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IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY2002High-RateCodesThatAreLinearinSpaceandTimeBabakHassibiandBertrandM.HochwaldMultiple-antennasystemsthatoperateathighratesrequiresimpleyeteffectivespace–timetransmissionschemestohandlethelargetrafficvolumeinrealtime.Atratesoftensofbitspersecondperhertz,VerticalBellLabsLayeredSpace–Time(V-BLAST),whereeveryantennatransmitsitsownindependentsubstreamofdata,hasbeenshowntohavegoodperformanceandsimpleencodinganddecoding.YetV-BLASTsuffersfromitsinabilitytoworkwithfewerreceiveantennasthantransmit HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME18057)satisfythefollowinginformation-theoreticoptimalitycri-—thecodesaredesignedtomaximizethemutualinfor-mationbetweenthetransmitandreceivesignals.WebrieflysummarizethegeneralstructureoftheLDcodes.Supposethatthereare transmitantennas, receiveantennas,andanintervalof symbolsavailabletousduringwhichthepropagationchannelisconstantandknowntothereceiver.Thetransmittedsignalcanthenbewrittenasa matrix thatgovernsthetransmissionoverthe duringtheinterval.Weassumethatthedatasequencehasbeenbrokeninto substreams(forthemomentwedonotspecify )andthat arethecomplexsymbolschosenfromanarbitrary,say -PSKor -QAM,constellation.Wecalla lineardispersioncodeoneforwhich obeys wheretherealscalars aredeterminedby ThedesignofLDcodesdependscruciallyonthechoicesoftheparameters , andthedispersionmatrices Tochoosethe weproposetooptimizeanonlinearinformation-theoreticcriterion:namely,themutualinformationbetweenthetransmittedsignals andthereceivedsignal.Wearguethatthiscriterionisveryimportantforachievinghighspectralefficiencywithmultipleantennas.Wealsoshowhowtheinformation-theoreticoptimizationhasimplicationsforperformancemeasuressuchaspairwiseerrorprobability.SectionIVhasseveralexamplesofLDcodesandperformancecomparisonswithexistingschemes.Wenowpresentthemultiple-antennamodelconsideredinthispaper.A.TheMultiple-AntennaModelInanarrow-band,flat-fading,multiple-antennacommunica-tionsystemwith transmitand receiveantennas,thetrans-mittedandreceivedsignalsarerelatedby (2)where denotesthevectorofcomplexreceivedsignalsduringanygivenchanneluse, denotesthevectorofcomplextransmittedsignals, denotesthechannelmatrix,andtheadditivenoise is unit-variance,complex-Gaussian)distributedthatisspatiallyandtemporallywhite.Thechannelmatrix andtransmittedvector areassumedtohaveunitvarianceentries,implyingthat and Sincetherandomquantities , ,and areindependent,the in(2)ensuresthat isthesignal-to-noiseratio(SNR)ateachreceiveantenna,independentlyof .Weoften(butnotalways)assumethatthechannelmatrix hasindependent Theentriesofthechannelmatrixareassumedtobeknowntothereceiverbutnottothetransmitter.Thisassumptionisrea-sonableiftrainingorpilotsignalsaresenttolearnthechannel,whichisthenconstantforsomecoherenceinterval.Thecoher-enceintervalofthechannelshouldbelargecomparedto Whenthechannelisknownatthereceiver,theresultingchannelcapacity(oftenreferredtoastheperfect-knowledgecapacity)is[2],[1] wheretheexpectationistakenoverthedistributionoftherandommatrix Thecapacity-achieving isazero-meancomplexGaussianvectorwithcovariancematrix ,where isthemaximizingcovariancematrixin(3).Whenthedistributionon isrotationallyinvariant,i.e.,when foranyunitary and (asisthecasewhen hasindependent entries),theoptimizingcovarianceis and(3)becomes Thisexpectationcansometimesbecomputedinclosedform(see,forexample,[22]).Whenthechannelisconstantforatleast channeluseswemaywrite sothatdefining and (wherethesuperscript denotes“transpose”),weobtain Itisgenerallymoreconvenienttowritethisequationinitstrans-posedform wherewehaveomittedthetransposenotationfrom simplyredefinedthismatrixtohavedimension .The isthereceivedsignal, isthetransmittedsignal,and istheadditive noise.In , ,and ,timerunsverticallyandspacerunshorizontally.Weareconcernedwithdesigningthesignalmatrix obeyingthepowerconstraint Equation(3)actuallyslightlygeneralizes[2],[1],whichassumethat IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY2002Wenotethat,ingeneral,thenumberof matrices neededinacodebookcanbequitelarge.Iftherateinbitsperchanneluseisdenoted ,thenthenumberofmatricesis Forexample,with transmitand receiveantennasthechannelcapacityat 20dB(with distributed )ismorethan12bitsperchanneluse.Evenwitharelativelysmallblocksizeof ,weneed matricesat .ThehugesizeoftheconstellationsgenerallyrulesoutthepossibilityofdecodingatthereceiverusingexhaustiveTheLDcodesthatwepresenteasilygeneratetheverylargeconstellationsthatareneeded.Moreover,becauseoftheirstruc-ture,theyalsoallowefficientreal-timedecoding.Inthenextsec-tion,webrieflydescribeandanalyzesomeexistingspace–timecodessothatwemaymotivatetheLDcodesandexplainhowtheyaredifferent.II.INFORMATIONNALYSISOFPACESincethecapacityofthemultiple-antennachannelcaneasilybecalculated,wemayaskhowwellaspace–timecodeperformsbycomparingthemaximummutualinformationthatitcansup-porttotheactualchannelcapacity.Thedistributionforthe matrix thatachieves(4)isindependent butwecannoteasilyusethisbyitselfasaguidelineforcon-structinganddecodinga(random)constellationof tricesbecauseofthesheernumberofmatricesinvolved.There-fore,aconstellationofmatricesthathassufficientstructureforefficientencodinganddecodingisgenerallyneeded.Onesuchstructureisthatofanorthogonaldesign,originallyproposedin[11]andlatergeneralizedin[12].A.MutualInformationAttainableWithOrthogonalDesignsAnorthogonaldesignisintroducedbyAlamoutiin[11]for andhasthestructure Thecomplexscalars and aredrawnfromaparticular -PSKor -QAM)constellation,butwemaysimplyassumethattheyarerandomvariablessuchthat Weshowthatthisparticularstructurecanbeusedtoachieveca-pacitywhenthereisonereceiveantennabutwhenthereismorethanone.Portionsofourargumentmayalsobefoundin[23],[24].1)OneReceiveAntenna( With ,(5)be- Thiscanberewrittenas Itreadilyfollowsthat Weeffectivelyhaveanequivalentmatrixchannel in(7)thatisascaledunitarymatrix.Maximum-likelihooddecodingof and is,therefore,decoupled[11].Wemayaskhowmuchmutualinformationtheorthogonaldesignstructure(6)canattain?Toanswerthisquestionweneedtocomputethemutualinformationbetweenthetransmittedandreceivedvectors and intheequivalentchannelmodel(7)andcompareitwiththecapacityofan , antennasystem. hasthepowerconstraint ,themaximummutualinformationin(7)issimplythechannelcapacitythatisobtainedwiththestructuredchannelmatrix .Ifwedenotethismaximummutualinformationby ,using(3)weobtain wherethefactor infrontoftheexpectationnormalizesforthetwochannelusesspannedbytheorthogonaldesign.Since,sub-jecttoatraceconstraint,thedeterminantofanypositive-definitematrixismaximizedwhenitseigenvaluesareallequal,itiseasytoseethatthemaximizingcovariancematrixis ,sothatweobtain Theexpressionontherightsymbolicallydenotesthecapacityattainedbyasystemwith transmitantennasand receiveantennasatSNR .Thisequationimpliesthattheor-thogonaldesign(6)canachievethefullchannelcapacityofthe , system,andthereisnolossincurredbyas-sumingthestructure(6)asopposedtoageneraltransmitma- 2)TwoorMoreReceiveAntennas( With receiveantennas,(5)becomes whichcanbereorganizedas Wenowreadilysee Aswith ,maximum-likelihoodestimationof and isdecoupled.However,unlikewith ,theorthogonaldesignstructureprohibitsusfromachievingchannelcapacity. HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME1807 Fig.1.Maximummutualinformationachievedbyorthogonaldesign(6)comparedtoactualchannelcapacityforthesystem.Solidline:maximummutualinformationfororthogonaldesign.Dashedline:capacityoftheToseethis,wecomparethemaximummutualinformationbe- and in(10)with ,theactualchannelcapacityforthesystem.Asbefore,themaximummutualinformationin(10)issimplythechannelcapacityforthestructuredchannelmatrix .De-notingthismaximummutualinformationby ,weob- Thelastequationimpliesthattheorthogonaldesign(6)isre-strictiveanddoesnotallowustoachievethefullchannelca-pacityofthe , system,butratherthecapacityof , systemattwicetheSNR.Thus,when wetakealosswiththestructure(6).TheamountofthislossissubstantialathighSNRandisdepictedinFig.1whichshowstheactualchannelcapacitycomparedtothemaximummutualinformationobtainedbytheorthogonaldesign(6).For receiveantennas,theanalysisissimilarandisomitted.Wesimplystatethatfor transmitantennas receiveantennasthe orthogonaldesignallowsustoattainonly ,ratherthanthefull 3)OtherOrthogonalDesigns:Theprecedingsubsectionfocusesonthe orthogonaldesignbuttherearealsoorthogonaldesignsfor .Thecomplexorthogonaldesignsfor arenolonger“full-rate”(see[12])andthereforegenerallyperformpoorlyinthemaximummutualinformationtheycanachieve,evenwhen .Wegiveanexampleofthesenonsquareorthogonaldesigns[12],[25].For ,wehave,forexample,therate orthogonaldesign Thefactor ensuresthat .Itcanbeshownthatmaximum-likelihoodestimationofthevariables isdecoupled.Againusinganargumentsimilarto IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY2002 Fig.2.Maximummutualinformationachievedbyorthogonaldesign(13)comparedtoactualchannelcapacity.Solidlines:maximummutualinformationorthogonaldesignforreceiveantennas.Dashedlines:capacityoftheonepresentedfor ,itisstraightforwardtoshowthatthemaximummutualinformationattainablewith(13)with receiveantennasis whichis(much)lessthanthetruechannelcapacity .WeomittheproofandreferinsteadtoFig.2whichshowstheactualchannelcapacitycomparedtothemaximummutualinformationobtainedbytheorthogonaldesign(13).B.MutualInformationAttainableWithV-BLASTWeshowedinSectionII-Athat,eventhoughorthogonalde-signsallowefficientmaximum-likelihooddecodingandyield“full-diversity”(theappearanceofthesumofthe inthemutualinformationformulasatteststothis),orthogonaldesignsgenerallycannotachievehighspectralefficienciesinamul-tiple-antennasystem,nomatterhowmuchcodingisaddedtothetransmittedsignalconstellation.Thisisespeciallytruewhenthesystemhasmorethanonereceiveantenna.Anexaminationofthemodel(10)(anditscounterpartsforotherorthogonalde-signs)revealsthattheorthogonaldesigndoesnotallowenough“degreesoffreedom”—thereareonlytwounknownsin(10)butfourequations.Wecanconcludethatorthogonaldesignsarenotsuitableforvery-high-ratecommunications.Ontheotherhand,aschemethatisproventobesuitableforhighspectralefficienciesisV-BLAST[7].InV-BLASTeachtransmitantennaduringeachchannelusesendsanindependentsignal(oftenreferredtoasasubstream).Thus,overablockof channeluses,the transmitmatrixtakesontheform ............ whereeach isanindependentsymboldrawnfromacomplexconstellation.Sincethetransmittedsymbolsarenotdispersedin isarbitrary.(Wecould,forexample,take .)When (thenumberofreceiveantennasisatleastaslargeasthenumberoftransmitantennas),thereexistefficientschemesfordecodingtheV-BLASTmatrices.Thesearebasedon“successivenullingandcanceling”[7],anditsmoreefficientvariants[18],aswellasmorerecentlyonspheredecoding[19].Allthesedecodingschemesessentiallysolveawell-conditionedsystemoflinearequations.Successivenullingandcancelingprovidesafastapproximatesolutiontothemaximum-likeli-hooddecodingproblemwiththebenefitofcubiccomplexityinthenumberoftransmitantennas .Spheredecoding,ontheotherhand,findstheexactmaximum-likelihoodestimateandbenefitsfromavoidinganexplicitexhaustivesearch.Recentwork[20]hasshownanalyticallythatforawiderangeofSNRs,theexpectedcomputationalcomplexityofspheredecodingisalsoroughlycubicinthenumberoftransmitantennas. HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME1809Becausethereisnorestrictiononthetransmittedmatrix in(14),themaximummutualinformationthatcanbeachievedbytransmittingV-BLAST-likematricesisindeedthefullmul-tiple-antennachannelcapacity.Nevertheless,V-BLASTsuffersfromtwodeficiencies.First,nullingandcancelingfailswhentherearefewerreceiveantennasthantransmitantennas,sincethedecoderisconfrontedwithanunderdeterminedsystemoflinearequations.Althoughspheredecodingcanstillbeusedtofindthemaximum-likelihoodestimate,thecomputationalcom-plexityisexponentialin .Second,becauseV-BLASTtransmitsindependentdatastreamsonitsantennasthereisnobuilt-inspatialortemporalcoding,andhencenoneoftheerrorresilienceassociatedwithsuchcoding.Weseektoremedythesedeficienciesinthenextsection.III.LPACEInthissection,weproposeahigh-ratecodingschemethatretainsthedecodingsimplicityofV-BLAST,handlesanycon-figurationoftransmitandreceiveantennas,andhasmanyofthecodingadvantagesofschemes,suchastheorthogonaldesigns,withoutsufferingthelossofmutualinformation.Wecallalinear-dispersion(LD)codeoneforwhich (15)where arecomplexscalars(typicallychosenfrom -PSKor -QAMconstellation)andwherethe and fixed complexmatrices.Thecodeiscompletelyde-terminedbythesetofdispersion ,whereaseachindividualcodewordisdeterminedbyourchoiceofthe Weoftenfinditmoreconvenienttodecomposethe theirrealandimaginaryparts andtowrite (16)where and .Thedispersion alsospecifythecode.Theinteger thedispersionmatricesare,forthemoment,unspecified.Withoutlossofgenerality,weassumethat and havevariance andareuncorrelated.Otherwise,wecanalwaysreplacethemwithappropriatelinearcombina-tionsthathavethisproperty—thissimplyleadstoaredefini-tionofthe sand ’s.Thus, areunit-varianceanduncorrelated.RecallfromourmodelinSectionI-AthattheWeremarkthatitisalsopossibletodefine =  = ...,sothattheLDcodesbecome  A wherethescalars arereal.transmitsignal isnormalizedsuchthat .Thisinducesthefollowingnormalizationonthematrices : Thedispersioncodes(16)subsumeasspecialcasesbothor-thogonaldesignsandV-BLAST.Forexample,the onaldesign(6)correspondsto and whereasV-BLASTcorrespondsto and (20)where and are -dimensionaland columnvectorswithoneinthe thand thentries,respec-tively,andzeroselsewhere.NotethatinV-BLASTeachsignal istransmittedfromonlyoneantennaduringonlyonechanneluse.WiththeLDcodes,however,thedispersionmatricespotentiallytransmitsomecombinationofeachsymbolfromeachantennaateverychanneluse.Thiscanleadtodesirablecodingproperties.Beforewespecifygoodchoicesofthedispersionmatrices,wediscussA.DecodingAnimportantpropertyoftheLDcodes(16)istheirlinearityinthevariables ,leadingtoefficientV-BLAST-likedecodingschemes.Toseethis,itisusefultowritetheblock inamoreconvenientform.Wedecomposethematricesin(21)intotheirrealandimaginarypartstoobtain where and .Equivalently, IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY2002Wedenotethecolumnsof , , , , ,and by , , , , ,and ,anddefine (22)where .Wethengathertheequationsin and toformthesinglerealsystemofequations ... .. .. wheretheequivalent realchannelmatrixisgivenby ............... Wehavealinearrelationbetweentheinputandoutputvectors and wheretheequivalentchannel isknowntothereceiverbecausetheoriginalchannel ,andthedispersionmatrices areallknowntothereceiver.Thereceiversimplyuses(24)tofindtheequivalentchannel.Thesystemofequationsbetweentransmitterandreceiverisnotunderdeterminedaslongas Wemay,therefore,useanydecodingtechniquealreadyinplaceforV-BLAST,suchassuccessivenullingandcanceling[7],itsefficientsquare-rootimplementation[18],orspheredecoding[19].Themostefficientimplementationsoftheseschemesgen-erallyrequire computations,andhaveroughlyconstantcomplexityinthesizeofthesignalconstellation B.DesignoftheDispersionCodesAlthoughwehaveintroducedtheLDstructure wehavenotyetspecified orthedispersionmatrices and .Wehavetheinequality Intuitively,thelarger is,thehigherthemaximummutualinformationbetween and issincethematrixsignal moredegreesoffreedom.(Recallthatorthogonaldesignsgen-erallyhavelowmutualinformationbecausetheydonothaveenoughdegreesoffreedom.)Ontheotherhand,thesmaller is,themoreofacodingeffectweobtainsincetheequivalent becomes“skinnier”andthesystemofequationsin(23)becomesmoreoverdetermined.Asageneralpractice,wefinditusefultotake sincethistendstomaximizethemutualinformationbetween and whilestillhavingthebenefitofcodinggain.Weareleftwiththequestionofhowtodesignthedisper-sionmatrices.Wemayfirstexaminehowsensitivetheperfor-manceoftheLDcodesistothechoiceofthedispersionma-trices.Experimentswithchoosingrandomdispersionmatricessubjecttothenormalizationconstraint(18),orthemorestrin-gentconstraints for suggestthattheperformancefor“average” isnotgen-erallyverygood.Fig.3showsthebit-errorrateofan , antennasystemwithrandomlychosenversusoptimized(accordingtoacriterionwespecifyshortly)dispersionmatrices.Thedifferenceisdramatic;itisimportanttochoosethedisper-sionmatriceswisely.OnepossiblewayofdesigningthespreadingmatricesistostudythepairwiseprobabilityoferroroftwodifferentLDcode-words,say and Theworstcasepairwiseerrorisgenerallyobtainedwhen and differinonlyoneelement.Wecanthenseektochoosethedispersionmatricesthatminimizestheprobabilityofthiserror.Themaindrawbackofthisstrategyisthatitleadstoacriterionontheindividualcolumnsofthematrix ,ratherthanonthematrixinitsentirety.Therefore,itisconceivablethatdesignsbasedonthiscriterioncouldleadtoa(near)singular ,leadingtootherformsoferrors.Finally,itisnotclearwhateffectminimizingpairwiseerrorprobabilityhasontheoverallerrorprobability,especiallyforahigh-ratesystem.Therefore,thisstrategyforchoosingthedispersionmatricesdoesnotappeartobepromising.Wecanalsostudytheaveragepairwiseerrorprobability,ob-tainedbychoosingGaussian in(25)andaveragingthepair-wiseerrorobtainedbetweenanindependent and .WeshowinAppendixBthattheaveragepairwiseerrorhasupperbound pairwise Wecanthenseektominimizetheupperboundwithanap-propriatechoiceof and .Eventhough(27)isasimpleformula,suggestingthatitcanpossiblybeminimized,wedonotattempttodosohere.Themainreasonisthefollowing.SincemultipleantennasareusedforveryhighAthighSNR,thecapacityofthemultiple-antennasystemgrowsasmin(M;N)log,suggestingthatweneed=min(M;Ndegreesoffreedomperchanneluse. HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME1811 Fig.3.Bit-errorperformancecomparisonforarandom( drawnfromacomplexGaussiandistributionandnormalized)andanoptimizedLDcodefortransmitandreceiveantennafor,andratebits/channeluse(obtainedbytransmitting64-QAMon ; rates,thepairwiseerrorprobabilityforanytwosignalsisex-tremelysmall.InSectionI-Awearguethatevenforthesmalltest-caseof transmitand receiveantennas,wecouldtheoreticallyhaveaconstellationsizeofasmanyas signal-matricesat 20dB.Itisthereforeconceivablethatthepairwiseerrorprobabilitybetweenanytwocouldberoughly .Tryingtominimizeaquantitysuchas(27)thatisalreadysosmallcanbenumericallyquitedifficult.Fortunately,informationtheorysuggestsanaturalalternativethatisconnectedwithminimizing(27)butismorefundamental.RecallfromSectionII-Athatorthogonaldesignsaredeficientinthemaximummutualinformationtheysupportfor or .Wethereforechoose tomaximizethemu-tualinformationbetween and in(23).ThisguaranteesthatwearetakingthesmallestpossiblemutualinformationpenaltywithintheLDstructure(16).Weproposetodesigncodesthatare“blessed”bythe“logdet”formula(3).Weformalizethedesigncriterionasfollows.TheDesignMethod1)Choose (typically, 2)Choose thatsolvetheoptimizationproblem foranSNR ofinterest,subjecttooneofthefollowing ii) , iii) , where isgivenby(24)withthe havingindependent Notethat(28)iseffectively(3)with ;asmentionedinSectionIII,wemaytaketheentriesof (the ’sand ’s)tobeuncorrelatedwithvariance .Moreover,becausetherealandimaginarypartsofthenoisevector in(23)alsohavevariance ,theSNRremains .Wealsonotethat(28)differsfrom(3)bytheoutsidefactor becausetheeffectivechannelisreal-valuedandtheLDcodespans channeluses.Wenowmakesomeremarks.1)Clearly, 2)Theproblem(28)canbesolvedsubjecttoanyofthecon-straintsi)–iii).Constrainti)issimplythepowerconstraint(18)thatensures .Constraintii)ismorerestrictiveandensuresthateachofthetransmittedsignals and aretransmittedwiththesameoverallpowerfromthe antennasduringthe channeluses.Finally,constraintiii)isthemoststringent,sinceitforcesthesym- and tobedispersedwithequalenergyinallspatialandtemporaldirections. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY20023)Sinceconstraintsi)–iii)aresuccessivelymorerestric-tive,thecorrespondingmaximummutualinformationsobtainedin(28)arenecessarilysuccessivelysmaller.Nevertheless,wehavefoundthatconstraintiii)generallyimposesonlyasmallinformation-theoreticpenaltywhilehavingtheadvantageofbettercoding(ordiversity)gains.Usingsymmetryargumentsonemayconjecturethattheoptimalsolutiontotheproblemwithconstrainti)shouldautomaticallysatisfyconstraintii).Butwehavenotexperimentedsufficientlywithconstrainti)toconfirmthis;weinsteadusuallyrestrictourattentiontoconstraintsii)andiii).Wehaveempiricallyfoundthatoftwocodeswithequalmutualinformations,theonesatisfyingthemorestringentconstraintgiveslowererrorrates.ExamplesofthisphenomenonappearinSectionIV.4)Thesolutionto(28)subjecttoanyoftheconstraintsi)–iii)ishighlynonunique:simplyreorderingthe givesanothersolution,asdoespre-orpost-multiplyingallthematricesbythesameunitarymatrix.However,thereisalsoanothersourceofnonuniquenesswhichismoresubtle.Equation(23)showsthatwecanalwayspre-mul-tiplythetransmitvector bya orthogonalmatrix toobtainanewvector withen-triesthatarestillindependentand -distributed.Thus,wemayrewrite(23)as Defining , ,and asin(22)allowsustowritethenewequivalentchannel asshownin(29)atthebottomofthepage.Sincetheentriesof and havethesamejointdistribution,themaximummutualinformationobtainedfromtheequivalentchannels and arethesame.Thisimpliesthatthetransformationfromthedispersionmatrices to (30)where isanorthogonalmatrix,pre-servesthemutualinformation.Thus,thetransformation(30)isanothersourceofnonuniquenesstothesolutionofThisnonuniquenesscanbeusedtoouradvantagebecauseajudiciouschoiceoftheorthogonalmatrix allowsustochangethedispersioncodethroughthetransformation(30)tosatisfyothercriteria(suchasspace–timediversity)withoutsacrificingmutualinfor-mation.ExamplesofthisappearinRemark7,whereweconstructunitary fromtherank-oneV-BLASTdispersionmatrices(20),andinSectionIVinsomeofthetwoandthree-antennaLDcodeconstructions.5)Theconstraintsi)–iii)areconvexinthedispersionma- sincetheycanberewrittenas ii¢) , iii¢) , allofwhichareconvex.However,thecostfunction isneitherconcavenorconvexinthevariables .Therefore,itispossiblethat(28)haslocalmaxima.Nevertheless,wehavebeenabletosolve(28)withrelativeeaseusinggradient-basedmethodsanditdoesnotappearthatlocalmaximaposeagreatproblem.TableIinSectionIV-Agathersthemaximummutualinformationsobtainedviagradient-ascentforavarietyofdifferent , ,and TheresultsshowthatmaximummutualinformationsobtainedarequiteclosetotheShannoncapacity(whichisclearlyanupperboundonwhatcanbeachieved)andsotheysuggestthatthevaluesobtained,ifnottheglobalmaxima,arequiteclosetothem.(Forconvenience,weincludethegradientofthecostfunction(28)inAppendixA.) ......... ......... ...... (29) HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME18136)Weknowthatfor , , ,onesolutionto(28),foranyoftheconstraintsi)–iii),istheorthogonaldesign(19).Thisholdssimplybecausethemutualinfor-mationofthisparticularorthogonaldesignachievestheactualchannelcapacity .Wenotethattherearealsomanyothersolutionsthatworkequally7)When and onesolutionto(28),subjecttoeitherconstraintsi)orii),isgivenbytheV-BLASTmatrices(20)sincetheseachievethefullcapacityofthemultipleantennalink.TheV-BLASTmatrices,however,arerank-oneandthereforedonotsatisfyconstraintiii).Butitisalsopossibletoobtainanexplicitsolutionto(28)subjecttoiii).For ,onesuchsetofmatricesisgivenby (31)where ...... ......... TheabovecodecanbeconstructedbystartingwiththeV-BLASTmatrices(20)andapplyingthetransformation(30)withasuitable .Wedonotgivethefull here,andonlymentionthat,for ,thetransformationis withsimilarexpressionsforthe .Itcanbereadilycheckedthatthematrix constructedfromthecoef-ficientsrelating to isorthogonal.Fig.4inSectionIVpresentsaperformancecomparisonoftheLDcode(31)withV-BLAST.8)Theblocklength isessentiallyalsoadesignvariable.Althoughitmustbechosenshorterthanthecoherencetimeofthechannel,itcanbevariedtohelptheoptimiza-tion(28).Wehavefoundthatchoosing oftenyieldsgoodperformance.9)AlthoughtheSNR isadesignvariable,wehavefoundthattheoptimization(28)isnotsensitivetoitsvalueforlarge ( 20dB).Oncetheoptimizationisperformed,theresultingLDcodegenerallyworkswelloverawiderangeofSNRs.10)Itdoesnotappearthat(28)hasasimpleclosed-formsolutionforgeneral , , ,althoughweseeinSec-tionIVthat,insomenontrivialcases,itcanleadtoso-lutionswithsimplestructure.Wehavefoundthattheso-lutionto(28)oftenyieldsanequivalentchannelmatrix thatis“asorthogonalaspossible.”Althoughcompleteorthogonalityappearsnotalwaystobepossible,ourex-periencewithoptimizing(28)showsthatthedifference canbemadequitesmallwithaproperchoiceof and (seeTableIinSec-tionIV-A).Thus,thereappearstobeverylittlecapacitypenaltyinassumingtheLDstructure(16).11)Whentheequivalentchannelmatrix isorthogonal,maximum-likelihooddecodingandtheV-BLAST-likenulling/canceling[7]performequallywellbecausetheestimationerrorsof aredecoupled.12)Thedesigncriterion(28)dependsexplicitlyonthenumberofreceiveantennas ,boththroughthechoice andthroughthematrix in(24).Hence,theoptimalcodes,foragiven , ,and ,aredifferentfordifferent Nevertheless,acodedesignedfor receiveantennascanalsoeasilybedecodedusingnulling/cancelingorspheredecodingwith antennas.Hence,ifwewishtobroadcastdatatomorethanoneuser,wemayuseacodedesignedfortheuserwiththefewestreceiveantennas,witharatesupportedbyalltheusers.13)Theultimaterateofthecodedependsonthenumberofsignalssent ,theblocklengthofthecode ,andthesizeoftheconstellationfromwhich arechosen.Weassumethattheconstellationis -PSKor Thentherateinbitsperchanneluseiseasilyseentobe 14)Astandardgray-codeassignmentofbitstothesymbolsofthe -PSKor -QAMconstellationmaybeused.15)Weseethattheaveragepairwiseerrorprobability(27)andthedesigncriterion(28)haveasimilarexpression.Byinterchangingtheexpectationandlogin(28),weseethatmaximizing(28)hassomeconnectionstominimizingOntheotherhand,ourdesigncriterionisnotdirectlyconnectedwiththediversitydesigncriteriongivenin[9]and[10],whichisconcernedwithmaximizing Aconstellationattainsfulldiversityif(33)isnonzero.Thiscriteriondependsonlyonmatrixpairs,andthere-foredoesnotexcludematrixdesignswithlowspectralefficiencies.Athighspectralefficiencies,thenumberofsignalsintheconstellationofpossible matricesisroughlyex-ponentialinthechannelcapacityatagivenSNR.Thisnumbercanbeverylarge—inSectionIVwepresenta IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY2002codefor and thateffectivelyhas matrices.Therelationbetweenthediversitycriterionandtheperformanceofsuchalargeconstellationisverytenuous.Evenif formanypairsof ,ourprobabilityofencounteringoneofthesematricesmaystillbeexceedinglysmall,andtheconstellationperformancemaystillbeexcellent.Forsuchalargeconstellationitisprobablymoreimportantforthematricesinthisconstellationtobedistributedinthespaceofmatricesaccordingtothedistributionthatat-tainscapacity;themutualinformationcriterionattemptstoachievethisdistribution.AstheSNRisallowedtoincrease,theperformanceofsomegivenspace–timecodewithsomegivenratebe-comesmoredependentonthediversitycriterionsincemakingadecodingerrortoa“nearestneighbor”becomesrelativelymoreimportant.Chernoffboundcomputationsin[10]showthatthepairwiseerrorfallsoffas ,where istherankof .However,byincreasingtheSNRandkeepingthecode(andhencerate)fixed,weareeffectivelyreducingtherelativespectralefficiencyofthecodeascomparedwiththechannelcapacity.Wearethere-foreledagaintotheconclusionthatdiversityplaysasec-ondaryroleathighspectralefficiencies.InSectionIV,wepresentacomparisonofcodesthatsatisfyvariouscombi-nationsofthemutualinformationanddiversitycriteria.Thecodethatsatisfiesbothcriteriaperformsbest,fol-lowedbythemutualinformationcriteriononly,followedbythediversitycriteriononly.16)Althoughthedispersionmatrices can,ingen-eral,becomplex,wehavefoundthatconstrainingthemtoberealimposeslittle,ifany,penaltyintheoptimizedmutualinformation.17)Ourmutualinformationcalculationsanddesignexam-plesassumethatthechannelmatrix hasindependent entries,butdesignsforotherchanneldistribu-tionsusingthemutualinformationcriterionarealsopos-IV.EXAMPLESOFLDCODESANDInthissection,wepresentsimulationsthatcomparetheper-formanceofLDcodestoV-BLASTandorthogonaldesignsoverawiderangeofSNRsandvariouscombinationsofreceiveandtransmitantennas.AlltheLDcodesaredesignedforatargetSNRof 20dB(seeRemark9inSectionIII-B).LDVersusV-BLAST: , , Welookfirstatan , systematrate compareV-BLASTwithanLDcode.InV-BLAST, andthematricesaregivenby(20).TodesignanLDcodewealsochoose butusethematricesgivenby(31)thatsatisfyconstraintiii)in(28).Toachieve ,wetransmitquaternaryphase-shiftkeying(QPSK)on .There-sultscanbeseeninFig.4,wherethebiterrorsarecompared.EventhoughbothV-BLASTandtheLDcodesupportthefullchannelcapacity,whichis11.28bits/channeluseatSNR dB,theLDcodehasbetterperformance;thiscanprobablybeattributedtothespatialandtemporaldispersionofthesymbolsthatV-BLASTlacks.Sincewearetransmittingatarate ourspectralefficiencyislowrelativetothechannelcapacity,andwemaythereforeanticipatesignificantcodingadvantagesfromalsosatisfyingthediversitycriterion(33)—seeRemark15inSectionIII-Bforanexplanationoftherelativeimportanceofdiversityatlowspectralefficiencies.TheLDcode(31)maybemodifiedasinRemark4inSectionIII-B,withoutchangingitsmutualinformation,bypremultiplyingthetransmittedsignalvectorbyanorthogonalmatrix .In[26],atwo-antennacodeisdesignedusingthefulldiversitycriterion.Thiscodealsohappenstosupportthefullcapacityofthechannel,andwemayputitintoourLDcodeframeworkbychoosing tobetheblock-diagonalmatrixshownin(34)atthebottomofthepage(wherethesubscript“ ”denotesrealpart,and“ ”denotesimaginarypart)andwhere and .Theresultisacodethatsatisfiesboththemutualinformationcriterionanddiversitycriterion;itisalsodisplayedinFig.4andhasthebestperformance.Althoughthecodesinthefigureallsatisfythemutualinformationcondition,theimportanceofalsosatisfyingthediversitycriterionatrelativelylowspectralefficienciesisunderscored.Thenextexampleshowsthatsatisfyingthemutualinformationconditionismostimportantathigherspectralefficiencies.LDVersusOD: , WeshowinSectionII-A2thatthe orthogonaldesignisdeficientinmutualinformationwhen .Thisdeficiencyshouldbereflectedinitsperformancewith .Wetestitsperformancewhen at versustheLDcodegivenby(31)for and .TheresultcanbeseeninFig.5whichclearlyshowsthebetterperformanceoftheLDcodeoverawiderangeofSNRs.Toachieve ,weseefrom(32)thattheorthogonaldesignneedstochoose and a256-QAMconstellation,whiletheLDcodecanchoosefroma16-QAMconstellationbecauseithasfoursymbols Wenotethattheorthogonaldesignhasgooddiversity(33)[12] (34) HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME1815 Fig.4.TheuppertwocurvesarethebiterrorperformanceofV-BLAST(20)withnulling/canceling(upper),andwithmaximum-likelihooddecoding(lowThelowertwocurvesaretheLDcodegivenby(31)for(upper)andthecode(34)with (lower).Forboththesecodes,spheredecodingisusedtofindthemaximum-likelihoodestimates.Therateis,andisobtainedbytransmittingQPSKon ; butachievesonly7.47-bits/channelusemutualinformationat 20dB,whiletheLDcodeachievesthefullchannelcapacityof11.28bits/channeluse.TheorthogonaldesignandLDcodearemaximum-likelihooddecoded(usingthespheredecoderinthecaseoftheLDcode).Theorthogonaldesigniseasiertode-codethantheLDcodebecause and maybedecodedsep-arately,anditsperformanceisbetterforSNR 35dB(wherespectralefficiencyislowcomparedwithcapacity).ButwemayobtainacodethatisuniformlybetteratallSNRsbyusing(34)toimprovethediversityof(31)withoutchangingitsmutualinformation.Asshownin[26],setting isagoodchoicewhentransmitting16-QAM.TheperformanceofthisconstellationisalsoshowninFig.5.ItsperformanceisbetterthantheunmodifiedLDcodeathighSNR.Clearly,thebestcodesatisfiesboththemutualinformationanddiversitycriteria,ifpossible.LDVersusOD: , , Wepresentacodefor transmitantennasand receiveantennasandcompareitwiththeorthogonaldesignpre-sentedinSectionII-A3withblocklength .Theorthog-onaldesign(13)iswrittenintermsof and as Itturnsoutthatthisorthogonaldesignisalocalmaximumto(28)for and .Itachievesamutualinformationof5.13bits/channeluseat 20dB,whereasthechannelcapacityis6.41bits/channeluse.TofindanLDcodewiththesameblocklength,wefirstob-servethat mustobeytheconstraint ,with and .Therefore, ,andtoobtainthehighestpossiblemutualinformationwechoose .Afteroptimizing(28)usingagradient-basedsearch(AppendixA)andconvergingtoalocalmaximumat 20dB,wefind(36)asshownatthebottomofthefollowingpage.Thiscodehasamutualinforma-tionof6.25bits/channeluseat 20dB,whichismostofthechannelcapacity.Thematrix hassomeinterestingfeatures.First,ithasorthogonal(butnotorthonormal)columns;second,itscorresponding matrixisnonzeroinonly12ofits56off-diagonalentries.Fig.6comparestheperformanceoftheorthogonaldesign(35)withtheLDcode(36)atrate .(From(32),therateofeithercodeis ;weachieve havingtheorthogonaldesignsend256-QAM,andtheLDcodesend64-QAM.)Thedecodinginbothcasesismaximumlike-lihood,whichinthecaseoftheLDcodeisaccomplishedwiththespheredecoder,andinthecaseoftheorthogonaldesignissimplebecause aredecoupled.Wealsocomparede-codingwithnulling/canceling,whichappearstobeonlyslightlyworsethanmaximumlikelihood(thisisperhapsbecausethecolumnsoftheLDcodeareorthogonal—seeRemark11in IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY2002 Fig.5.Biterrorperformanceoftheorthogonaldesign(19)andtheLDcodegivenby(31)for(dashedline)andthecode(34)with (lowersolidline).Therateisimplyingthattheorthogonaldesigntransmits256-QAMwhereastheLDcodestransmit16-QAM.Thedecodingismaximumlikelihoodinbothcases.SectionIII-B).WeseefromFig.6thattheLDcodeperformsuniformlybetter. , LDCodeFrom OrthogonalDessign , , LDcode(36)isobtainedviaagradientsearchandhasmutualinformation6.25bits/channeluseat 20dB.However,thisislessthanthefull , capacityof ,andwewouldliketoclosethegapalittle.WeshouldbeabletomakeanLDcodewithmutualinformationatleastaslargeasthemutualinformationofthetwo-antennaorthogonaldesign(6),whichis .Wedonotresumeourgradientsearchsincethevalue appearstobealocalmaximum,butrathertryaslightlydifferentapproach.Webeginwiththetwo-antennaorthogonaldesignandcreateathree-antennaLDcodethatpreservesitsmutualinformation.Onepossiblecodeisobtainedbysymmetricallyconcate-natingthree orthogonaldesigns(normalizedtoobeythepowerconstraint) WhenviewedasanLDcode,(37)hasthedeficiencythat and areonlynonzerofortwo-channelusesandnotforthefullsix-channeluses.Moreover, and haveranktwo,rather (36) HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME1817 Fig.6.Block(dashed)andbit(solid)errorperformanceoforthogonaldesign(35)andtheLDcode(36)forantennas.Therateis6bits/channeluse,obtainedintheorthogonaldesignbytransmitting256-QAMon ... andobtainedintheLDcodebytransmitting64-QAMon ... .Theuppermostblockandbitcurvesaretheorthogonaldesign,decodedwithmaximumlikelihood.Thelowertwoblockandbitcurves(veryclosetooneanother)aretheLDcodedecodedwithnulling/canceling(upper)andmaximumlikelihood(lower).Thecomparisonofblockerrorismeaningfulherebecausetheblocksizeinallcasesisthantheirfullpossiblerankofthree.Consequently,constraintiii)inSectionIII-Bisnotsatisfied,andaswepointoutinRe-mark3,oftwocodesthathavethesamemutualinformation,theonesatisfyingthestrongerconstraintgenerallyperformsbetter.Itisclearthatthecode(37)isreallyonlyatwo-antennacodeincrudedisguiseandperformsworsethan(36),eventhoughitsmutualinformationisslightlyhigher.Toimproveitsperformance,weseektomodifyitsothatcon-straintiii)issatisfiedwithoutchangingitsmutualinformation.OnepossiblemodificationisdescribedinRemark4inSec-tionIII-B.(See,inparticular,thetransformationinvolving (30).)Let denotethe discreteFouriertransform(DFT)matrix,andchoose tobethe realorthogonalmatrixob-tainedbyreplacingeachelement of bythe realmatrix .Thetransformationof tonewdispersionmatrices is Theresulting matrixisshownin(39)atthebottomofthefollowingpage.Eachdispersionmatrixspansallsixchannelusesandisunitary .Thus,constraintiii)issatisfied.Becausethetransformation(38)isaspecialcaseofthetransformation(30),themutualinformationisstill6.28bits/channeluse( 20dB).WeseeinFig.7thatthiscodeperformsverywell:displayedis(37)(whichhasthesameperformanceasthe onaldesign)andtheLDcodes(36)and(39)forrate (ThesymbolconstellationishenceQPSK.)Thecode(37)hastheworstperformance.TheLDcode(36)with hasbetterperformance,despiteitslowermutualinformation,becauseitsatisfiesconstraintiii).Thebestperformer,however,is(39),becauseitsmutualinformationishigherthan(36)( versus ),itsatisfiesconstraintiii),andperhapsalsobecauseithasalongerblocklength( versus TwoLDCodes: , , Fig.8demonstratesthedramaticimprovementofincreasingthenumberoftransmitterantennasfrom to with .AnLDcodewasdesignedfor , ,and thatattains11.84bits/channeluseatSNR 20dB,whereasthechannelcapacityis12.49bits/channeluse.Wedonotexplicitlypresentthecodebecause andtherearetherefore24 and matrices.(Thereadermayobtainthecodebycontactingtheauthors.)WecomparethiscodewiththebestLDcodewehavefor transmitantennas((31)modifiedwith(34)where ). IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY2002 Fig.7.Biterrorperformancefor,and.Thetopcurveis(37),whoseperformanceisidenticaltoaorthogonaldesign.ThemiddlecurveistheLDcode(36),andthelowercurveistheLDcode(39).Inallcases,thetransmittedsymbolsareQPSK,decodedviamaximumlikelihood.LDCode: , , ThelastexampleisanLDcodefor and atrate 16bits/channelusedisplayedinFig.9.Itisworthnotingthatthecapacityofan , systemat20dBis24.94-bits/channeluse.WethereforerestrictourattentioninthisfiguretorelativelyhighSNR.TheLDcodewasdesignedusinggradientsearchappliedto(28)untilalocalmaximumwasobtainedat 20dB.Thecodeattainsamutualinformationof23.10-bits/channeluse,with andhas .To ,wechoose froma16-QAMcon-stellation.Becauseofthesheernumberofmatricesinvolved,weagaindonotexplicitlypresenttheLDcodehere.Wein-cludethisexampletodemonstratethatveryhighratesarewellwithinthereachofthesecodes,evenwithmaximum-likelihooddecoding.Thefigurecomparestheperformanceofnulling/can-celingversusmaximum-likelihooddecodingwiththespherede-coder,andmaximumlikelihoodperformsfarbetter.Itisremark-ablethatthespheredecodersucceedsatallinobtainingthemax-imum-likelihoodestimate,sinceafullexhaustivesearchwouldneedtoexamine hypotheses.A.TableofMutualInformationsTableIsummarizesthemutualinformationsofsomeLDcodesthatwegenerated,includingtheexamplesfromthe (39) HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME1819 Fig.8.BiterrorperformanceofthebestLDcodefor((31)modifiedwith(34))andanLDcodefor,andTherateisandisobtainedbytransmitting16-QAMoneachsymbol.Thedecodinginbothcasesismaximumlikelihood.TheLDcodeforachieves11.84bits/channelusemutualinformationat20dBversusthechannelcapacityof,andbenefitsdramaticallyfromthetwoextratransmitantennas.TABLEIUTUALNFORMATION ;T;M;NBTAINEDVIAPTIMIZATIONOFTHE(28),COMPAREDTOTHECTUALAPACITY;M;NALUESOF20dB previoussection,andtheactualchannelcapacitiesat 20dB.Ascanbeobserved, isverycloseto ;thereislittlepenaltyinthelinearstructureofthedispersioncodes.Whenstudyingthistable,weshouldbearinmindthattheentriesfor arenotnecessarilythebestachievablesince(28)wasmaximizedviagradientascent.Ourmaximaarethereforequitepossiblylocal.Further-more,thevaluesof areforcodeswithblocklengthsobeying .Conceivably,increasing couldalsoyieldhighervaluesfor V.CThelineardispersioncodeswehaveintroducedaresimpletoencodeanddecode,applytoanycombinationoftransmitandreceiveantennas,subsumeasspecialcasesmanyearlierspace–timetransmissionschemes,andsatisfyaninformation-theoreticoptimalityproperty.Wehavearguedthatcodesthataredeficientinmutualinformationcanneverbeusedtoattaincapacity.Wealsohaveshownthatinformation-theoreticopti-malityhasatheoreticalconnectionwithlowpairwiseerrorprob-abilityandgoodperformanceathighspectralefficiencies.TheLDcodesaredesignedtobelinearwhilehavinglittle(ifany)penaltyinmutualinformation,andadditionalchannelcoding canbecombinedwithanLDcodetoattainmost(ifnotall)ofthechannelcapacity.WehavegivensomespecificexamplesoftheLDcodes,andpresentedarecipeforgeneratingmorecodeswithinthislinearstructureforanycombinationoftransmitandreceiveantennas.Oursimulationsindicatethatcodesgeneratedwiththisrecipecomparefavorablywithexistingspace–timeschemesintheirgoodperformanceandlowcomplexity.Wehavearguedthatthediversitycriterioncommonlyusedtodesignspace–timecodesplaysasecondaryroletomutualinformationcriterionathighspectralefficiencies.Thediversitycriterionalonemayleadtocodedesignsthatcannotattaincapacity.Inoursimulations,wedecodedtheLDcodesbyeithermax-imumlikelihoodorbynullingandcanceling.Becauseofthelinearrelationbetween and in(25),themaximum-likeli-hoodsearchcouldbeaccomplishedefficientlyusingthespheredecodingalgorithm,which,forSNRsofinterest,haspolyno-mialcomplexityinthenumberofantennas. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY2002 Fig.9.BiterrorperformanceofanLDcodefor=32withnulling/canceling(uppercurve)andmaximum-likelihooddecoding(lowercurve).Therateis=16andisobtainedbytransmitting16-QAMoneachsymbol.TheLDcodeachievesmutualinformation23.10-bits/channeluseat20dBversusthechannelcapacityofWehaveusedtheaveragecapacity(acrossdifferentchannelrealizations)asourdesignmetric,ratherthananoutagecapacity(forone-channelrealization),becausechannelsarerarelystaticforverylongandbecausetheoutagecapacitydoesnothaveasimpleclosed-formexpression.Itisreasonabletoexpectthatcodesthatworkwellonaverageshouldalsohelpreducetheprobabilityofoutageevents,butthisremainstobeexplored.ItwouldbeinterestingtoseeiftheLDcodesthatsatisfy(28)possessanygeneralalgebraicstructure.Thiswouldleadtobettertheoreticalunderstandingofthecodes,aswellastopossiblyfasterandbetterdecodingalgorithms.Thecodes(36)and(39),forexample,arelocalmaximaofthecostfunctionandyethavesimplestructure.Itwouldalsobeimportanttocharac-terizetheoreticallyhowclose canbemade —whatisthepenaltyincurredbytheLDstruc-ture?Ourexperiencesuggeststhatthepenaltyissmall.Perhapsthepenaltyapproacheszeroas Finally,therearepotentiallymanywaystooptimizethecostfunction(28),andthegradientmethodwechoseisonlyoneofthem.MoresophisticatedoptimizationtechniquesmayalsobeOMPUTATIONInallthesimulationspresentedinthispaper,themaxi-mizationofthecostfunctionin(28),neededtodesigntheLDcodes,isperformedusingasimpleconstrained-gradient-ascentmethod.Inthisappendix,wecomputethegradientofthecostfunctionin(28)thatthismethodrequires.Moresophisticatedoptimizationtechniquesthatwedonotconsider,suchasNewton–Raphson,scoring,andinterior-pointmethods,canalsousethisgradient.Tohelpcomputethisgradient,werewritethecostfunctionin(28)asshownin(A1)atthebottomofthepage,where , ,and aredefinedin(22)for and Wewishtocomputethegradientofthecostfunctionin(A1)withrespecttothespreadingmatrices , , ,and .Tosimplifythegradientcalculation,weassumethelog-arithmin(A1)isbase .Thegradientwithrespectto computedhereexplicitly—theremainingthreegradientsfollowsimilarargumentsandaregivenattheendofthesection. ....... . . ...... (A1) HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME1821 thentryofthegradientofafunction is (A2)where and arethe -dimensionaland unitcolumnvectorswithoneinthe thand thentries,respectively,andzeroselsewhere.Definingthematrixappearinginthe of(A1)as ,sothat ;wethenobtain ......... .. ... ......... transpose higherorderterms.(A3) yields(A4),shownatthebottomofthefollowingpage,whereinthelaststepweuse .Thisnowleadsto ......... .. ... ......... ......... .. ... ......... ........ wherewehavedefinedthe matrix as .. ...... Thisconcludesthegradientcalculationwithrespectto Similarexpressionscanbefoundforthegradientswithre-spectto , ,and .With ... ......... thesegradientsare (A8) (A9) (A10) VERAGEAIRWISEROBABILITYOFRRORThesignalmodel(25)is where isagivenreal matrix.Wecomputethepairwiseerrorbyconditioningon pairwise pairwise IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.48,NO.7,JULY2002Weassumethattwoindependent signalvectors and arechosenwithentriesfromindependentzero-meanrealGaussiandensitieswithvariance .Theentriesoftheadditive noisevector arealsochosenfromthisdensity.Wewanttheprobabilitythatamaximum-likelihooddecodermistakes for ,giventhat istransmitted,averagedover ,andconditionedon .Because hasaGaussiandistribution,weequivalentlywant pairwise transmitted (B12) Equation(B12)followsbecause and areindependentGaussianvectorsandwehavereplacedthedifferenceofthetwovectorsbyasinglevectorwithtwicethevariance.Tocompute(B13),welookatthecharacteristicfunctionofthescalar Wecanwrite Thecharacteristicfunctionof is Weusetheformula forpositivereal ,where isareal vector,toconcludethat Thisimpliesthat pairwise (B14) ......... ... ... ......... transpose higherorderterms ........ ... ... ......... transpose higherorderterms(A4) HASSIBIANDHOCHWALD:HIGH-RATECODESTHATARELINEARINSPACEANDTIME1823Wewishtoswitchtheorderofintegration,andwehavetomakesurethat iswell-behavedasafunctionof and .Wecanensureitsgoodbehaviorbyshiftingthepathofintegrationto .(Theformalargumentforwhythisdoesnotaffectthevalueoftheintegralisomitted.)Weobtain pairwise Wenowcompute byfirstcomputingtheeigen-valuesof .Theeigenvaluesof arethesolutionsto asapolynomialin .Usingastandarddeterminantidentity[27],wecanwritethisequationas where aretheeigenvaluesof .Solvingthislastequationyields zeroeigenvalues,withthe eigenvaluesgivenby Therefore, and(B15)becomes pairwise Togetanupperboundonthepairwiseerrorprobability,weignorethesecondappearanceof in(B16)toobtain pairwise Itfollowsthat pairwise Applyingaunionboundtothisaveragepairwiseprobabilityoferroryieldsanupperboundonprobabilityoferrorofasignalconstellation.Supposethatthetransmissionrateis ,sothatthereare elementsinourconstellationfor ;then [1]G.J.Foschini,“Layeredspace–timearchitectureforwirelesscommu-nicationinafadingenvironmentwhenusingmulti-elementantennas,”BellLabs.Tech.J.,vol.1,no.2,pp.41–59,1996.[2]I.E.Telatar,“Capacityofmulti-antennaGaussianchannels,”Europ.Trans.Telecommun.,vol.10,pp.585–595,Nov.1999.[3]A.Wittneben,“Basestationmodulationdiversityfordigitalsimulcast,”Proc.IEEEVehicularTechnologyConf.,1991,pp.848–853.[4]N.SeshadriandJ.Winters,“Twosignalingschemesforimprovingtheerrorperformanceoffrequency-division-duplex(fdd)transmissionsys-temsusingtransmitterantennadiversity,”inProc.IEEEVehicularTech-nologyConf.,1993,pp.508–511.[5]A.Wittneben,“Anewbandwidth-efficienttransmitantennamodulationdiversityschemeforlineardigitalmodulation,”inProc.IEEEInt.Com-municationsConf.,1993,pp.1630–1634.[6]J.Winters,“ThediversitygainoftransmitdiversityinwirelesssystemswithRayleighfading,”inProc.IEEEInt.CommunicationsConf.,1994,pp.1121–1125.[7]G.D.Golden,G.J.Foschini,R.A.Valenzuela,andP.W.Wolniansky,“DetectionalgorithmandinitiallaboratoryresultsusingV-BLASTspace–timecommunicationarchitecture,”Electron.Lett.,vol.35,pp.14–16,Jan.1999.[8]G.J.Foschini,G.D.Golden,R.A.Valenzuela,andP.W.Wolniansky,“Simplifiedprocessingforhighspectralefficiencywirelesscommuni-cationemployingmulti-elementarrays,”J.Select.AreasCommun.,vol.17,pp.1841–1852,Nov.1999.[9]J.-C.Guey,M.P.Fitz,M.R.Bell,andW.-Y.Kuo,“SignaldesignfortransmitterdiversitywirelesscommunicationsystemsoverRayleighfadingchannels,”inProc.IEEEVehicularTechnologyConf.,1996,pp.[10]V.Tarokh,N.Seshadri,andA.R.Calderbank,“Space–timecodesforhighdataratewirelesscommunication:PerformancecriterionandcodeIEEETrans.Inform.Theory,vol.44,pp.744–765,Mar. 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