A Soft detector with good performance/complexity trade-off for a MIMO system

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EURASIPJournalonAppliedSignalProcessing2004:9,1266c 2004HindawiPublishingCorporationASoftDetectorwithGoodPerformance/ComplexityTrade-OffforaMIMOSystemJianhuaLiuDepartmentofElectricalandComputerEngineering,UniversityofFlorida,P.O.Box116130,Gainesville,FL32611-6130,USAEmail:jhliu@dsp.u.eduJianLiDepartmentofElectricalandComputerEngineering,UniversityofFlorida,P.O.Box116130,Gainesville,FL32611-6130,USAEmail:li@dsp.u.eduReceived30May2003;Revised23November2003Wepresentahybridsoftdetectorthathasagoodperformance/complexitytrade-o foramultiple-inputmultiple-output(MIMO)wirelesscommunicationsystemwithknownchannelinformation.Thenewsoftdetectorcombinesthemeritsofasimpleunstruc-turedleast-squaresLS)-basedsoftdetectorandalistspheredecoderLSD)-basedsoftdetectorfordatabitdetection.Theformeriscomputationallymuchmoree cientthanthelatteratthecostofpoorerperformance.Thepoorperformanceoftheformeroccursmainlywhenthechannelmatrixisill-conditioned.Wheneverthishappens,weusetheLSD-basedsoftdetectorinthehy-bridsoftdetector;otherwise,weusetheLS-basedone.Moreover,weprovideatightradiusforaspheredecoder,aharddetector,viausingtheoutputofanLS-basedharddetector.ThesetwoharddetectorsareneededtodetermineifLSorLSDshouldbeusedinthehybridsoftdetector.Asanapplicationexample,weconsiderdoublingthemaximumdatarateoftheIEEE802.11aconformablewirelesslocalareanetworksbyaMIMOsystemwithtwotransmitandtworeceiveantennas.Forthisapplication,thenewsoftdetectorisabout10timesfasterthantheLSD-basedoneandisabout10timesslowerthantheLS-basedone.YetthepacketerrorrateduetousingthenewsoftdetectorisquiteclosetothatofusingtheLSD-basedone.Keywordsandphrases:BLAST,MIMO,softdetector,convolutionalcodes,OFDM,WLAN.1.INTRODUCTIONHightransmissiondatarateisofparticularimportanceforfuturewirelesscommunicationservices.Onepromisingwayofincreasingthetransmissiondatarateistodeploymul-tipleantennasatboththetransmitterandreceiverendstoexploitthehugechannelcapacityo eredbysuchasysteminamultipath-richenvironment[1 2 ].Thecorrespond-ingsystemisreferredtoasamultiple-inputmultiple-output(MIMO)wirelesssystem.Inpracticalcommunicationsystems,forwarderrorcor-rectioncodes,suchastheconvolutionalcode,areoftenusedtolowerthetransmissionerrorratetoanacceptablelevel[ 3 4 ]byaddingredundancyinthetransmission.Softdetec-torshavebeenpreferredtoharddetectorssincetheformercanleadtobetterdetection/decodingperformance.Forthesingle-inputsingle-out(SISO)systems,softdetectorshavebeenwellstudied[3 4 ].Lately,muchattentionhasbeenpaidtothesoftdetectorsfortheMIMOsystems.Thespace-timebit-interleavedcodedmodulation(STBICM)scheme[5 6 ]seemstobethebest(intermsofperformance)softdetectorforaBell-lablayeredspace-time(BLASTsystem[7 8 9 ],anespeciallyattractiveformoftheMIMOsystems.However,STBICMcanonlybeimple-mentedviatheextremelyine cientbrute-forthsearch.Inpractice,softdetectorswithgoodperformance/complexitytrade-o saredesired.Amongtheotherexistingsoftdetectors,thefollowingtwoareparticularlyattractive.Oneistheunstructuredleast-squaresLS)-basedsoftdetectorof[10 ],whichfocusesmoreonthecomputationale ciencyside.TheotheristhelistspheredecoderLSD)-basedsoftdetectorof[11 ],whichfo-cusesmoreontheperformanceside.Theformerisverysimplesince,forexample,itdecouplesamultidimensionalQAMsymboldetectionintomultipleone-dimensionalQAMsymboldetections.However,theperformanceofthisdetec-torcanbepoor,especiallywhenthechannelmatrixisill-conditioned.ThelatterhasaperformanceclosetothatofSTBICMwithasignicantlyimprovedcomputationale ciency;itisbasedontheSTBICMprinciplebutsearchesinasmallsphere,viamodifyingthespheredecoderSPD),whichisaharddetector[12 ].(SPDisane cientalgorithmtoimplementthecomputationallyexpensivemaximum-likelihood(ML)harddetector.)However,theLSD-basedsoft

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AMIMOSoftDetectorwithGoodPerformance/Complexity1267 detectorstillrequiresordersofmagnitudewithmorecom-putationsthanitsLS-basedcounterpart.Inthispaper,wecombinethemeritsoftheLS-andLSD-basedsoftdetectorstoobtainanewsoftdetector,referredtoasthehybridsoftdetector,whichhasabetterperformancethantheLS-basedoneandahighercomputationale ciency thantheLSD-basedone.ThepoorperformanceoftheLS-basedsoftdetectorismainlyduetoprovidingpoorsoftin-formationtotheViterbidecoderasaresultofthechannelmatrixbeingill-conditioned.Wheneverthishappens,weusetheLSD-basedsoftdetectorinthenewhybridsoftdetec-tor;otherwise,weusetheLS-basedone.TodecideifLSorLSDshouldbeusedinthehybriddetector,wechecktoseewhetherornottheoutputoftheLS-basedharddetectoristhesameastheoutputofSPD.Ifso,wechooseLS;otherwise,weuseLSD.Tofurtherimprovethecomputationale ciency,weprovideatightradiusforSPDbasedontheoutputoftheLS-basedharddetector.Asanexample,weconsiderdoublingthemaximumdatarateoftheIEEE802.11a[13 ]conformablewirelesslocalareanetworksWLANs)byaBLASTsystemwithtwotransmitandtworeceiveantennas.Atthereceiver,weusesoftdetec-torsfordatabitdetection.WecomparetheperformanceofthenewhybridsoftdetectorwiththatoftheLS-andLSD-basedsoftdetectors.Thehybriddetectorisabout10timesfasterthantheLSD-basedoneandisabout10timesslowerthantheLS-basedone.YetthepacketerrorratePER)duetousingthehybridsoftdetectorisquiteclosetothatofusingtheLSD-basedone.Theremainderofthispaperisorganizedasfollows.Section2describesthechannelencodinganddecodingforaBLASTsystemthatemploystheconvolutionalencoder.Section3givesthedatamodelandformulatesthesoftin-formation,thatis,bitmetric.Section4presentsthepro-posednewhybridsoftdetector.Simulationresultsaregivenin Section5.Finally,weprovideourcommentsandconclu-sionsinSection6. 2.CHANNELCODINGConsideraBLASTsystemwithM transmitandN N M receiveantennas,asshowninFigure1.Figures2 and 3 ,re-spectively,showthediagramsoftheBLASTtransmitterandreceiver.Atthetransmitter,aconvolutionalencoderCC)isemployed,andaninterleaverisusedtobreakthememoryof badchannelsofthetransmission.Atthereceiver,adein-terleaverisusedbeforetheconvolutionalchannel)decoder,forexample,theViterbialgorithm,torecovertheorderofthecodedbitsequence.A1:M DEMUXandM :1MUXpairisusedatthetransmitterandreceiver,respectively,toaccommodatetheBLASTscheme.Atthetransmitter,asshowninFigure2,theCC,whichhasaconstraintlengthK C ,takesablock(alsocalledpacket)ofbitsd ={ d 1 d 2 ... d K 1,+1} 1 K [withK C 1) satthetailtoresettheCC]asitsinputandgivesalargerblockofbitsu = C( d ={ u 1 u 2 ... u K 1,+1} 1 K asitsoutput,where 1and+1denotethebinarydigits0 M H N Transmitter. Receiver. Figure1:DiagramofaMIMOsystem.and1,respectively.TheCCcodingrateisthendenedasR C = K/ K .WecanpuncturetheCCoutputblocku toobtainasmallerblockofbitsv ={ v 1 v 2 ... v K 1,+1} 1 K K< K toincreasethetransmissiondatarate.Thepunctur-ingrateisR P = K/ K ,andthecodingrateofthepuncturedCCisR = R C /R P = K/ K .Theoutputv ofthe(punctured)CCisthenfedtotheinterleaverwhoseoutputisdenotedas v ={ v (1) v (2) ... v K 1,+1} 1 K .LetK = K/Mbe aninteger.Thentheoutputsofthe1:M DEMUXareM independentlayers,denotedas v m ={ v (1) m v (2) m ... v K m 1,+1} 1 K m = 1,2,... M .Themodulatormapseachlayerofthebitsintodatasymbolsthroughthemappingf : 1,+1} 1 B C ,whereC denotesthedatasymbolcon-stellationandB = log 2 | C | isthenumberofbitsrepresentedbyadatasymbol.Let K = K /B beaninteger,whichisthenumberofdatasymbolsineachlayer.Thentheoutputsofthemodulatorscanbedenotedas x m ={ x (1) m x (2) m ... x K m } m = 1,2,... M .Finally,theM 1datasymbolvectorx k = x k 1 x k 2 x k M T ,where T denotesthetrans-pose,istransmittedthroughtheM transmitantennasatthesametime,with k denotingthetimeindex, k = 1,2,... K Thebitscorrespondingtox k aredenotedasaBM 1vec-torb k = b k 1 b k 2 b k BM T ,withb k i 1,+1} i = 1,2,... BM .Notethatx k isaone-to-onemapofb k andifneededitcanbewrittenasx k = x b k )tostressitsdependenceonb k Atthereceiver,asshowninFigure3,thesoftdetectorrstgeneratesthebitmetrics{ l k 1 l k 2 ... l k BM } ,withl k i beingthebitmetriccorrespondingtob k i i = 1,2,... BM ,attime k = 1,2,... K .Thesoftdetectorthenrearrangesthebitmetricstoobtain{ v ([ k 1] B +1) m v ([ k 1] B +2) m ... v kB m } forthebits { v ([ k 1] B +1) m v ([ k 1] B +2) m ... v kB m } ,whichweremappedtothedatasymbolx k m .Let v m ={ v (1) m v (2) m ... v K m } m = 1,2,... M ,denotethebitmetricsequencecorrespondingtothem thtransmittedlayer.TheM bitmetricsequencesarecombinedintoonelongerbitmetricsequence v = { v (1) v (2) ... v K } bytheM :1MUX.Passingtheabovebitmetricsequence v throughthedeinterleaver,weobtainthedeinterleavedbitmetricsequence v ={ v 1 v 2 ... v K } ForthepuncturedCCcodes,weneedthebitmetricforeachpuncturedbitaswellbeforeusingtheViterbialgo-rithm.Thiscanbedoneeasilybyusingzeroasthebitmet-ricforeachpuncturedbit.Oncewegetthebitmetricse-quence u ={ u 1 u 2 ... u K } correspondingtotheCCoutputu ,wecanusetheViterbialgorithmtoobtaintheestimate d ={ d 1 d 2 ... d K } ofthesourcebitsequenced

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1268EURASIPJournalonAppliedSignalProcessing Data sourced Conv.encoderv Interleaver v 1toM DEMUX v 1 x 1 Modulator. v M Modulator. x M Figure2:DiagramofaBLASTtransmitteremployingconvolutionalchannelcoding. Soft-detector v 1 v M M to1MUX v Deinterleaver v ViterbialgorithmData sink d Figure3:DiagramofaBLASTreceiveremployingViterbidecodingforconvolutionalcodes.Inthesequel,wefocusonthecalculationofthebitmet-ricsforthebitsintheQAMsymbol,duetoourWLANap-plication. 3.DATAMODELANDBITMETRICWenowgivethedatamodelandformulatethebitmetricfortheBLASTsystem.3.1.DatamodelThechannelmatrixofaMIMOtime-varyingRayleigh-fadingchannelattime k canbewrittenasH k = h k 1,1 h k 1,2 h k 1, M h k 2,1 h k 2,2 h k 2, M . h k N ,1 h k N ,2 h k N M C N M ,(1whereh k n m isthegainfromthem thtransmitantennatothen threceiveantennaattime k ,whichisassumedtobeknown.Withx k = x k 1 x k 2 x k M T denotingtheM 1QAMsymbolvectorbeingsentattime k ,thereceivedsignalcanbewrittenasy k = H k x k + n k C N 1 k = 1,2,... K ,(2wheren k N 0 2 k I N istheadditivezero-meanwhitecir-cularlysymmetriccomplexGaussiannoise.Withanappropriatepairofinterleaveranddeinterleaver,theMIMOchannelcanbeassumedtobeblockRayleighfading[14 15 ],thatis,H k isconstantattime k forthetrans-missionofx k butchangesindependentlyfromonetimein-dextoanother.Inthesequel,wefocusonobtainingthesoftinformationgiventhatweknowthechannelmatrixH k ,thenoisevariance 2 k ,andthereceiveddatavectory k .Forno-tationalconvenience,wedropthesuperscript k in2 )togetthedatamodely = Hx + n C N 1 ,(3or y = Hx b )+ n (4) 3.2.BitmetricThebitmetric(alsoknownastheL-value)forthei thbit,i = 1,2,... BM ,isdenedasl i = log P b i = +1 y H P b i 1 y H (5) AssumingequalprobabilityforeachdatabitsandusingBayestheorem,thebitmetriccanbewrittenasl i = log b B i ,+1 P y b H b B i 1 P y b H ,(6whereB i ,+1 and B i 1 arethesetof2BM 1 bitvectorsb withb i being+1and 1,respectively.Withtheassumptionofadditivezero-meanwhitecir-cularlysymmetriccomplexGaussiannoiseforthereceiveddata,theaboveequationcanbewrittenasl i = log b B i ,+1 e (1 2 y Hx b 2 b B i 1 e (1 2 y Hx b 2 ,(7which,byusingthemax-logapproximation[16 ],canbewrittenasl i max b B i ,+1 1 2 y Hx b 2 max b B i 1 1 2 y Hx b 2 = 1 2 min b B i 1 y Hx b 2 min b B i ,+1 y Hx b 2 (8)

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AMIMOSoftDetectorwithGoodPerformance/Complexity1269 Thisisinfacttheoptimalbutextremelyine cientSTBICMsoftdetector.Inthesequel,wepresentthehybridsoft-detector,whichhasagoodperformance/complexitytrade-o ,forcalculatingthebitmetric.4.THEPROPOSEDSOFTDETECTORTheproposedsoftdetectorisbasedonthecombinationoftheLSD-andLS-basedsoftdetectors.Asaresult,beforepre-sentingthenewsoftdetector,wesummarizeandcommentonthemeritsofthesetwoexistingdetectors,whicharedif-ferentapproximationsof8 )withdi erentfocusesontheperformance/complexitytrade-o 4.1.TheLSD-basedsoftdetectorTheLSD-basedsoftdetectorfocusesmainlyontheper-formancesideoftheperformance/complexitytrade-o .ItmaintainstheframeworkoftheSTBICMdetectorandim-provesthee ciencyof8 bysearchinginmuchsmallersub-sets B i ,+1 B i ,+1 and B i 1 B i 1 with| B i ,+1 |b 2 BM 1 and | B i 1 |b 2 BM 1 .TheLSD-basedsoftdetectorisimple-mentedinthefollowingtwosteps.StepSD1.Obtaintheset B ofvectorsb whichsatises y Hx b d l b B ,(9byusingthemodiedSPDalgorithmthathasaxedsphereradiusd l ,determinedbythean-tennanumbersandnoisevariance[11 ]. StepSD2.Foreachi = 1,2,... BM ,calculate B i ,+1 = B i ,+1 B and B i 1 = B i 1 B andobtainthebitmetricbyl (SD) i = 1 2 min b B i 1 y Hx b 2 min b B i ,+1 y Hx b 2 (10) Atthecostofsomeperformancedegradation,theLSD-basedsoftdetectorimprovesthecomputationale ciencyoftheSTBICMdetectorsignicantlyduetolimitingthesearchoverthemuchsmallersets.Wedonotknowtheexactdegra-dationforourWLANapplicationsincetheSTBICMdetectoristooslowtomakeareasonablecomparison.)However,theLSD-basedsoftdetectorisnotase cientasSPDduetothefollowingreasons:(a)LSDinStepSD1usesxedspherera-diuswhereasSPDuseschangingsphereradiusthatshrinkswiththendingofanewpointinthespherewithashorterdistanceand(b)thebitmetriccalculationinStepSD2needsadditionalcomputations.4.2.TheLS-basedsoftdetectorTheLS-basedsoftdetectorfocusesmainlyonthecomputa-tionalcomplexitysideoftheperformance/complexitytrade-o .WhiletheLSD-basedsoftdetectorimprovesthee ciencyof8 bylimitingthesearchonsmallersets,theLS-basedsoftdetectordecreasesthecomputationof8 )byde-couplingthedistance y Hx 2 intoM separatedistances,thatis,itdecouplesaMIMOchannelintomultipleSISOchannelsthatareprocessedindependentlyofeachother.TheLS-basedsoftdetectorhasthefollowingtwomainsteps.StepLS1.Ignorethediscreteconstellationofx toobtainanunstructuredLSsymbolestimatex (LS) of x as x (LS) = H H H 1 H H y = x + H H H 1 H H n = x + e (11) StepLS2.Forj = 1,2,... B and m = 1,2,... M ,obtainthebitmetricforeachbitusingthescheme(similarto8 ),butfortheSISOcase)givenin[ 17 ] l (LS) j m = 1 2 m min b m B m j 1 x (LS) m x b m 2 min b m B m j ,+1 x (LS) m x b m 2 (12) whereB m j ,+1 and B m j 1 arethesetof2B 1 bitvectorsb m 1,+1} B 1 withthej th bitbeing+1and 1,respectively,x (LS) m is the m thelementofx (LS) x b m C ,and 2 m = 2 [( H H H 1 ] m m with[A ] m m denoting them m )thelementofmatrixA WeremarkthatfortheSISOsystems,weusuallycon-sideranordinaryQAMsymbolastwoPAMsymbols(e.g.,a64-QAMsymbolcanbeconsideredastwo8-PAMsym-bols)duetotheorthogonalitybetweentherealandimagi-narypartsofaQAMsymbolaswellastheindependencebe-tweentherealandimaginarypartsoftheadditivecircularlysymmetricGaussiannoise.ThesameistruefortheBLASTsystemsemployingtheLS-basedsoftdetectorsincetherealandimaginarypartsofe m ,them thelementofe in12 ), m = 1,2,... M ,areindependentofeachother,asshownbelow:E ee T = H H H 1 E nn T H H H H 1 T = 0 ,13)wherewehaveusedthefactthatE[nn T ] = 0 TheLS-basedsoftdetectorisordersofmagnitudemoree cientthantheLSD-basedsoftdetectorduetothedecou-pling,aswillbeanalyzedlater.However,theperformanceoftheformerisworsethanthelatter(morethan2dBfortheM = N = 2caseforourWLANapplication,tobeshownbythesimulationexampleslater).Byroundingx (LS) m m = 1,2,... M ,totheclosestpointintheconstellationC ,weobtaintheoutputoftheLS-basedharddetector,whichwillbeusedlatter.Notethattheminimummean-squarederror(MMSE)-basedsoftdetectorisoftendeemedtobebetterthantheLS-basedone[18 ].Althoughthiscanbetruefortheconstant-modulusconstellations,suchasPSK,itisnotnecessarilytrueforQAMsymbols,assuggestedbyoursimulationre-sults(notprovidedhereduetothedi erentpowerlevels

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1270EURASIPJournalonAppliedSignalProcessing oftheQAMsymbols.HencetheLS-basedsoftdetectorismorepreferablethantheMMSE-basedonesincetheformerisslightlymorecomputationallye cientthanthelatter.4.3.ThehybridsoftdetectorTheabovetwosoftdetectorsprovidedi erentperfor-mance/complexitytrade-o sfordatabitdetection,withtheLSD-basedonefocusingontheperformanceandtheLS-basedoneonthecomputationale ciency.Inpractice,itisdesirabletohaveasoftdetectorthatisbetterthantheLS-basedoneinperformanceandfasterthantheLSD-basedoneincomputationalcomplexity.Weshowthatthiscanbedonebycombiningthesetwosoftdetectors,andthecorrespond-ingnewdetectorisreferredtoasthehybridsoftdetector.Now,weexaminewhathinderstheperformanceoftheLS-basedsoftdetector.WecanreadilyseethatwhenH H H isclosetoascaledidentitymatrix,thebitmetricsfromtheLS-basedsoftdetectorwillnotbeworsethanthosefromtheLSD-basedone.However,whenH H H becomesill-conditioned,thebitmetricsfromtheformerwillbemuchworsethanthosefromthelatter,becauseofthefollowingreasons:(a)someelementsofthenoisevectore in11 )aremagnieddrasticallyduetothepoorchannelsandb)usefulinformationislostduetothedecoupling.Hence,these(bad)bitmetricscorrespondingtotheill-conditionedchannelscanbeseenasthebottleneckfortheperformanceoftheLS-basedsoftdetector.Ifwecanidentifythesebadbitmetricsandre-placethembythosefromtheLSD-basedsoftdetector,wecanimprovethedetectionperformancesignicantly.WeidentifythebadbitmetricsbycomparingtheLS-basedharddetectoroutput x (LS) andtheSPDoutput x (SPD) Iftheyarenotthesame, x (LS) ismorelikelytohaveerror(s)since x (SPD) isanMLestimate,whichisbetterthanthefor-mertheoretically.Inthiscase,thecorrespondingbitmetricsfromtheLS-basedsoftdetectorareconsideredbad;other-wise,thesebitmetricscanbeconsideredreliable.Inviewoftheabove,wehavethefollowingstepsforthehybridsoftdetector.StepHY1.ObtaintheLSsymbolestimatex (LS) byusing 11 )ofStepLS1.StepHY2.DeterminetheLSharddetectionresult x (LS) StepHY3.CalculatetheSPDdetectionresult x (SPD) StepHY4.Checktheharddetectionresultsif x (LS) = x (SPD) ,thengotoStepHY5;otherwise,gotoStepHY6.StepHY5.Obtainbitmetricsby12 )ofStepLS2basedon x (LS) fromStepHY1andthenstop.StepHY6:obtainbitmetricsbyperformingStepsSD1andSD2andthenstop.Thecomputationalcomplexityofthehybridsoftdetec-torisdominatedbySPDandtheLSD-basedsoftdetector,thatis,StepsHY3andHY6.TospeedupthecalculationofSPDinStepHY3,weneedtoconsiderthedeterminationofitsinitialradius,whichisacrucialissueforSPD.Iftheinitialradiusistoosmall,therewillbenopointx inthesphereSPDcannotndtheMLsolution.Ontheotherhand,iftheinitialradiusistoolarge,SPDwillbeveryslowduetotheun-necessaryadditionalsearches.Thenumberoftheadditionalsearchescanbereducedbyusingamodiedsearchingap-proachgivenin[19 ].However,itcomplicatesthealgorithmsitself.Here,wegiveatightsphereradius,basedontheLS-basedhard-detectoroutput x (LS) ,byusingd r = y H x (LS) + d ,14)where d > 0isaverysmallvalue.NotethatthisradiuswillcontainatleastonepointtheoutputoftheLS-basedhard-detector.Notealsothat,formostcases(98outof100forthesignal-to-noise-ratios(SNRs)ofinterestinourWLANappli-cation,aswillbeshownbythesimulationresultsinthenextsection),thisradiuscontainsonlyonepoint.Byusingthistightradius,ourpreliminarysimulationresultsshowthatSPDcanbease cientastheinterferencecancellationandnullingalgorithm[8 ]andusesonly5timesasmanypsastheLS-basedsoftdetector.Thecomputationalcomplexity,intermsofops,foreachstepoftheLSD-basedsoftdetector,canbeestimatedasfol-lows.WeassumeM = N forconvenience.)StepHY1:O M 3 formatrixmultiplicationsandinver-sion.Forexample,acalculationusingMatlabindicatesthatthenumberofpsis444forthe M = 2case.StepHY2:Negligible.StepHY3:O M 3 )toO M 6 forSPD,dependingontheSNRandB [ 12 20 ].Forexample,preliminarycalculationsusingMatlabshowthat,byusingthetightradius,SPDusesonly5timesasmanypsasLSinStepHY1for64-QAM,M = 2, andtheSNRsofinterest.StepHY4:Negligible.StepHY5:NegligiblebytablecheckingforthePAMsym-bols. StepHY6:(a)O M 3 )toO M 6 forLSD,which,asshownbysimulationresults,usestypically2to10timesasmanopsasSPDinStepHY3,thatis,10to50timesasmanypsasLSinStepHY1.Weusetheaverage25inthesequel.)(b) O N 2 C BM )forbitmetriccalculation,whereN C isthenumberofcandidatesinthelistforLSDandtheoperationofndingtheminimumisperformedbyusingtheconven-tionalbubblingalgorithm;forexample,forM = 2, B = 6,andN C = 120(whichistypicalforagoodperformance),thisamountstoabout43200ps(assuming| B i ,+1 |= | B i 1 |= 60, i = 1,2,... ,12,forconvenience),whichisabout95timesasmanopsasLSinStepHY1.Aswillbeseenfromthesimulationresultsinthenextsection,lessthan2%ofthecaseshavedi erentSPDandLSharddetectionresults.Hence,wecanseethatthehybridsoftdetectorisabout1 LS +5 SPD +0 02 25 LSD +95 = 8 415)

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AMIMOSoftDetectorwithGoodPerformance/Complexity1271 OFDMpacketpreamble10 0 8 = 8 s1 6+2 3 2 = 8 s t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 GI 2 T 1 T 2 Signalld0 8+3 2 = 4 GI SignalOFDMdataeld0 8+3 2 = 4 GI OFDM symbol GI OFDM symbol Figure4:PacketstructureoftheIEEE802.11astandard.timesasslowastheLS-basedsoftdetector,whichindicatesthatthehybridsoftdetectorisabout 10timesslowerthantheLS-basedsoftdetector.WecanalsoseethattheLSD-basedsoftdetectorneeds120timesasmanyopsastheLS-basedone,whichmeansthatthehybridsoftdetectorisabout 10 timesfasterthantheLSD-basedone.ThiswillbeelaboratedinthenextsectionbasedontheparametersofourWLANapplication.)Notethatthenewhybridsoftdetectorismoree cientforhighSNRsthanforlowSNRssinceathighSNRstheprobabilitiesof x (LS) = x (SPD) arehighandthechancesofusingthecomputationallyexpensiveLSD-basedsoftdetectorarelow.Notealsothattheaboveanalysisofthecomplexityisonlyintendedtogiveafeelingaboutthee ciencyofthehybridsoftdetectorandisbynomeansveryaccurate.Moreaccurateanalysisofthecomplexities,includingthoseforSPDandLSD,isstillanopentopic.Weremarkthatthebadbitmetricscanalsobeidenti-byusingtheconditionnumberCN)ofH H H ,andtheresultingsoftdetectorcanbereferredtoastheCN-hybridsoftdetector.However,theCN-hybridsoftdetectorisinfe-riortoitshybridcounterpartduetothefollowingreasons.First,itishardtodetermineathresholdfortheCN.Ifthethresholdistoohigh,manybadbitmetricsfromtheLS-basedsoftdetectorwillbeusedinthehybridsoftdetector,whichwillleadtodegradedperformance.Ontheotherhand,ifthethresholdistoolow,thecomputationallyexpensiveLSD-basedsoftdetectorwillbeusedtoooften,whichwillre-sultinincreasedcomputationalcomplexity.Second,alargeCNdoesnotnecessarilyresultindetectiondi erencesbe-tweenSPDandtheLS-basedharddetectors.Neitherdoesasmallconditionnumberguaranteethesamedetectionresultforthetwoharddetectors.Aswillbedemonstratedusingthesimulationresultsinthenextsection,forapracticalchoiceofCN,say100,theCN-hybridsoftdetectorhasacomparable(0 06 (25+95)= 7 2timesasmanypsasLS)complex-itywiththehybridsoftdetector;yet,theperformanceoftheformerisinferiortothelatter.5.SIMULATIONRESULTSOurresultsobtainedunderthefadingchannelconditioncanbereadilyextendedtotheorthogonalfrequency-divisionmultiplexing(OFDM)-basedWLANsystemsoperatingoverfrequency-selectivefadingchannels[21 ].Thisisbecauseforeachsubcarrierthechannelisafadingone.Inoursimu-lations,wefollowtheIEEE802.11a5GHzbandhigh-speedWLANstandard[13 ]wheneverpossible.TheOFDM-basedWLANsystem,asspeciedbytheIEEE802.11astandard,usespacket-basedtransmission.Figure4showsthepacketstructurespeciedbythestandard.EachpacketconsistsofmanyOFDMsymbols.EachOFDMsymboloccupies64subcarriers,amongwhich48areusedfordatasymbolsand4forpilotsymbols.Therearealso12nullsubcarriers.TheOFDMsymbolsareobtainedviatakingtheinversefastFouriertransformFFTofthedata,pilots,andnullsonthesesubcarriers.ThenominalbandwidthoftheOFDMsignalis20MHzandtheI/Qsamplingintervalis50nanoseconds.Duetothefactthatthemodulationandde-modulationaredoneinthefrequencydomain,afrequencydomainbit-levelinterleaverisusedtosegmenttheencodedbitsequenceaccordingtothetransmissiondatarateandtoscatterthemoverthe48di erentdata-carryingsubcarriers.Beforeinterleaving,anindustrialstandardconstraintlength7andR C = 1 / 2CCisemployedtocodethesourcebitse-quence.IntheIEEE802.11astandard,themaximumtrans-missiondatarateis54Mbps;inthiscasethe64-QAMcon-stellationisusedandthechannelcodingrateisR = 3 / 4, whichcomesfrompuncturingtheR C = 1 / 2convolution-allyencodedsequencewiththepuncturingrateR P = 2 / 3. Thechannelisassumedtobeedduringthepackettrans-mission. Weconsiderdoublingthemaximum54Mbpstransmis-siondataratebyusingaBLASTsystemwithtwotrans-mitandtworeceiveantennas,thatis,M = N = 2.ThisOFDM-basedBLASTWLANsystemisbackwardcompat-iblewithitsSISOcounterpart,withthepacketstructureshowninFigure5.See[21 ]formoredetaileddescriptionoftheMIMOsystemdesign.)Inoursimulations,eachofthe MN = 4timedomainMIMOchannelsisgeneratedac-cordingtotheexponentialchannelmodel[22 ]withtheroot-mean-squarespreadingtimet rmsbeing50nanoseconds;the4channelsarestatisticallyindependentofeachother.AfterFFTatthereceiver,thechannelmatrixforeachsubcarrierhasthesameformasin1 ),withthe k beingthesubcar-rierindexinthiscase.Thissubcarrierindexisequivalenttothetimeindexfortime-varyingfadingchannelssincethechannelfortheOFDM-basedWLANsisassumedtobeedfortheentirepacket,withthechangesacrossthesubcarri-ersduetothedelaytimespreading.Notethattheintersym-bolinterferenceisavoidedintheOFDM-basedsystemsduetousingthecyclicpr[13 ].)Weconsiderthecaseofper-fectchannelknowledge,wherethecarrierfrequencyo set, symboltiming,channelresponse,andnoisevarianceareallknowninalloursimulations;inpracticalapplications,theseparameterscanbeestimatedviaapplyingthechannelparam-eterestimationmethods,suchasthosein[21 23 24 25 ],tothepacketpreambles.Duetothefactthat52outof64subcarriersareusedintheOFDM-basedSISOWLANsystem,theSNRusedhereinisdenedas52 2 / 64forthe64-QAMconstellationwhose

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1272EURASIPJournalonAppliedSignalProcessing IEEE802. 11acompatiblepacketpreambleShorttrainingsymbols10 0 8 = 8 s Longtrainingsymbolblock11 6+2 3 2 = 8 s t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 GI 2 T 1 T 2 GI 2 T 1 T 2 Signalld0 8+3 2 = 4 GI SignalGI SignalLongtrainingsymbolblock21 6+2 3 2 = 8 GI 2 T 1 T 2 GI 2 T 1 T 2 OFDMdataeld0 8+3 2 = 4 GI OFDMsymbolGI OFDMsymbolGI OFDMsymbolGI OFDMsymbol Figure5:PacketstructurefortheOFDM-basedBLASTWLANsystem. LS CN-hybridHybridLSD SISOreference2324252627282930SNRdB)10 2 10 1 10 0 PER Figure6:PERversusSNRcomparisonsforthesoftdetectors.averageenergyisnormalizedto1.FortheOFDM-basedBLASTWLANsystem,wekeepthesametotaltransmissionpowerandmaintainthesamesubcarrierstructureasitsSISOcounterpart.Wegivetwosimulationexamplestodemonstratetheper-formanceandcomputationalcomplexityofourhybridsoftdetector.Example1Performance).ThePERonepacketconsistsof1000bytes,whicharecontainedin19OFDMsymbols)isanimportantparameterfortheOFDM-basedWLANsys-tems.InanOFDM-basedWLANsystem,evenifonlyoneerroroccurs,theentirepacketisconsideredtobewrong.InFigure6,weshowthePERcomparisonfortheLS-basedsoftdetector,theCN-hybridsoftdetectorwithCNbeing100),thehybridsoftdetector,andtheLSD-basedsoftdetectorasafunctionofSNRfortheOFDM-basedBLASTWLANsystematthe108Mbpsdatarate.WealsogivethePERcurveofusingthesoftdetectorfortheSISOsystematthe54Mbpsdatarateasareference.WecanseefromthesimulationresultsthatfortheOFDM-basedBLASTWLANsystem,theperformanceofthehybridsoftdetectorisclosetothatoftheLSD-basedsoftdetector.WecanalsoseethatthehybridsoftdetectoroutperformsitsCN-hybridcounterpart.Moreover,thePERcurveofthehybridsoftdetectorhasnearlythesameslope CN-hybridHybrid2324252627282930SNRdB)10 2 10 1 ProbabilitiesofusingLSD Figure7:ProbabilitiesofusingLSDinthehybridandCN-hybridsoftdetectorsasSNRvaries.astheLSD-basedone,whichmeansthatathighSNRs,thehybridsoftdetectorcano ermuchbetterperformancethantheLS-basedone.Also,bycomparingthesolidlinewiththedashedline,wecanseethatifweusethehybridsoftdetectoratthereceiver,weneedabout1.5dBmoreSNRtokeepthesame10%PERwearecurrentlymostlyinterestedinPERsof10%)todoublethedataratewithM = N = 2.Notethatevenwiththeneedofthis1.5dBextraSNR,thatis,1.5dBmoretotaltransmissionpower,thePERperformanceoftheOFDM-basedBLASTWLANsystemwiththehybridsoftde-tectorisstillimpressivesinceevenifwewishtodoublethetransmissiondatarateusingtwoseparateSISOsystemsovertwodi erentphysicalchannelsbydoublingthebandwidth,westillneed3dBextraSNRortotaltransmissionpower.Ifweconsiderthecaseof1%PER,wecandoublethedataratewithabout0.5dBless totaltransmissionpower.Example2Complexity).TofacilitatetheanalysisofthecomplexityofthehybridandCN-hybrid(withCNbeing100)softdetectors,weprovideasimulationexampletodemonstratetheprobabilityofusingtheLSD-basedsoftde-tectorinthesesoftdetectors.WecanseefromFigure7that fortheSNRsofinterest,theprobabilityofusingtheLSD-basedsoftdetectorintheCN-hybridsoftdetectorisabout6%andlessthan2%inthehybridsoftdetector.

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AMIMOSoftDetectorwithGoodPerformance/Complexity1273 6.CONCLUDINGREMARKSWehaveproposedahybridsoftdetectorwithagoodperfor-mance/complexitytrade-o bycombiningtheLS-andLSD-basedsoftdetectors.ThecombinationisperformedbasedoncomparingtheoutputsofSPDandtheLS-basedhardde-tector.TospeedupthecomputationofSPD,wehavealsoprovidedatightsphereradiusthatcanbeusedtoguaranteethendingofatleastonesolution.SimulationresultshaveshownthattheperformanceofourhybridsoftdetectorisclosetothatoftheLSD-basedsoftdetectorinourWLANapplication.Thenewdetectorisabout10timesfasterthantheLSD-basedandabout10timesslowerthantheLS-basedsoftdetectors.ACKNOWLEDGMENTSWewouldliketothanktheanonymousreviewersfortheirhelpfulcommentswhichhavehelpedimprovethequalityofthispaper.ThisworkwassupportedinpartbytheNa-tionalScienceFoundationGrantCCR-0097114andtheIn-tersilCorporationContract2001056.REFERENCES [1]G.J.FoschiniandM.J.Gans,Onlimitsofwirelesscommu-nicationsinafadingenvironmentwhenusingmultiplean-tennas,WirelessPersonalCommunications,vol.6,no.3,pp.311,1998.[2]I.E.Telatar,Capacityofmulti-antennaGaussianchannels,EuropeanTransactionTelecommunications,vol.10,no.6,pp.585,1999.[3]S.LinandD.J.Costello,ErrorControlCoding,PrenticeHall,EnglewoodCli s,NJ,USA,1982.[4]J.G.Proakis,DigitalCommunications,McGraw-Hill,NewYork,NY,USA,3rdedition,1995.[5]A.M.Tonello,Space-timebit-interleavedcodedmodula-tionwithaniterativedecodingstrategy,inProc.IEEEVTS-FallVTC2000.52nd,vol.1,pp.473,Boston,Mass,USA,2000. [6]J.J.Boutros,F.Boixadera,andC.Lamy,Bit-interleavedcodedmodulationsformultiple-inputmultiple-outputchan-nels,inProc.6thIEEEInternationalSymposiumonSpreadSpectrumTechniquesandApplications,vol.1,pp.123,Parsippany,NJ,USA,September2000.[7]G.J.Foschini,yeredspace-timearchitectureforwirelesscommunicationinafadingenvironmentwhenusingmultipleantennas,BellLabsTechnicalJournal,vol.1,no.2,pp.41,1996. [8]G.D.Golden,G.J.Foschini,R.A.Valenzuela,andP.W.Wolniansky,Detectionalgorithmandinitiallaboratoryre-sultsusingV-BLASTspace-timecommunicationarchitec-ture,ElectronicsLetters,vol.35,no.1,pp.14,1999.[9]Z.LiuandG.B.Giannakis,yeredspace-timecodingforhighdataratetransmissions,inProc.IEEEMilitaryCommu-nicationsConferenceMILC01),vol.2,pp.1295,McLean,VA,USA,October2001.[10]J.LiuandJ.Li,Asimplesoft-detectorfortheBLASTsys-tem,inProc.SensorArrayandMultichannelSignalProcessingWorkshop(SAM02),pp.159,Rosslyn,Va,USA,August2002. [11]B.M.HochwaldandS.tenBrink,Achievingnear-capacityonamultiple-antennachannel,IEEETrans.Communications, vol.51,no.3,pp.389,2003.[12]O.Damen,A.Chkeif,andJ.-C.Belore,Latticecodedecoderforspace-timecodes,IEEECommunicationsLetters,vol.4,no.5,pp.161,2000.[13]IEEE802.11a-1999,IEEEStandardforinformationtechnologelecommunicationsandinformationexchangebetweensystemslocalandmetropolitanareanetworkspecicrequirementspart11:WirelessLANmediumac-cesscontrolMAC)andphysicallayer(PHY)specicationsamendment1:High-speedphysicallayerinthe5GHzband,IEEE,1999.[14]T.L.MarzettaandB.M.Hochwald,Capacityofamobilemultiple-antennacommunicationlinkinRayleighfad-ing,IEEETransactionsonInformationTheory,vol.45,no.1,pp.139,1999.[15]J.-C.Guey,M.P.Fitz,M.R.Bell,andW.-Y.Kuo,Signalde-signfortransmitterdiversitywirelesscommunicationsystemsoverRayleighfadingchannels,IEEETrans.Communications, vol.47,no.4,pp.527,1999.[16]J.Hagenauer,E.O er,andL.Papke,terativedecodingofbinaryblockandconvolutionalcodes,IEEETransactionsonInformationTheory,vol.42,no.2,pp.429,1996.[17]G.Caire,G.Taricco,andE.Biglieri,it-interleavedcodedmodulation,IEEETransactionsonInformationTheory,vol.44,no.3,pp.927,1998.[18]C.Z.W.H.Swetman,J.S.Thompson,B.Mulgrew,andP.M.Grant,AcomparisonoftheMMSEdetectoranditsBLASTversionsforMIMOchannels,inIEESeminaronMIMO:CommunicationsSystemsfromConcepttoImplementations, pp.19/1/6,London,UK,December2001.[19]A.M.ChanandI.Lee,Anewreduced-complexityspheredecoderformultipleantennasystems,inProc.IEEEInter-nationalConferenceonCommunications(ICC02),vol.1,pp.460,NewYork,NY,USA,2002.[20]M.O.Damen,K.Abed-Meraim,andM.S.Lemdani,ur-therresultsonthespheredecoder,inProc.2001IEEEInter-nationalSymposiumonInformationTheory,pp.1,Wash-ington,DC,USA,June2001.[21]J.LiuandJ.Li,AMIMOsystemwithbackwardcompatibil-ityforOFDM-basedWLANs,EURASIPJournalonAppliedSignalProcessing,vol.2004,no.5,pp.696,2004.[22]N.Chayat,Tentativecriteriaforcomparisonofmodulationmethods,Doc.IEEE802.11-97/96,September1997.[23]E.G.Larsson,G.Liu,J.Li,andG.B.Giannakis,ointsym-boltimingandchannelestimationforOFDMbasedWLANs,IEEECommunicationsLetters,vol.5,no.8,pp.325,2001.[24]J.Li,G.Liu,andG.B.Giannakis,Carrierfrequencyo set estimationforOFDM-basedWLANs,IEEESignalProcessingLetters,vol.8,no.3,pp.80,2001.[25]J.LiuandJ.Li,Channelparameterestimationanderrorre-ductionforOFDM-basedWLANs,toappearinIEEETrans-actionsonMobileComputing. JianhuaLiureceivedtheB.S.degreeinelec-tricalengineeringfromDalianMaritimeUniversity,Dalian,China,in1984,theM.S.degreeinelectricalengineeringfromtheUniversityofElectronicScienceandTech-nologyofChina,Chengdu,China,in1987,andthePh.D.degreeinelectronicengi-neeringfromTsinghuaUniversity,Beijing,China,in1998.FromMarch1987toFebru-ary1999,heworkedattheCommunica-tion,TelemetryandTelecontrolResearchInstitute,Shijiazhuang,China,wherehewasanAssistantEngineer,Engineer,SeniorEn-gineer,andFellowEngineer.FromMarch1995toAugust1998,

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1274EURASIPJournalonAppliedSignalProcessing hewasalsoaResearchAssistantatTsinghuaUniversity.FromFebruary1999toJune2000,heworkedatNanyangTechnologicalUniversity,Singapore,asaResearchFellow.SinceJune2000,hehasbeenaResearchAssistantintheDepartmentofElectricalandCom-puterEngineeringattheUniversityofFlorida,Gainesville,workingtowardsaPh.D.degreemajoringinelectricalengineeringandmi-noringinstatistics.Hisresearchinterestsincludewirelesscommu-nications,statisticalsignalprocessing,andsensorarrayprocessing. JianLireceivedtheM.S.andPh.D.degreesinelectricalengineeringfromTheOhioStateUniversity,Columbus,in1987and1991,respectively.FromApril1991toJune1991,shewasanAdjunctAssistantProfes-sorwiththeDepartmentofElectricalEngi-neering,TheOhioStateUniversity,Colum-bus.FromJuly1991toJune1993,shewasanAssistantProfessorwiththeDepartmentofElectricalEngineering,UniversityofKen-tucky,Lexington.SinceAugust1993,shehasbeenwiththeDe-partmentofElectricalandComputerEngineering,UniversityofFlorida,Gainesville,wheresheiscurrentlyaProfessor.Hercurrentresearchinterestsincludespectralestimation,arraysignalprocess-ing,andtheirapplications.Dr.LiisaMemberofSigmaXiandPhiKappaPhi.Shereceivedthe1994NationalScienceFoundationYoungInvestigatorAwardandthe1996O ceofNavalResearchYoungInvestigatorAward.ShewasanExecutiveCommitteeMem-berofthe2002InternationalConferenceonAcoustics,Speech,andSignalProcessing,Orlando,Florida,May2002.ShehasbeenanAs-sociateEditoroftheIEEETransactionsonSignalProcessingsince1999andanAssociateEditoroftheIEEESignalProcessingMag-azinesince2003.SheispresentlyaMemberoftheSignalProcess-ingTheoryandMethods(SPTM)TechnicalCommitteeoftheIEEESignalProcessingSociety.