsimplerulestoconstraintheaprioriinformation:R1: Thelargerthesphereradius,thelargerthemaximumallowableaprioriinformation,denotedaslmax.R2: Thelargerthe,thesmallerthemaximumallowableaprioriinformationlmax.R3: ThehighertheSNR,thelargerthemaximumallowableaprioriinforma-tionlmax.

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WechooseR1becauseforagivenSNRandagivennumberofcandidatesinasphere,thelargertheradius,thebetterthechannelcondition.Wegivemorefreedomorlargermaximumallowableaprioriinformationlmaxtoabetterchannelcondition.ThereasonofchoosingR2isthatthemorecomplicatedtheconstellationandthelargerthenumberoftransmittedsymbols,thelargerthebTlA.WeputmoreconstraintonthemaximumallowablevalueoflAtopreventthevalueofbTlAfromgrowingtoolarge.Finally,wefollowR3becausethehighertheSNRorthesmallerthenoisevariance2,themoretrustwecanputonthelargeraprioriinformationfromtheouterdecoder. Basedontheaforementionedrules,weconstrainthevalueoftheaprioriinfor-mationforaBLASTsystemwithtwotransmitantennasandtworeceiveantennasasfollows:lmax=d wheredistheradiusofthesphereandkCisaconstantrelatedtoortheconstellationsizeandthenumberoftransmitantennas.Forexample,for64-QAM,wechoosekC=442fortheaboveBLASTsystem,where42istheaveragepowerofthe64-QAMconstellation.Wechoosedsothatthespherecontains5candidates. Oncewehavedeterminedthemaximumallowableaprioriinformationlmax,wecanconstraintheaprioriinformationfromtheouterdecoderasfollows:l(C)A=8><>:lA;maxflAglmax;lmax Theconstrainingschemein(3.70)isbynomeansthebest;however,itisverysimpleandcanleadtoexcellentresultsinourapplications(seethesimulationexamplesbelow).

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Figure3{15:DiagramofthereceiverwithturboprocessingforanOFDM-basedBLASTsystem. TobeconservativeintermsoftheaboveProblem(a),westillclipthesoft-outputsoftheCLSD-basedsoft-detector(by50,insteadofthe8suggestedin[46])inadditiontoconstrainingtheaprioriinformation.3.9.2AlgorithmoftheTurboProcessing EstimatethechannelparametersbyusingthesequentialmethodofSection3.4.StepTP2: DetectallthedatabitscontainedinthepacketbytheLSD-basedsoft-detectorandthendecodetheentirepacketbytheBCJRalgorithm.StepTP3:

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Figure3{18showsthecomparisonoftheiterativeCLSD-andLSD-basedsoft-detectorswith4iterations,alongwithareferenceoftheLSDbasedsoft-detector

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Figure3{16:PERversusSNRforaBLASTsystemwithtwotransmitantennasandtworeceiveantennasusingtheiterativeLSD-basedsoft-detectorwithvariousiterationsfortime-invariantchannelswithtr=50nsatthe108Mbpsdatarate(withperfectchannelknowledge).withoutiteration.WeobservethatforlowtomoderateSNRs,theCLSD-basedsoft-detectorsignicantlyoutperformstheLSD-basedone|withabout1.4dBreductioninSNRatthePERbeing101.ForhighSNRs,oursimulationresultsshowthatCLSDisnotworsethanLSD.Example9.

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Figure3{17:PERversusSNRforaBLASTsystemwithtwotransmitantennasandtworeceiveantennasusingtheiterativeCLSD-basedsoft-detectorwithvariousiterationsfortime-invariantchannelswithtr=50nsatthe108Mbpsdatarate(withperfectchannelknowledge).Example10. Figure3{21showsthecomparisonoftheBLASTsystemandtheSISOsystemusingvariousprocessingalgorithmsandundervariousconditionsforchannelswithtr=50ns.WeobservefromthePERcurvesthat:

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Figure3{18:PERcomparisonoftheiterativeCLSD-andLSD-basedsoft-detectorswith4iterationsaBLASTsystemwithtwotransmitantennasandtworeceivean-tennasasafunctionofSNRfortime-invariantchannelswithtr=50nsatthe108Mbpsdatarate(withperfectchannelknowledge).foritsSISOcounterpart,sincewiththeGraycodedQAMsymbols,iterativedecodingdoesnothelptheSISOsystem[115].

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Figure3{19:PERversusSNRforaBLASTsystemwithtwotransmitantennasandtworeceiveantennasusingthenon-iterativeLSD-basedsoft-detectorwithvariousiterationsofchannelupdatingfortime-invariantchannelswithtr=50nsatthe108Mbpsdatarate.ofthesizeofthemobileterminals,itisnotpracticaltoenlargethespacingfortheantennasatthemobileterminals.Yet,itisfeasibletoenlargethespacingfortheantennasattheaccesspoints.Attheaccesspointswecaninstallthreeantennas,eachatanapexofanequilateraltriangle,orwecaninstallfourantennas,eachatanapexofasquare.Fortheup-linktransmission,allthethreeorfourreceiveantennasattheaccesspointscanbeusedtoimprovetheperformance.Forthedown-linktransmission,ontheotherhard,only2ofthetransmitantennasattheaccesspointswhichhavethelargestangularspacingwithrespecttothereceiveantennaswillbeusedforthetransmission.HerewedemonstrateusingansimulationexamplethatareasonableangularspacingcanyieldagoodperformancefortheBLASTsystems. Weconsiderdown-linktransmission.Allthesimulationconditionsarethesameasbefore,exceptthatthechannelsareRicianfadingchannels.TheRicianfadingchannelsfortheOFDM-basedMIMOsystemsaresimulatedasthesuperpositionoftheLOSchannelsandthemultipathschannels,withtheLOSchannelsbeingKdBhigherthanthemultipathchannels(hereKisreferredtoastheRicianfactor).TheRicianfadingchannelsarenormalizedtohavethesameaveragechannelgainasthe

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Figure3{20:PERversusSNRforaBLASTsystemwithtwotransmitantennasandtworeceiveantennasusingtheturboprocessingwithvariousiterationsfortime-invariantchannelswithtr=50nsatthe108Mbpsdatarate.Rayleighchannelsusedbefore.FortheLOSchannels,weusethestandardchannelmodelforantennaarray(see,e.g.,[66])withthespacingofthereceiveantennasbeinghalfthewavelengthandthetwotransmitantennasaresymmetricaltotheboarddirectionofthereceiveantennas.Forthemultipathchannels,weusethesamemodelasbefore.Example11.

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Figure3{21:PERversusSNRcomparisonofaBLASTsystemwithtwotransmitan-tennasandtworeceiveantennastoaSISOsystematrespectivemaximumdataratesusingvariousprocessingalgorithmsandundervariousconditionsfortime-invariantchannelswithtr=50ns. NotethatfortheLOStransmissioncases,beamformingschemes,suchasthoseproposedforsmartantennas[71],canalsobeusedtoimprovethetransmissionrobust-nessortoincreasethetransmissiondatarate.Thenewlyproposedrobustadaptivebeamformingschemes(see,e.g,[66])areespeciallyapplicablesincethenon-LOSre-ectionscanbeseenastheerrorofthesteeringvectorsoftheantennaarray;thesteeringvectorsareduetotheLOStransmissions.Forexample,astheBLASTsystem,therobustCaponbeamformer[66]canbeusedtoreceivetwopacketssimul-taneouslytransmittedfromtwotransmitantennastoresultinadoubleddataratebyexploitingthedierentpacketLOSarrivalangles.(TherobustCaponbeamformertriestosuppressothersignalswhilereceivingtheinterestedone.Thetwotransmit-tedpacketsaretreatedastheinterestedsignalinturn.)Yet,weobservefromoursimulationresults(notprovidedhere)thattherobustCaponbeamformerworkswellonlyforlowdataratecaseswhereSNRsarelow;forthehighdataratecaseswhereSNRsarehigh,therobustCaponbeamformerhasmuchworseperformancethantheBLASTsystemsduetolimitedcapabilityofinterferencesuppressioninthepresenceofsteeringvectorerrors.

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Figure3{22:PERversustransmitantennaangularspacingforaBLASTsystemwithtwotransmitantennasandtworeceiveantennasusingthenon-iterativesoft-detectorsfortime-invariantRicianfadingchannelswithtr=50nsandK=20(withestimatedchannelparameters)atthe30dBSNRandthe108Mbpsdatarate.3.11ConcludingRemarks

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Thenc=Hye~Hy~e;(3.72) and~c=~Hye+Hy~e:(3.73) Hence,wehaveEc~cT=EhHye~Hy~eeT(~Hy)T+~eT(Hy)Ti=EhHyeeT(~Hy)T~Hy~e~eT(Hy)Ti+EhHye~eT(Hy)T~Hy~eeT(~Hy)Ti=1 22hHy(~Hy)T~Hy(Hy)Ti=1 22Hy(~Hy)THy(~Hy)TT;(3.74) wherewehaveusedthefactthatE[eeT]=E[~e~eT]=2

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WeconsiderthegeneralcaseofMIMOsoft-detectionwherethedimensionsofHin(3.12)isNMandxisM1witheachelementofxcorrespondingtoBdatabits.Let=BM.Thenxcanbemappedtob=[b1b2:::b]T,withbi2f1;+1g,i=1;2;:::;.(Notethatxcanbeexpressedasx(b)tostressitsdependenceonb.)Byusingournotation,theMAPMIMOsoft-detectorof[12,46,116]canbereformulatedasfollows. Thebitmetric(alsoknownastheL-value[39])fortheithbit,i=1;2;:::;,isdenedaslD(i)=lnP(bi=+1jy;H) which,byusingtheBayes'theorem,canbewrittenaslD(i)=lnP(bi=+1;yjH)=P(yjH) {z }lE(i)+lnP(bi=+1jH) {z }lA(i)=lnP(bi=+1) Here,lA(i)andlE(i)arereferredtoastheaprioriandextrinsicinformation,re-spectively.NotethatwehaveexploitedthefactthatthetransmitteddatabitbiisindependentofthechannelH.Thisfactwillalsobeusedinthesequel. ThenumeratoroflE(i)canbewrittenas:P(yjbi=+1;H)=XP(y;fbj=bjgj6=ijbi=+1;H)=Xb2Bi;+1P(yjb;H)P(b=bjbi=+1);(3.77)

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whereBi;+1isthesetof21bitvectorswithbibeing+1.Duetotheemploymentoftheinterleaver(s),thedatabitsinbcanbeseenasindependentofeachother,andtheaboveequationcanbeexpressedas:P(yjbi=+1;H)=Xb2Bi;+1P(yjb;H)Yj6=iP(bj=bj):(3.78) WecanwritethedenominatoroflE(i)inasimilarway.Hence,wehavelE(i)=lnPb2Bi;+1p(yjb;H)Qj6=iP(bj=bj) Withboththenumeratorandthedenominatorin(3.79)dividedbyYj6=iq andbyexploitingthefactthatP(bj=bj) wehavelE(i)=lnPb2Bi;+1p(yjb;H)exp1 2bTlA1 2lA(i) Pb2Bi;1p(yjb;H)exp1 2bTlA+1 2lA(i);(3.82) wherelA=[lA(1)lA(2):::lA()]T.Withtheassumptionoftheadditivezero-meanwhitecircularlysymmetriccomplexGaussiannoiseforthereceiveddata,theaboveextrinsicinformationcanbewrittenas(bydroppingthebaroverbforconvenience),lE(i)=lnPb2Bi;+1exp1 2bTlA1 2lA(i) Pb2Bi;1exp1 2bTlA+1 2lA(i)=lnXb2Bi;+1exp1 2bTlA1 2lA(i)lnXb2Bi;1exp1 2bTlA+1 2lA(i):(3.83)

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Byusingthethemax-logapproximation[39],theaboveextrinsicinformationcanbewrittenas:lE(i)t1 2maxb2Bi;+11 2maxb2Bi;11 Forthecaseofnon-iterativedetectionwheretheaprioriinformationisneglected,thesoft-informationcanbewrittenas:lE(i)t1 2maxb2Bi;+11 2maxb2Bi;11

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TheSISOsystemsovertime-varyingchannelsconsideredhereinincludetheDSRC[4]systemsandtheMBWA[56]systems,twotypicalkindsofmobilebroad-bandwirelesscommunication(MBWC)systems.Duetothesimilarityoftheabovesystems,wewillexemplifyourpresentationusingtheE2213-02[4]conformableSISODSRCsystems. Aswementionedearlier,E2213-02isverysimilartoIEEE802.11a.LikeIEEE802.11a,E2213-2employspacket-basedtransmissionwhichusesapreambletoobtainchannelparametersandusescoherentdetectiontodetectthedatabitscontainedinthepayload.Tocontrolthepacketerrorrate,convolutionalchannelcoding/decodingisusedintheDSRCsystems.Thepacket-basedtransmissioniswellsuitedfortime-invariantchannelsbutwillsuerfromsevereperformancedegradationsfortime-varyingchannels,whichcanoccur,forexample,duetousermobility.Thelackofmechanismfortrackingthetime-varyingchannelsforcoherentdetectionisthemaincausefortheperformancedegradation.Asaresult,theSISODSRCsystemsconformingtothisstandardcanonlybeusedwithlowtransmissiondataratesandshortpackets,i.e.,whenthethroughputisrelativelylow. Inthischapter,weconsidersystemdesignandsignalprocessingissuesthatad-dresstheperformancedegradationproblemforthehightransmissiondataratecasesfortheSISOsystemsoverthetime-varyingchannels.Weproposeanewschemeforchannelupdating/trackingthatcansignicantlyoutperformalltheexistingap-proachesincludingthedierentialonesforhighratetransmission.Ournewschemecombinespacketdesignwithsignalprocessing(atthereceiver)toaddressthechanneltrackingproblem.Forthepacketdesign,wesegmentanentirepacketintomultiple132

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OurnewschemecanbeeasilyusedtomodifytheE2213-02standardsothattheDSRCsystemsconformingtothemodiedstandardcanbeusedfortime-varyingchannels.ThesamemodicationcanbeusedintheWiMaxstandardaswellastheWi-FistandardsincludingIEEE802.11a,IEEE802.11g,andHIPER/LAN2,whichareallpacket-andOFDM-based,toaccommodatemobility.Moreover,thenewschemecanbereadilyadoptedintotheMBWAstandardswhichareunderdevelop-ment. Theremainderofthischapterisorganizedasfollows.Section4.1describesthepacketdesignanddatamodelfortheOFDM-basedDSRCs.Section4.2presentsthesignalprocessingalgorithmsforchanneltracking.NumericalexamplesareprovidedinSection4.3.Finally,weconcludeinSection4.4.4.1PacketDesignandDataModel

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Figure4{1:ThepacketstructureforaSISOsystemasspeciedbytheE2213-02standard.thetruechannelsfortheOFDMsignalingfortime-varyingchannels.Finally,weestablishthedatamodelfortheOFDM-basedDSRCs.4.1.1E2213-02Standard Weuseasimilarideaasin[89]totrackthechannelfortheOFDM-basedDSRCsystems.Thatis,werstdetect/decodethedatabitscontainedinthekthOFDMdatasymbolandthenusethedecodeddatabitsasknowntrainingbitstoupdatethechannelresponseestimateforthekthOFDMdatasymbol.Herek=1;2;:::;K1withKbeingthenumberofOFDMdatasymbolsinapacket.Thismethodrequiresagooddetection/decodingresultforeachOFDMdatasymbol.Forthecurrentstandard,theentiredatablockisconvolutionallyencodedasawhole(sixzerosareattachedattheendofthedatablocktoresettheencoder),andtheperformanceofdecodingonlyoneOFDMdatasymbolusingtheViterbialgorithm(VA)willbe

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Figure4{2:SubpacketstructureoftheOFDMDATAelddesignedfortheSISOsystemsovertiming-varyingchannels.poorduetoeliminatingthefutureinputsintotheVA.Toimprovethedecodingperformance,wecansimplyterminatethecodingofeachOFDMdatasymbolbyattachingsixzerostothesub-blockofdatabitscorrespondingtoanOFDMdatasymboltoresettheconvolutionalchannelencoder,assuggestedin[89]andshowninFigure4{2.Inthisway,wecanrefertoeachOFDMdatasymbolasasubpacketandthecorrespondingtransmissionassubpacket-basedtransmission.Thereisactuallynoerrorpropagationproblematallinoursystem.Ifasubpacketiswrong,itmakesnosensetocontinuethedetecting/decodingprocessofthenextsubpacketssincewewilldiscardtheentirepacketandaskforaretransmission. WecouldusemultipleOFDMdatasymbolsinasubpacket.Theadvantageisthatwecanreducethetotalnumberofzerobitsusedtoresettheconvolutionalchannelencoder.However,thisusagewillreducethefrequencythechannelisupdatedsinceonemustwaituntilallthedatabitswithinasubpacketaredetected/decoded.Wedonotconsiderthisoptionhereafter.4.1.3ChannelModel Forconvenience,weconsiderhereinachannelmodelforquasi-statictime-varyingfrequency-selectivechannels.Duringthetimeintervalcorrespondingtoasubpacket,i.e.,anOFDMdatasymbol(withCP),thechannelisassumedtime-invariantand

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First,weconsiderachannelmodelforatime-invariantandfrequency-selectivechannelwithinonesubpacket.Fornotationalconvenience,wedropthedependenceonthesubpacketindex.A(baseband)time-domain(TD)analogchannelimpulseresponseofatime-invariantandfrequency-selective(multipath)fadingchannelcanbewrittenash(t)=PMXp=0p(ttp);(4.1) wherePM1isaninteger,(t)istheDiracDelta,andpandtp=ptS(01:::PM,tS=100ns)arethecomplexgainandtimedelayofthepthpath,respectively.Thischannelisoftenapproximated,i.e.,modeled,byanexponentiallydecayingFIRlterwithlengthLF
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NotethatwhilefDislargeenoughtocausechannelvariations,thevariationoffDisusuallyquitesmallcomparedtothefrequencyspacingbetweentwoadjacentsubcarriersoftheOFDMsignaling.Hence,theeectofinter-carrierinterference(ICI)causedbyfDisnegligible.Forexample,fDis703.2HzforRFbeing5.9GHzandtherelativemovingspeedbeing80mph,anditsvariation(causedbythechangeofthemovingspeed)canatmostbe100Hzifthepacketisnottoolong.ComparingtothefrequencyspacingbetweentwoadjacentsubcarriersforDSRC,whichis156.2kHz,thevariationoffDisatleastthreeordersofmagnitudesmallerandcanbeneglected.(Ofcause,fDitselfcanbetakencareofbyCFOcorrection.)4.1.4DataModel bethecorrespondingfrequency-domain(FD)channelresponsevector(ontheNSsubcarriers)ofhe(t)givenin(4.2).Thereceiveddatavectorisdenotedaszk=znek+wk,wherewkN(0;(2=NS)INS)istheadditivezero-meanwhitecircularlysymmetriccomplexGaussiannoisewithvariance2=NS.TheFFToutputofzkcanbewrittenas[124]yk=WNSzk=diagfhkgxk+WNSwk4=diagfhkgxk+ek2CNS1;(4.5)

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wherediagfhkgdenotesthediagonalmatrixformedfromhkandekN(0;2INS).Thedatamodelin(4.5)canalsorepresenttheOFDMsymbolsintheSIGNALeldandthepacketpreamble. Equation(4.5)canalsobewrittenasyk=diagfxkghk+ek:(4.6) Notethat(4.5)isusefulforthedatasymbol(ordatabit)detection,whereas(4.6)canbeusedfortheFDchannelresponseestimation.4.2SignalProcessingAlgorithmsforChannelTrackingattheReceiver Thechannelresponsetrackingcanbedoneinthefollowingthreesteps:(T1)updatingtheFDchannelresponseforthekthsubpacketbyusingdetected/decodeddatabitsfromthekthOFDMdatasymbolviatheDFFEmethod,(T2)improvingthechannelestimateobtainedbyaDFFEmethodviausingweightedpolynomialtting

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Figure4{3:Real-timechannelresponseupdatingbyusingthedetected/decodeddatabitsinasubpacket.method,and(T3)predictingtheFDchannelresponseforthe(k+1)stsubpacket,correspondingtothe(k+1)stOFDMdatasymbol,asshowninFigure4{3.Wewilldetailallthreestepsbelow.4.2.1DecisionFeedbackFDChannelResponseEstimation whereXk=diagfxkg. ForthenSthsubcarrier,nS=1;2;:::;NS,wehave:yk;nS=xk;nShk;nS+ek;nS;(4.8) whereyk;nS,xk;nS,andhk;nSdenotethenSthelementofyk,xk,andhk,respectively,andek;nSistheadditivenoise.From(4.8),thechannelresponseforthenon-zero

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Byusingthesubpacket-basedtransmissionanddetection/decodingschemefortime-varyingchannels,theerrorinxk;nScanbegreatlyreducedduetotheterminationoftheconvolutionalcoding.Also,wemaydetermineifthedetection/decodingiscorrectornotforanOFDMdatasymbolbycheckingthefollowingdistance:kykXk^h(DFFE)kk2;(4.10) theEuclideannormoftheresiduevectorforthekthOFDMdatasymbolafterchannelresponseupdating. Yet,comparedtotheFDchannelresponseestimateobtainedusingthetrainingsymbolsinthepacketpreamble,theaccuracyoftheupdatedFDchannelresponsecanbepoor.Thiscanbeseeninthefollowing.Undertheassumptionthatxk;nSiserror-free,theadditivenoisein(4.8)canbewrittenasek;nSN(0;2),whichleadstoEn^h(DFFE)k;nSo=hk;nS;(4.11)varn^h(DFFE)k;nSo=2=jxk;nSj2:(4.12) Forthetrainingusingthepreamble,thevarianceoftheFDchannelresponsees-timateis2=2sincetwolongOFDMtrainingsymbolsareusedtogetherandforeachsubcarrierthetrainingsymbolisBPSK,whichhasunitenergy[77].FortheupdatedFDchannelresponseusing64-QAMsymbols,thevariancecanbeaslargeas422=2forsubcarrierswithsymbolshavingtheminimumenergy.(Considera64-QAMconstellationwithunitenergy.Thenthesymbolscanbeexpressedasf(2kr1)=p

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Inthefollowing,weprovideamethodtoimprovetheaccuracyoftheupdatedFDchannelestimate.4.2.2WeightedPolynomialFitting Piecewisepolynomialttinghasbeenusedin[123]toimprovetheFDchannelresponseestimationaccuracybasedontheFDchannelresponseestimateobtainedbytraining,wherethevarianceoftheestimatesonallthesubcarriersisassumedtobethesame.HereweusepiecewiseweightedpolynomialttingtoimprovetheFDchannelresponseestimationperformancesince,asmanifestedin(4.12),thevarianceoftheupdatedchannelestimatebyDFFEoneachsubcarrierdependsonthedatasymboltransmittedonthesubcarrier. Forthepiecewiseweightedpolynomialtting(PF),wedividethenon-zerosub-carriersintothefollowingfourgroups,eachwith13subcarriers:f1;2;:::;13g,f14;15,:::;26g,f38;39;:::;50g,andf51;52;:::;63g.WetapolynomialoforderPtoeachofthesegroups.Thedependenceonthegroupindexisdroppedbelowfornotationalconvenience.Letv=[6;5;:::;6]TandV=[v0;v1;:::;vP]T,where()pdenotestheelement-wisepowertotheorderp.Let~h(DFFE)kbetheupdatedFDchanneles-timatebyDFFEforoneoftheabovegroups[asubvectorof^h(DFFE)kin(4.10)],kbethevectorofpolynomialcoecients,and~ekbethenoisevectorwhichcontainsestimationerrorduetoadditivenoise[cf.(4.7)]andpolynomialmodelerror.With

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Notethatthemodelerrorpartofthenoise~ekcanbeignoredifPissucientlylarge,andhencewehaveE(~ek~eHk)=2(~XHk~Xk)1;(4.14) where~Xkisthediagonalmatrixformedbythereconstructeddatasymbolscorre-spondingto~h(DFFE)k[i.e.,~XkisasubmatrixofXkin(4.7)]. Basedon(4.13)and(4.14),theMarkovestimateofkis[108]:^k=VT~XHk~XkV1VT~XHk~Xk~h(DFFE)k=VT~XHk~XkV1VT~XHk~yk;(4.15) where~ykisformedfromyk[cf.(4.7)]inthesamewayas~h(DFFE)kformedfrom^h(DFFE)k.ThentheupdatedFDchannelestimatebasedonPFcanbewrittenas~h(PF)k=VVT~XHk~XkV1VT~XHk~yk:(4.16) NotethatalargePcanhaveasmallmodelerrorandalargecoecientestimationerror,andviceversaforasmallP.AbalancedchoiceofPcanleadtoagoodchannelestimationperformance,aswillbeshownusingthesimulationresultsinthefollowingsection. Themodelerrorforacertainchannelisaectedbythesymboltimingestimate,andanappropriatesymboltimingcompensationcanleadtoimprovedFDchannelestimationperformance,asdetailedbelow.4.2.3WeightedPolynomialFittingwithSymbolTimingCompensation

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Aswillbeshownusingthesimulationresultsbelow,theabovemodiedPF(MPF)cangreatlyoutperformPF,whichinturnsignicantlyoutperformDFFEforupdatingthechannelresponse^hk;nSwithxk;nS.Thepurposeofchannelupdatingforthekthsubpacketistofacilitatethedetection/decodingofthedatabitscontainedinthesubsequentOFDMdatasymbol,sk+1.Bypredictingthechannelofthe(k+1)stsubpacket,wecanimprovethedetection/decodingperformanceforsk+1.4.2.4ChannelResponsePrediction Herein,weuseapolynomialtting-basedpredictionapproach.WeusePLvaluesofchannelresponseonthenSthsubcarrier,beforethe(k+1)stsubpackettopredictthechannelresponseonthenSthsubcarrierforthe(k+1)stsubpacket.WeagaindropthedependenceonnSfornotationalconvenience.Letv=[PL+1PL+2:::0]T(4.17)

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andV=[v0v1:::vP](4.18) withPbeingtheorderofpredictionpolynomial.LethbeavectorofFDchannelresponseestimatesonthenSthsubcarrierfromthePLsubpacketsaheadofthe(k+1)stone.Thenthepolynomialcoecientvectorcanbecomputedas:^=(VTV)1VTh;(4.19) andthepredictionresultfortheFDchannelresponseonthenSthsubcarrierforthe(k+1)stsubpacketisthesumoftheelementsof^.WerefertousingthispredictionapproachwithMPFasMPF/P.4.3NumericalExamples

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Figure4{4:MSEsofchannelestimationforaSISODSRCsystemusingPFandMPFasfunctionsofPfortime-invariantchannelswithtr=100nsatvariousSNRsatthe27Mbpsdatarate.forthepolynomialmodels.HerethesymboltimingisestimatedusingthesequentialparameterestimationmethodpresentedinChapter2.Figure4{4showstheMSEcurves,wherethetwodashedonesshowtheMSEforMPFwithSNRbeing25and35dB,respectively,andthetwodash-dottedonesshowtheMSEforPF.WeseethatMPFoutperformsPFuniformlyforallP.Wealsonotethatfortr=100ns,P=4isagoodchoiceforMPF,andP=6isagoodchoiceforPF.Example2. thesymboltimingisestimatedasinExample1,(2) Weobserve,fromFigure4{5,thatMPF/P>MPF>PF>DFFE,intermsofFDchannelresponseupdatingortrackingperformance.WealsoobservethatathighSNRs,MPF/PdemonstrateslargerimprovementoverMPF.OurnewMPF/PFD

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Figure4{5:MSEsofchannelestimationforaSISODSRCsystemusingDFFE,PF,MPF,andMPF/PasfunctionsofSNRfortime-invariantchannelswithtr=100nsatthe27Mbpsdatarate.channelresponsetrackingmethodsignicantlyoutperformstheconventionalDFFEupdatingmethod.Example3. Sofar,wehaveshownthatournewschemeisveryeectiveinthattheMSEofchannelestimationcanbegreatlyreduced.Inwhatfollows,wewilldemonstratetheoverallsystemperformance,intermsofpacketerrorrate(PER),ofaSISODSRCsystemusingournewscheme.Inoursimulations,onepacketconsistsof1000bytes,whichisthesameasthepreviouschapters.Coherentdetection(CD)withsoft-information,i.e.,thesoft-detectionschemegiveninAppendix2.5.4,isusedinall

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Figure4{6:MSEsofchannelestimationforaSISODSRCsystemusingMPFandMPF/Pasfunctionsofthemovingspeedforquasi-statictime-varyingchannelswithtr=100nsatvariousSNRsatthe27Mbpsdatarate.ofthefollowingexamples.Dierentialmodulation/detection(DD)with64-DAPSKisalsoconsideredinsomeoftheexamplesasareference.FortheDDcase,weusethesamedatacarryingsubcarriersasfortheCDcaseandthesoft-informationisobtainedusingamethodsimilartotheonegivenin[87].Example4. estimatedchannelparameters(CFO,symboltiming,andchannelresponseareallestimatedusingthepreamble,asin[77])andquasi-statictime-varyingfrequency-selectivefadingchannels(tr=100ns,themovingspeedis20mph,andtheRFis5.9GHz);Case2: thesameasforCase1exceptthatthemovingspeedis5mph;Case3: thesameasforCase1exceptthatthemovingspeedis0mph|thechannelsaretime-invariant;Case4: thesameasforCase3exceptthatthechannelknowledgeisperfectlyknown(withCFO,symboltiming,andchannelresponseallassumedknown).

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Figure4{7:PERversusSNRforaSISODSRCsystemconformingtotheE221302standardforvariousmovingspeedsforquasi-statictime-varyingfrequency-selectivefadingchannelswithtr=100nsatthe27Mbpsdatarate. WeobservefromFigure4{7thatevenifthemovingspeedisaslowas5mph,theperformanceofanE2213-02conformablesystemisgreatlydegraded,andforamovingspeedof20mph,thesystembasicallyfails. NotethatthePERcurvesforCases3and4inthisexampleprovidethecompari-sonbenchmarkfortheoverallsystemperformance.Simulationresults,whicharenotgivenherein,alsoshowthatforthecaseoftime-invariantchannels,aSISODSRCsystemusingthesubpacket-basedtransmissionhasthesamePERperformanceasforCases3and4inFigure4{7,providedthattheformerdoesnotusethechanneltrackingalgorithmsbutusesthesamechannelparametersestimatedfromthepacketpreambleasthelatter.Example5.

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Figure4{8:PERversusSNRforaSISODSRCsystemusingthesegmentedpacketdesignandchannelupdating/trackingbyDFFE,PF,MPF,andMPF/Pfortime-invariantchannelswithtr=100nsatthe27Mbpsdatarate.byusingthepreamble,asinCase3ofthepreviousexample.WealsogivethecorrespondingPERcurveforDDDDwith64-DAPSKasareference.NotefromthesimulationresultsthatourMPFchannelupdatingapproachsignicantlyoutperformstheDFFEchannelupdatingmethodandcanhavecomparableperformanceastheDDscheme.Moreover,ourMPF/PchanneltrackingmethodsignicantlyoutperformstheDDschemeandisabout7dBbetterthantheDFFEmethod.Example6.

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Figure4{9:PERversusSNRforaSISODSRCsystemusingthesegmentedpacketde-signandchannelupdating/trackingbyMPF/Pforvariousquasi-statictime-varyingchannelswithtr=100nsandvariousmovingspeedsatthe27Mbpsdatarate.channelupdating/trackingbyMPF/PcansignicantlyoutperformitscounterpartusingDD,especiallyforthecasesofhighmovingspeedsandhighSNRs.4.4ConcludingRemarks

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TheMIMOsystemsovertime-varyingchannelsconsideredhereinincludetheSTBCandBLASTsystemstargetingattheapplicationsofDSRCandBMWA.WewillexemplifyourpresentationusingtheMIMOsystemsresemblingtheE2213-02conformableSISODSRCsystems. Duetotheproblemswithdierentialmodulation/detectionfortheMIMOsys-temsdiscussedinChapter1,weusechannelupdating/trackingtodealwiththetime-varyingchannelproblemfortheMIMOsystems.Hence,weconsiderhereinvarioussystemdesignandsignalprocessingissuesfortheMIMOsystemsemployingchannelupdating/tracking. Specically,rst,weconsidertheMIMOpreambledesignfortheseMIMOsys-tems.TheMIMOpreambledesignweintroducedinChapter3fortheMIMOsystemsisforthetime-invariantchannels.Forthetime-varyingchannels,however,thepream-bleistoolong.WedesignaMIMOpreamblewhichisshorterthantheonegiveninChapter3andprovidecorrespondingsignalprocessingalgorithmstoestimatetheMIMOchannelresponses. Second,weconsiderthechannelupdating/trackingproblemfortheMIMOsys-tems.AsfortheSISOsystemsoverthetime-varyingchannels,weusesegmentedsubpacketstoprovideamechanismforchannelupdating/tracking.Thechannelup-dating/trackingmethodsproposedfortheSISOsystemscanbereadilyextendedtotheMIMOsystems,andtheMIMOdetectorsdevisedforthetime-invariantchannelcasecanbeusedheredirectlyaswell.WeusetheSTBCsystemstoexemplifyourpresentation.151

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Simulationexamplesareprovidedtoshowtheeectivenessofoursystemdesignsandsignalprocessingalgorithms. Theremainderofthischapterisorganizedasfollows.Section5.1describestheMIMOpreambledesignandMIMOchannelresponseestimationbasedonthispreambledesign.Section5.2demonstratesthepacketdesignandsignalprocessingalgorithmsforchannelupdating/trackingfortheSTBCsystems.Finally,weconcludeinSection5.3.5.1MIMOPreambleDesignandChannelEstimation ForsystemswithClassItrainingsymbols,onetransmitantenna(e.g.,Tx1)usesonehalfofallthesubcarriersappropriatedfortrainingandtheother(i.e.,Tx2)usestheremaininghalf,andthetransmissionsfromthetwotransmitantennasdonotinterferewitheachother|thetrainingsymbolsfromthetwotransmitantennasareorthogonaltoeachotheratthesubcarrierlevel.TheClassItrainingsymbolswereproposedbasedontheassumptionthatthechannelforOFDMsignalingcanberepresentedbyanFIRlter,andanML-basedestimationmethodisusedwiththesesymbolstoestimatetheFIRchannelresponseinthetimedomain;see[57]forsuchanexistingchannelestimationapproach,whichisreferredtoastheML-FIRapproachherein.AnadvantageofML-FIRisthattheresultingClassIOFDMtrainingsymbolscanbeofthesamelengthasthoseforaSISOsystem.Yet,aspointedoutin[118,77],theFIRchannelmodelisonlyapoorapproximationoftherealisticOFDMchannels.

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ForsystemswithClassIItrainingsymbols,eachtransmitantennausesallthesubcarriersappropriatedfortraining,andthetransmissionsfromthetwotransmitantennasareseparatedusingon/otransmission(see,e.g.,[121])orplus/minustrans-mission(see,e.g.,theMIMOpreamblepresentedinChapter3)|thetrainingsymbolsareorthogonaltoeachotheratthesymbollevel.AnadvantageofusingtheClassIItrainingsymbolsisthatthesimplesubcarriertrainingmethod,whichisthesameasfortheSISOcase[77],canbeusedforchannelresponseestimationinthefre-quencydomain(FD),whichcanobviatethenecessityoftheFIRmodelassumption.Yet,thelengthoftheresultingClassIIOFDMtrainingsymbolsisusuallytwiceaslongasthatforaSISOsystem,andcanleadtoperformancedegradationsforthetime-varyingchannels. OurfocushereinistoimprovethechannelestimationperformancefortheMIMOsystemsoverthetime-varyingchannels.WedeviseanewMIMOpreamblewhichisbasedonamodiedversionoftheClassItrainingsymbols.BasedonthisMIMOpreambledesign,weproposeanewchannelestimationapproachwhichexploitsthepolynomialapproximationoftheFDchannelresponse.Thisapproachisreferredtoasthepolynomialtting/interpolation(PF/I)approach.ThePF/IapproachestimatesthechannelresponseinFDbyrstapplyingpolynomialttingtothesubcarrierscontainingthetrainingsymbolsandthenperforminginterpolationtocoverthesub-carrierswithouttrainingsymbols.Aswillbedemonstratedbysimulationresults,PF/IcansignicantlyoutperformML-FIRforrealisticOFDMchannels.Thissug-geststhatthepolynomialmodelissuperiortotheFIRmodelforchannelresponseestimationfortheOFDMsignaling.

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Table5{1:Frequency-domaintrainingsequencesforthemodiedClassItrainingsymbolsfortheMIMOsystemsoverthetime-varyingchannels. Tx1 0x0x...x00x0...0x01 NotethattheOFDMsymbolfortheSIGNALeldisalsogeneratedusingthesubcarrierlevelorthogonality,thesameasforthelongOFDMtrainingsymbols;thisisneededformaintainingthebackwardcompatibility.Notealsothat,asmentionedearlier,weneedtoaddanotherSIGNALeldtodierentiatetheSTBCtransmissionfromtheBLASTtransmission.ThisadditionalSIGNALeldcanfollowtheexistingone,andtheOFDMsymbolforthisnewSIGNALeldisgeneratedusingthesamesubcarrierlevelorthogonality.WiththisadditionalSIGNALeld,thechannelcanbebettertracked.5.1.2ChannelEstimation

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whereh(n;m)nSdenotesthechannelgainfromtransmitantenna(Tx)mtoreceivean-tenna(Rx)nonthenSthsubcarrier. Toestimatethechannelresponsesonthosesubcarrierswithouttrainingsymbols,wemustexploitthechannelcorrelationsamongsubcarriers.OnesimplewayistoresorttochannelmodelstoapproximatethechannelsfortheOFDMsignaling.TheFIRmodelof[57,68]issuchanexampleandiswidelyused.Analternativemodel,whichweadvocatehere,isthepiecewisepolynomialmodelwehaveusedinChapter4.AsseeninChapter4,thepolynomialmodelcanbesuccessfullyusedtoimprovethechannelresponseestimationaccuracyforSISOOFDMchannels.ItispreferredoveritsFIRcounterpartforrealisticOFDMchannelssince:(a)theFDchannelresponseofanFIRltermustbeperiodicinNSwhereastheFDchannelresponseofarealisticOFDMchannelcannotbeperiodicunlesstlF=0foralllF,whichisclearfrom(2.12)andFigure2{4,and(b)theFDchannelresponsedescribedbyapolynomialcanbeexible(withoutthisperiodicityconstraint).Anadditionaladvantageofthepolynomialapproximationisthatitisnotsensitivetotheresponsesoftheltersusedatthetransmitterandreceiversincetheeectsoftheselterscanbeeasilyabsorbedintothepolynomials. ConsidertheestimationofthechannelsfromTx1andTx2toRxn,n=1;2;:::;N.AsinChapter4,weusepiecewisepolynomialstoapproximatetheFDchannelresponses.Wegroupthesubcarriersinto4sets,witheachsetcorrespondingtoapolynomial.Thisgroupingdependsonthetransmitantenna:forTx1,the4sets

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ApolynomialoforderPcanbeusedtoapproximatetheFDchannelresponseoneachoftheabovesets.Forexample,forchannelh(n;1)nSonS1;1(27[nS1]NS13),wehave:h(n;1)nS=PXp=0(n;1;1)p[nS1]NSf(n;1;1)p+e(n;1;1)nS;P;(5.2) where(n;1;1)p,p=0;1;:::;P,isthepth(complex)polynomialcoecient,[nS1]NSisequaltonS1fornSNS=2andnS1NSfornS>NS=2,f(n;1;1)=20isusedtoshifttheinitiallocationofthepolynomial,ande(n;1;1)nS;PisthemodelerrordependingonP.(Fortheabovethree-letter-superscriptindexing,thersttwoareusedtorefertothechannelbetweenTx1andRxn,andthelasttwo|(1;1)|thesubcarrierset.) Now,letusconsiderthenewPF/ImethodfortheMIMOsystemsbyconsideringh(n;1)nSonS1;1asanexample.Letv1=[75311357]TandV1bethe8(P+1)matrixwiththeithcolumnbeingv(i1)1(element-wisepower).Let^h(n;1;1)C2C81bethechannelgainestimatesonthesubcarrierswithtrainingsymbols,i.e.,subcarriersf27;25;23;21;19;17;15;13g,obtainedviasubcarriertraininginthesamewayasfortheSISOsystem.Thenthepolynomialcoecientsobtainedviapolynomialtting(PF)canbewrittenas:^(n;1;1)=[^(n;1;1)0^(n;1;1)1:::^(n;1;1)P]T=(VT1V1)1VT1^h(n;1;1)C:(5.3)

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Let^h(n;1;1)FbethePFcounterpartof^h(n;1;1)C.Then^h(n;1;1)Fcanbeobtainedusing:^h(n;1;1)F=V1^(n;1;1):(5.4) Letv=[6420246]TandVbethematrixformedfromvinthesamewayasV1fromv1.Let^h(n;1;1)Ibethechannelresponseestimateonsubcarriersf26;24;23;20,18;16;16gviapolynomialinterpolation.Then,wehave:^h(n;1;1)I=V^(n;1;1):(5.5) Combining^h(n;1;1)Fand^h(n;1;1)I,weobtaintheestimateofthechannelresponsefromTx1toRxnonS1;1. Inexactlythesameway,i.e.,usingthecalculationsgivenin(5.3)to(5.5),wecanestimatethechannelresponsesfromTx1toRxnonS1;2andS1;4aswellasthosefromTx2toRxnonS2;1,S2;3,andS2;4.ForthechannelsfromTx1toRxnonS1;3andfromTx2toRxnonS2;2,theresponsecanbeobtainedsimilarly.Letv2=[85;311357]TandV2bethematrixformedfromv2inthesamewayasV1fromv1.Let^(n;1;3),^h(n;1;3)C,^h(n;1;3)F,and^h(n;1;3)Ibethecounterpartsof^(n;1;1),^h(n;1;1)C,^h(n;1;1)F,and^h(n;1;1)I.Thenwehave,correspondingto(5.3),(5.4),and(5.5),respectively,^(n;1;3)=(VT2V2)1VT2^h(n;1;3)C,^h(n;1;3)F=V2^(n;1;3),and^h(n;1;3)I=V^(n;1;3).5.1.3NumericalExamples

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Figure5{2:MSEsofchannelestimationforaMIMODSRCsystemusingPF/IandML-FIRasfunctionsofPwithashortMIMOpreambledesignfortime-invariantrealisticOFDMchannelswithtr=100nsatvariousSNRs(withperfectsymboltiming).(a)thechannelswithtr=100nsforDSRCareequivalenttochannelswithtr=50nsforWLANsincethebandwidthoftheformerisonehalfofthatofthelatter;(b)forML-FIR,PisthelengthofFIRchannelmodel;and(c)tojointlyshowthetwoMSEcurvescorrespondingtoPF/IandML-FIR,weuse4PandPtomarkthehorizontalaxisforPF/IandML-FIR,respectively.)Weassumetheknowledgeofperfectsymboltiminghere.Figure5{2showstheMSEcurves,wherethetwodashedonesshowtheMSEsforML-FIRwithSNRbeing25and35dB,respectively,andthetwosolidonesshowtheMSEsforPF/I.WeseethatPF/IoutperformsML-FIRwithreasonablyselectedP.Wealsonotethatfortr=100ns,P=5isagoodchoiceforPF/I.Example2.

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Figure5{3:MSEsofchannelestimationforaMIMODSRCsystemusingPF/I(withP=5)andML-FIR(withP=19)asfunctionsofSNRwithashortMIMOpreambledesignfortime-invariantrealisticOFDMchannelswithtr=100ns(withperfectsymboltiming).channels,especiallyathighSNRsthataredesiredforhighdatarateMIMOsystems.ThisalsosuggeststhatthepolynomialmodelissuperiortoitsFIRcounterpartforchannelestimationfortheOFDMsignaling.5.2ChannelUpdating/TrackingfortheMIMOSystems

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bethereconstructeddatamatrixforthenSthsubcarrier,andletYk;nS=266666664y1;2k1;nSy1;2k;nSy2;2k1;nSy2;2k;nS......yN;2k1;nSyN;2k;nS377777775(5.7) bethereceiveddatamatrixonthenSthsubcarrier,whereyn;2k1;nSandyn;2k;nSarethereceiveddataonthenSthsubcarrierattimes(2k1)and2k,respectively,fromthenthreceiveantenna.LetHk;nS=266666664h1;1;k;nSh1;2;k;nSh2;1;k;nSh2;2;k;nS......hN;1;k;nShN;2;k;nS377777775(5.8) bethechannelmatrixforthekthpairofOFDMdatasymbols,where^hn;m;k;nSistheestimatedchannelresponseonthenSthsubcarrierfromthemthtransmitantennatothenthreceiveantennaattimek.Then,parallelto(4.8)ofChapter4,wehaveYk;nS=Hk;nSXk;nS+Ek;nS;(5.9)

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whereEk;nSisthenoisematrix.Formtheaboveequation,thechannelresponsematrixforthenon-zerosubcarrierscanbeupdatedas:^H(DFFE)k;nS=Yk;nSXHk;nS(Xk;nSXHk;nS)1=Yk;nSXHk;nS=(jx2k1;nSj2+jx2k;nSj2):(5.10) Then,theerrormatrixofthechannelupdatecanbewrittenas^H(DFFE)k;nS4=Ek;nSXHk;nS=(jx2k1;nSj2+jx2k;nSj2):(5.11) Now,letusconsider^h(DFFE)n;k;nS,thenthrowof^H(DFFE)k;nS.Wehave,whentherecon-structeddatamatrixiserror-free,En^h(DFFE)n;k;nSo=0;(5.12) andE^h(DFFE)n;k;nSH^h(DFFE)n;k;nS=2 WecanseefromtheaboveequationthatthevarianceofeachelementofthechannelmatrixisdeterminedbytheaverageenergyofthetwodatasymbolssentinthetwotimeintervalsonthenSthsubcarrier. Then,basedontheDFFEchannelupdates,wecanrenethechannelupdatesusingthemodiedweightedpolynomialtting(MPF)method.ThisisdoneforeachoftheMNchannels,inthesamewayasdiscussedinChapter4. Next,wecanpredicatethechannelusingtheMPF/PmethodpresentedinChap-ter4.NotethatduetothedoubledlengthofeachsubpacketcomparedtotheSISOcase,wechoosePL=4and~P=1fortheSTBCsystems.5.2.3NumericalExamples

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onewholepacketasinChapter3,estimatedchannelparametersusingtheshortMIMOpreamble,andMLphasetrackingusedasinChapter3;Case2: subpacket,estimatedchannelparametersusingtheshortMIMOpreamble,andMLphasetracking;Case3: subpacket,estimatedchannelparametersusingtheshortMIMOpreamble,andaveragingthetwoMLphasetrackingresultswithinapairofOFDMdatasymbols;Case4: subpacket,estimatedchannelparametersusingtheshortMIMOpreamble,andchannelupdating/trackingwithMPF/P(thepairofSIGNALeldsareusedtofacilitatethechannelupdating/tracking). WeobservefromFigure5{4,whichalsohasaPERcurvefortheSTBCWLANsystem(withthelongMIMOpreamble)forreference,that: Wecanalsosee,aswepointedoutearlierinChapter3,thataveragingthetwoMLphasetrackingresultswithinapairofOFDMdatasymbolscangreatlyimprovethedetectionperformancefortheSTBCsystem.

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Figure5{4:PERversusSNRforanSTBCDSRCsystemwithtwotransmitantennasandonereceiveantennausingvariouscombinationsofoursystemdesignsandsignalprocessingalgorithmsfortime-invariantchannelswithtr=100nsatthe27Mbpsdatarate.Example4.

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Figure5{5:PERversusSNRforanSTBCDSRCsystemwithtwotransmitanten-nasandonereceiveantennausingacombinationofoursystemdesignsandsignalprocessingalgorithmsforvariousquasi-statictime-varyingchannelswithtr=100nsandvariousmovingspeedsatthe27Mbpsdatarate.5.3ConcludingRemarks

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Inthischapter,werstprovideasummaryofconsideredissuesandthecontri-butionsofthisdissertationandthenrecommendthefuturework.6.1Summary First,wehaveconsideredthechannelmodelingfortheOFDMsignaling.Wehaveclariedthefact,whichwasrstrevealedin[118]butlargelyoverlookedeversince,thattheFIRchannelmodelisonlyanapproximatemodelfortheOFDMsignaling.WehaveconsideredachannelmodelusefulforcharacterizingtherealisticchannelsfortheOFDMsignaling.ThischannelmodelisespeciallyusefulforassessingtheperformanceofthechannelparameterestimationmethodsdevisedforOFDM-basedwirelesscommunicationsystems. Second,wehaveconsideredthechannelparameterestimationproblem.Wehaveprovidedasequentialparameterestimationmethodthatcanbeusedtoestimateallof166

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SISO STBC BLAST TIC TVC TIC TVC TIC TVC Preambledesign S1 P2 P P SIGNALelddesign S S P P P P OFDMdatasymbol S P P D3 Ddesign Pilotsymboldesign S N/A4 N/A P N/A CFOandsymbol P D D D D Dtimingestimation Channelestimation P D P P P P MLphasetracking P N/A P N/A P N/A LSphasetting P N/A D N/A D N/A Samplingclock P N/A D N/A D N/Asynchronization Semi-blindchannel P N/A P N/A P N/Aestimation Channelupdating N/A P N/A P N/A D Softdetection P D P D P35 S:conformingtothestandard;P:designoralgorithmpresentedinthecontext;D:designoralgorithmdonebutnotpresentedinthecontextduetosimilaritytoitspresentedcounterpart;N/A:notapplicable;P3:threedierentmethodspresented.thechannelparameters,includingCFO,symboltiming,aswellaschannelresponse,fortherealisticchannels.ThesequentialmethodfullyexploitsthestructureofthepacketpreambleasspeciedbytheIEEE802.11astandard. Finally,wehaveconsideredthefollowingerrorreductionproblems.First,wehavegivenanMLphasetrackingapproachusingpilottonestoestimate(andcor-rect)theresidueCFOinducedphaseerror(CPE)foreachreceivedOFDMdatasymbol.Second,wehavesuppliedanLSphasettingapproachusingtheestimated

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A.Aghamohammadi,H.Meyr,andG.Ascheid.AdaptivesynchronizationandchannelparameterestimationusinganextendedKalmanlter.IEEETrans-actionsonCommunications,37(11):1212{1219,November1989.[2] S.M.Alamouti.Asimpletransmitdiversitytechniquesforwirelesscommuni-cations.IEEEJournalonSelectedAreasinCommunications,16(8):1451{1458,October1998.[3] L.Armstrong.DSRCapplications.Availableathttp://www.leearmstrong.com/DSRC/DSRCHomeset.htm,lastaccessed:07/23/2004.[4] ASTMStandardE2213-02.Standardspecicationfortelecommunicationsandinformationexchangebetweenroadsideandvehiclesystems|5GHzbandded-icatedshortrangecommunications(DSRC)mediumaccesscontrol(MAC)andphysicallayer(PHY)specications.November2002.[5] L.Bahl,J.Cocke,F.Jelinek,andJ.Raviv.Optimaldecodingoflinearcodesforminimizingsymbolerrorrate.IEEETransactionsonInformationTheory,20(2):284{287,March1974.[6] C.Berrou,A.Glavieux,andP.Thitimajshima.NearShannonlimiterror-correctingcodinganddecoding:Turbo-codes.ProceedingsoftheIEEEInter-nationalConferenceonCommunications,2:1064{1070,May1993.[7] J.A.C.Bingham.Multicarriermodulationfordatatransmission:anideawhosetimehascome.IEEECommunicationsMagazine,28(5):5{14,May1990.[8] R.S.Blum,Y.Li,J.H.Winters,andQ.Yan.Improvedspace-timecodingforMIMO-OFDMwirelesscommunications.IEEETransactionsonCommunica-tions,49(11):1873{1878,November2001.[9] R.S.Blum,Q.Yan,Y.Li,andJ.H.Winters.Improvedtechniquesfor4transmitand4receiveantennaMIMO-OFDMforwirelesscommunications.ProceedingsoftheIEEEVTS53rdVehicularTechnologyConference,2:1298{1302,May2001.[10] M.Bossert,A.Donder,andV.Zyablov.ImprovedchannelestimationwithdecisionfeedbackforOFDMsystems.IEEElectronicsLetters,34(11):1064{1065,May1998.172

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