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

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A Soft detector with good performance/complexity trade-off for a MIMO system
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EURASIP Journal on Applied Signal Processing
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Liu, Jianhua
Li, Jian
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We present a hybrid soft detector that has a good performance/complexity trade-off for amultiple-inputmultiple-output (MIMO) wireless communication systemwith known channel information. The new soft detector combines the merits of a simple unstructured least-squares (LS)-based soft detector and a list sphere decoder (LSD)-based soft detector for data bit detection. The former is computationally much more efficient than the latter at the cost of poorer performance. The poor performance of the former occurs mainly when the channel matrix is ill-conditioned. Whenever this happens, we use the LSD-based soft detector in the hybrid soft detector; otherwise, we use the LS-based one. Moreover, we provide a tight radius for a sphere decoder, a hard detector, via using the output of an LS-based hard detector. These two hard detectors are needed to determine if LS or LSD should be used in the hybrid soft detector. As an application example, we consider doubling the maximum data rate of the IEEE 802.11a conformable wireless local area networks by a MIMO system with two transmit and two receive antennas. For this application, the new soft detector is about 10 times faster than the LSD-based one and is about 10 times slower than the LS-based one. Yet the packet error rate due to using the new soft detector is quite close to that of using the LSD-based one.
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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.