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IEEE TRANSACTIONS ON INFORMATION THEORY VOL IEEE TRANSACTIONS ON INFORMATION THEORY VOL

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IEEE TRANSACTIONS ON INFORMATION THEORY VOL - PPT Presentation

46 NO 7 NOVEMBER 2000 2567 Differential SpaceTime Modulation Brian L Hughes Member IEEE Abstract Spacetime coding and modulation exploit the presence of multiple transmit antennas to improve performance on multipath radio channels Thus far most wor ID: 29177

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IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.46,NO.7,NOVEMBER2000DifferentialSpace–TimeModulationBrianL.Hughes,Member,IEEESpace–timecodingandmodulationexploitthepresenceofmultipletransmitantennastoimproveperformanceonmultipathradiochannels.Thusfar,mostworkonspace–timecodinghasassumedthatperfectchannelestimatesareavailableatthereceiver.Incertainsituations,however,itmaybedifficultorcostlytoestimatethechannelaccurately,inwhichcaseitisnaturaltoconsiderthedesignofmodulationtechniquesthatdonotrequirechannelestimatesatthetransmitterorreceiver.Weproposeageneralapproachtodifferentialmodulationfor atboththetransmitterandreceivercanincreasespectraleffi- overcomparablesingle-an-tennasystems[7].Space–timecodingandmodulationstrate-gies,whichexploitthepresenceofmultipletransmitantennas,haverecentlybeenadoptedinthird-generationcellularstan-dards(e.g.,CDMA2000[34]andwidebandCDMA[33],[15]),ManuscriptreceivedAugust1,1999;revisedMarch1,2000.ThisworkwassupportedinpartbytheNationalScienceFoundationunderGrantCCR-9903107,andbytheCenterforAdvancedComputingandCommuni-cation.ThematerialinthispaperwaspresentedinpartattheIEEEWirelessCommunicationsandNetworkingConference,NewOrleans,LA,September27–30,1999andatthe33rdAsilomarConferenceonSignals,Systems,andComputers,PacificGrove,CA,October24–27,1999. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.46,NO.7,NOVEMBER2000Inthispaper,weproposeanewandgeneralapproachtodif-ferentialmodulationformultipletransmitantennasbasedongroupcodes.Thisapproachcanbeappliedtoanynumberoftransmitandreceiveantennas,andanysignalconstellation.Wealsoderivelow-complexitydifferentialreceivers,errorbounds,andmodulatordesigncriteriaforthecasewherethenumberoftransmitantennasequalstheblocklengthofthegroupcode.Wethenusethedesigncriteriatoconstructoptimaldifferentialmodulationschemesfortwotransmitantennas.Theseschemescanbedemodulatedwithorwithoutchannelestimates.Thispermitsthereceivertoexploitchannelestimateswhentheyareavailable.Performancedegradesbyapproximately3dBwhenestimatesarenotavailable.Whenchannelestimatesareavail-able,thegroupcodesderivedinthispapercanalsobeusedasspace–timeblockcodes,asin[1],[30].Whilethispaperwasunderreview,welearnedofindependentworkbyHochwaldandSweldens[13]whichproposesasim-ilarapproachtodifferentialspace–timemodulation.Althoughtherearedifferencesintheproposedreceiversandthegeneralityoftheformulation,thedifferentialencodingmethodandmod-ulatordesigncriteriain[13]areessentiallythesameasours.However,thederivationofoptimalmodulationschemesfortwotransmitantennasisuniquetothispaper.Therestofthepaperisorganizedasfollows.InSectionII,weintroducethechannelmodelandprovidesomenecessarybackgroundonspace–timecodingwithandwithoutchanneles-timatesatthereceiver.InSectionIII,weintroduceourapproachtodifferentialmodulationformultipletransmitantennas,andderivelow-complexityreceivers,errorbounds,anddesigncri-teria.Finally,optimalmodulationschemesfortwotransmitan-tennasaregiveninSectionIV,andourmainconclusionsaresummarizedinSectionV.II.PRELIMINARIESA.ChannelModelConsiderawirelesschannelinwhichdataaresentfrom transmitantennasto receiveantennas[7],[8],[28],[32].Atthetransmitter,dataareencodedusing parallelencoders,oneforeachtransmitantenna.Theresultingencodedsymbolsaremappedintoaunit-energyconstellation andmodulatedontoapulsewaveformofduration fortransmissionoverthechannel. denotetheconstellationpointselectedbytheen-coderoftransmitantenna attime Thesignalthatarrivesateachofthe receiveantennasisasuperpositionofthe fadingtransmittedsignalsandnoise,asillustratedinFig.1.Weassumethatthedelayspreadofthemul-tipathissmallandthatthereceiverhasobtainedsymbol,butnotphase,synchronization.Moreover,weassumethat issmallcomparedwiththechannelcoherencetime,sothatfadingcon-ditionscanbeconsideredconstantover symbols.Ateachre-ceiveantenna,ademodulatorsynchronouslysamplestheoutputofafiltermatchedtothepulsewaveform,therebyproducing decisionstatisticsineachsymbolinterval.Underthesecon-ditions,therelationshipbetweenthedecisionstatisticsandthetransmittedsignalsisgivenby Fig.1.Aflat-fadingchannel. isthecomplexfadingpathgainfromtransmitantenna toreceiveantenna and isanoisevariable.Here where isthesignal-to-noiseratio(SNR)perreceiveantenna.Weassumethattheelementsinthetransmitandreceivearraysarespacedsoastoproduceindependentfadingbetweeneachpairoftransmitandreceiveantennas.Thepathgains noisevariables arethereforeindependentandidenticallydistributed,complexGaussianrandomvariableswithproba-bilitydensityfunction(pdf) Definingthecodematrix ............ wecanrecastthischannelinanequivalentmatrixform (1)where isthe receivematrix, isthe fadingmatrix,and isthe noisematrix.Wedistinguishbetweentwocommunicationsituationsforthischannel.Wesaythereceiverhasperfectchannelstateinfor-(CSI),ifthereceiver(butnotthetransmitter)hasaper-fectestimateofthefadingmatrix .Ifneitherthetransmitternorthereceiverknowtheoutcomeof ,wesaythereisnoCSIInthispaper,weareprimarilyinterestedinmethodsfortrans-mittingdatawithoutCSI.Inordertoshowwhythesemethodswork,however,wemakeuseofresultsoncommunicationwithperfectCSI,whicharesummarizedinthenextsection.B.PerfectCSIattheReceiverMostworkonspace–timecodinghasassumedthatperfectCSIisavailableatthereceiver.Wenowsummarizeresultsonoptimalreceivers,errorbounds,anddesigncriteriaforthissit-uationfrom[9],[28].Aspace–timecodefortheconstellation consistsofacollec-tionofcodematrices ,where .When isknownatthereceiver,thepdfofthereceivedmatrixgiventhat istransmittedis HUGHES:DIFFERENTIALSPACE–TIMEMODULATIONwhere“ ”isthetraceand denotestheconjugatetranspose.Ifthecodematricesareequallylikely,theoptimalreceiveristhemaximum-likelihood(ML)detector([18,p.72]),whichreducestotheminimumEuclideandistancedetector Here“ ”denotesanyargumentthatachievesthemaximum(orminimum).Let bethepairwiseerrorproba-bilityofthisreceiver,i.e.,theprobabilityofincorrectlydecoding as ,inacodeconsistingofonlythesetwomatrices.TheChernoffboundonthiserrorprobabilitytakestheform([28,eq. (3)where istheidentitymatrixand denotesthedeterminant.Forlarge ,thisboundbehavesas where and dependonthedifferencematrix .Theparameter isequaltotherankof andcanbeinterpretedasthediversityadvantageofthecodepair[9].Themaximumdiversityistherefore ,provided Thequantity canbeinterpretedasthecodingadvantage,andisgivenby (4)where denotestheproductofthenonzeroeigenvaluesof ,includingmultiplicities.Thisisclearlyamatrixanalogoftheproductdistance[4],[35],whicharisesinsingle-antennafadingchannels.When ,notethattheproductdistance(4)reducesto Also,notethat ifandonlyif Forlarge ,theperformanceofanyspace–timecode isdeterminedprimarilybytheminimumdiversity andtoalesserextentbytheminimumcodingadvantage, Ifweareinterestedonlyincodeswith ,however,notethatwecansimplyusethesingle-performancecriterion whichispositiveonlyif ,inwhichcase Forexample,considerthetransmitdiversityschemeproposedbyAlamoutiin[1],inwhich antennasareusedtosendtwosymbols bytransmittingthecodematrix Fortheunit-energyquaternary-phaseshiftkeying(QPSK)con- ,itiseasytoverifythatthe16codematricesinthisschemehaveminimumdistance Therefore,thediversityis andtheminimumproductdistanceis C.NoCSIattheReceiverIntheabsenceofCSIatthereceiver,HochwaldandMarzetta[11]havearguedheuristicallythatthecapacityofthemulti-antennachannel(1)canbeapproachedforlarge or bycodematriceswithequal-energy,orthogonalrows.Accordingly,theyfocusedattentiononcodeswiththeproperty forall whichtheycalledunitaryspace–timecodes.Inthissection,wesummarizeresultsonoptimalreceivers,errorbounds,andde-signcriteriaforunitarycodesfrom[11].Wepresentthisworkinadifferentformthan[11],however,inordertomoreclearlyrelateittotheresultsoftheprevioussection.Atfirstglance,itmayappearthattheseresultsshouldalsofollowasaspecial ofthosein[29];however,thechannelismodeledasmemorylessin([29,eq.(2)]),whichisinconsistentwithourassumptionthat isfixedfor .When istransmittedand isunknown,thereceivedma- in(1)isGaussianwithconditionalpdf where .Notethatthematrixidentity andtheunitaryproperty(7)implythat doesnotdependon .Furthernotethat whichfollowsfromtheidentity Giventheseresults,theMLdetectorforaunitarycodere-ducestoaquadraticreceiver AChernoffboundonthepairwiseerrorprobabilityofthisreceiverforunitarycodeswasderivedin[11,eq.(18)].Wecanrewritethisboundinacompactmatrixformas Asintheprevioussection,wecanextractusefulinsightsoncodedesignbyexaminingtheasymptoticsofthisbound.Tothebestofourknowledge,thefollowingobservationsarenew,unlessotherwiseindicated.Forlarge ,theboundin(9)behavesas ,where and nowdependonthecross-productmatrix .ThediversityadvantageAsshownin[11],theboundsin(3)and(9)canbothbesharpenedbyafactoroftwo,omittedhereforsimplicity. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.46,NO.7,NOVEMBER2000 isequaltotherank HochwaldandMarzetta[11]haveobservedthatthemaximumdiversityis ,whichisachievedwhen isnotasingularvalueof .Observingthat weseethat ifandonlyiftherowsof and arelinearlyindependent,whichispossibleonlyif Thecodingadvantage isgivenbyaquantityanalogoustotheproductdistance(4) (11)When andthevectors and arereal,thisquan-tityreducesto ,where istheanglebetween and .Wethereforeproposetocallthisquantitytheangulardis- and .Angulardistanceprovidesadesigncriterionforspace–timecodingwithoutCSI,whichisanalogoustotheproductdistance(4)forperfectCSI.Fromtheidentity ,weseethat issymmetricin and .For ,theangulardistancereducesto Forlarge ,theperformanceofthecode withreceiver(8)isdeterminedmainlyby and Onceagain,ifweareinterestedonlyincodeswith ,wecanusethesingle-performancecriterion whichispositiveonlyif ,inwhichcase Asanexample,considertheperformanceofthecode(6)intheabsenceofCSI.If isanyunit-energyphase-shiftkeying(PSK)constellation,then forall .Thusthecodeisunitaryandtheresultsaboveapply.Foranycodematrices and ,theseidentitiesalsoimply ,where istheall-zeromatrix.Weconcludethat ,andhencethecodeisessentiallyuselessintheabsenceofCSI.III.DPACEODULATIONForasingletransmitantenna,oneofthesimplestandmosteffectivenoncoherentmodulationtechniquesisDPSK.Differ-entiallyencodedPSKcanbedemodulatedcoherentlyornonco-herently.Moreover,thenoncoherentreceiverhasasimpleformandperformswithin3dBofthecoherentreceiveronRayleighfadingchannels([19,p.774]).Itisnaturaltoconsiderexten-sionsofthistechniquetomultiantennachannels.Recently,TarokhandJafarkhani[31]haveproposedadiffer-entialmodulationschemefor transmitantennasbasedonAlamouti’scode(6).ThisschemesharesmanyofthedesirablepropertiesofDPSK:itcanbedemodulatedwithorwithoutCSIatthereceiver,achievesfulldiversityinbothcases,andthereex-istsasimplenoncoherentreceiverthatperformswithin3dBofthecoherentreceiver.However,theschemealsohassomelimi-tations.First,theencodingproceduresignificantlyexpandsthesignalconstellationfornonbinarysignaling(e.g.,fromQPSKto9QAM).Second,theapproachdoesnotseemtoextendtocom-plexconstellationsfor ,orrealconstellationsfor withoutapenaltyinrate.Asnotedin[31],theschemereliesonthefactthat(6)isacomplexorthogonalblockdesign,andsuchdesignsdonotexistfor [30].(Realorthogonaldesignsfor and werederivedin[30],andthesecanbeusedtocon-structBPSKmodulatorsforuptoeighttransmitantennas[31].)Inthissection,wepresentanewapproachtodifferentialmod-ulationformultiple-transmitantennasbasedongroupcodes.Thisapproachcanbeappliedtoanynumberofantennasandanyconstellation.Thegroupstructuregreatlysimplifiestheanalysisoftheseschemes,andmayalsoleadtosimplerandmoretrans-parentmodulationanddemodulationprocedures.A.UnitaryGroupCodesOurapproachtodifferentialmodulationisbasedonanewclassofspace–timeblockcodeswhichpossessagroupstruc-ture.Considerasystemwith transmitantennasandconstel- .Forany ,let beanygroupof matrices( forall ),andlet bea matrixsuchthat forall .Wecallthecollectionofmatrices (multichannel)groupcodeoflength overtheconstellation .Therateofthiscodeisgivenby b/s/Hz,where denotesthecardinalityof MultichannelgroupcodesareageneralizationofSlepian’sgroupcodes[26]tomultipleantennasandcomplexconstella-tions.For andgroupsofrealorthogonalmatrices,SlepianconsideredtheuseofsuchcodesontheGaussianchannelandshowedthattheypossessahighdegreeofsymmetry:eachcode-wordhasthesameerrorprobability,theMLdecodingregionsareallcongruent,andthesetof(Euclidean)distancesfromacodewordtoallofitsneighborsisthesameforeachcodeword.In[6],Forneyintroducedgeometricallyuniformcodes,whichextendSlepian’sideatogroupsofarbitraryisometrieswithre-specttoEuclideandistance.Sincetheresultsin[6],[26]arerootedinEuclideandistance,however,theydonotapplydi-rectlytofadingchannelslike(1).Forthepurposesofthispaper,however,allthatwerequireisthegroupstructure.Example1:For -aryPSKisagroupcode and ,where Example2:For , isagroupcodeover ,with and ,where isthe right-shiftmatrix ........... HUGHES:DIFFERENTIALSPACE–TIMEMODULATIONExample3:For ,thepair isagroupcodeovertheQPSKconstellation Theclassofgroupcodesisapparentlyveryrich,andincludespolyphasecodes[39],permutationcodes[25],codesfromre-flectiongroups[17],allbinarylinearcodeswithBPSKmodula-tion[6],[24],andblock-circulantunitarycodes[12].Fromthis,itisclearthatgroupcodescanbeconstructedforanynumberoftransmitantennasandanyconstellation .Wecanalways tobea matrixin andlet beanygroup permutationmatrices.Whilepermutationgroupscanalwaysbeused,mostcomplexconstellationshavesymmetrypropertieswhichpermittheuseofawidervarietyofunitarymatrixgroups.Thecoreideaofthispaperisthatgroupcodescanbedif-ferentiallyencodedinawaysimilartoPSK.Forsimplicity,letusconsider tobethesetofpossiblemessages.Toinitializetransmission,thetransmittersends .Thereafter,mes-sagesaredifferentiallyencoded:tosend inblock transmittersends Thegroupstructureensuresthat whenever .Moreover,therateofthecodeisessentially forlarge Inthispaper,weconsiderthestructureandperformanceofdifferentiallyencodedgroupcodes,subjecttotwoadditionalre-strictions.First,weassumethat isaunitarycode,asin(7),sothattheresultsofSectionII-Capply.Clearly, isunitaryifandonlyif .Second,weassumeforsimplicity .Alloftheresultspresentedhereextendinanat-uralwayto andtosingle-antennasystems;however,thisextensionrequiresadditionaltoolsandintroducessomecompli-cations,andsowillbetreatedelsewhere.Notethatunitarygroupcodescanbeusedinseveraldifferentways.First,ifweencodemessagesby ratherthan(13),then isessentiallyaspace–timeblockcode,asin[1],[12],and[30].Inthiscase,theresultsofSectionII-Bapplywhenperfectchannelestimatesareavailableatthere-ceiver,andtheresultsofSectionII-Capplywhenestimatesarenotavailable.Second, canalsobedifferentiallyencoded,asin(13).WhenperfectCSIisavailable,wecanstillapplytheresultsofSectionII-Btodecodingthesequence ,andthenrecover from and .Weexpecttheerrorprobabilityofthisschemetobeapproximatelytwicetheerrorprobability withoutdifferentialencoding,sinceanerrorin toresultintwoerrorsinthemessagesequence.(Asimilarphe-nomenonoccurswithdifferentially-encodedPSK[19,p.274].)Inthispaper,wearemainlyinterestedinthefinalpossibility,inwhich isdifferentiallyencodedandCSIisabsentatthereceiver.Here,unitarycodematricesareusedinadifferentwaythanin[11],sotheresultsofSectionII-Cdonotapplydirectly.Inthefollowingsections,wederivenewreceivers,errorbounds,andcodedesigncriteriaforthissituation.B.ADifferentialReceiverWenowderiveareceiverfordifferentiallyencodedunitarygroupcodeswith .IntheabsenceofCSI,theMLdetectorforthesequence(13)consistsofthequadraticreceiver(8)ap-pliedtotheentirereceivedsequence ,where Evenformoderatevaluesof and ,thisreceiverisquitecom-plex.We,therefore,seekasimplersuboptimalreceiver.GiventheexampleofDPSK,itisnaturaltolookforareceiverthatestimates usingonlythelasttworeceivedblocks When ,thecodematricesthataffect are Notethat and imply .Fromthis,wecaneasilyshowthat forall .Itfollowsthatthe matrices satisfy forall ,andcanthereforeberegardedasaunitaryblockcodeoflength .If wereknownatthereceiver,theoptimaldecoderforthisblockcodewouldbethequadraticreceiver(8),whichde-pendsonlyonthecross-productmatrices Sincethesematricesdonotdependon ,however,there-ceiverdoesnotrequireknowledgeofthepastinordertodecodethecurrentmessage.Moreover,thisreceiverreducestoasimpleandelegantform,asshowninFig.2 where“ ”denotestherealpartofthetrace,andthelaststepfollowsfromtheidentity .InFig.2, denotesaone-blockdelay.AlthoughthisreceiverismuchsimplerthanMLdetectionbasedon ,itscomplexitygrowsexponentiallywith and ,since arerequired.Thisreceiverhasanestimator–correlatorinterpretation.Ifthereceiverknewboth andthefadingmatrix ,thentheoptimaldetectorwouldbetheminimumEuclideandistancerule(2).Forunitarycodes,thisreducestoacorrelationreceiver Wenowrecognize(16)asacorrelationreceiverinwhich isestimatedbythepreviousreceivedblock ThusthedifferentialreceiverhasthesameformasthereceiverforperfectCSI,anddiffersonlyinthequalityofitschannelestimate.Moregenerally,thissuggeststhatthesamereceivercanbeusedwithnoisychannelestimatesderivedfromothersources,whichliebetweenthesetwoextremes. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.46,NO.7,NOVEMBER2000 Fig.2.Adifferentialreceiver.C.ErrorBoundsandDesignCriteriadifferentialspace–timemodulation(DSTM),wemeanthedifferentialencoder(13)combinedwiththedifferentialreceiver(16).Inthissection,wederiveaboundonthepairwiseerrorprobabilityofDSTMandcriteriaforoptimallydesigning and .Asintheprevioussection,weassume and Consideragainthedetectionof in(13)basedonlyonthe .Recallthat isaunitaryblockcodeoflength ,andnotethat isaunitaryblockcodeoflength Whenchannelestimatesarenotavailableatthereceiver,theoptimaldetectorfortheblockcode is(16).Thustheper-formanceofDSTMisthesameastheperformanceoftheblock withMLdetection(8).Hencethepairwiseerrorprob-abilityisboundedby(9).From(15)andtheunitaryproperty(7),wehaveforall and Thus(9)canbewrittenas Forlarge ,thisboundtakestheform ,where and representthediversityandcodingadvantageofthepair .FromSectionII-C,weknowthat isequaltotherankoftheleftsideof(18),whichclearlyequalstherankof .FromSectionII-B,wethereforehave ,whichisthediversityadvantage and forperfectCSI.FromSectionII-C,thecodingadvantage isequaltotheangulardistance whichfrom(18)reducesto (20)where istheproductdistance(4).Recallthat codingadvantagewhenperfectCSIisavailableatthereceiver. ,wecanexpressthisintermsofthedistancebetweenthemessages Theseresultshaveimportantimplicationsforthetheoryanddesignofdifferentialspace–timemodulation.First,DSTMbasedonthegroupcode achievesmaximumdiversityintheabsenceofCSIifandonlyif achievesmaximumdiversityforperfectCSI.Second,thecodingadvantageofDSTMwithoutCSIisexactlyhalfthecodingadvantageof forperfectCSI.Third,thedesigncriteriaforDSTMarethesameasinSectionII-B:choose sothat suchthat isaslargeaspossible.From(21),wecanclearly tobeanymatrixthatsatisfies ,sincethechoicedoesnotaffectperformance.Comparingwith(3)for ,weseethat(19)isessen-tiallythesameastheChernoffboundforperfectCSI,exceptfora3-dBlossin .Thissuggeststhatthepairwiseerrorprob-abilityofDSTMsuffersa3-dBlossrelativetotheperformanceoftheblockcode withthecorrelationreceiver(17)andper-fectCSI.Thisconclusioncanbeverifiedbyexaminingtheexactpairwiseerrorprobabilitiesforlarge .Thisperformancelossisduemainlytothesuboptimalreceiver(16),whichusesonlythetwomostrecentreceivedblockstoestimate ,ratherthantheentirereceivedsequence.IV.ONITARYROUPSectionIIIprovidesageneralframeworkfordifferentialspace–timemodulationbasedonunitarygroupcodes.Inthissection,wecharacterizeallunitarygroupcodeswith and ,andweidentifythosethatareoptimalinthesenseofachievingthelargestminimumproductdistance .Allofthecodespresentedherecanbeusedintwodistinctways:First,whenperfectCSIisavailableatthereceiver,wecanusethemasspace–timeblockcodeswithencoder(14)anddecoder(2),asinSectionII-B.Second,whenCSIisabsent,wecanusethedifferentialencoder(13)anddetector(16),asinSectionIII-B.AsshowninSectionIII-C,thedesigncriteriaforthesetwoapplicationsarerelatedby and For ,thechoiceof affectstheconstellationbutnotthedistancestructureof ,asshownby(21).Wecanthereforechoose tobeanymatrixthatsatisfies Inparticular isaconvenientchoiceforallofthecodespresentedbelow.Wesaythattwocodes, and ,areifthereisaunitarymatrix suchthat .FromSectionII-Band(21),itiseasytoseethatequivalentcodeshavethesamedi-versity andminimumdistance .ThroughoutthissectionandtheAppendix,wespecifygroupsbytheirgenerators.LetInprinciple,thecodesinthissectioncouldalsobeusedasspace–timeblockcodeswithoutCSIatthereceiver,asinSectionII-C.Since ,however,thesecodesprovidenodiversityinthiscontext. HUGHES:DIFFERENTIALSPACE–TIMEMODULATION denotethegroupconsistingofalldistinctprod-uctsofpowersof Supposethat .IntheAppendix,weshowthatallunitarygroupcodeswith and (cf.AppendixD).Forodd ,the cyclicgroupcodeisdefinedby(22)and (23)where .Thiscodetakesvaluesinthe -PSKconstellationandhas codematrices,diversityadvantage ,andminimumproductdistance whichispositiveforallodd (cf.AppendixB).Forex-ample,the cyclicgroupcodehas and .Forall ,thedicyclicgroupcodeisgivenby(22)and whichtakesvaluesin -PSKandhas codematrices,di-versityadvantage ,andminimumproductdistance .Forexample,thecodeinExample3isdi-cyclicwith and .IntheAppendix,weshowthateveryunitarygroupcodewith and equivalenttoan cycliccodeorthedicycliccode.Thusthereareatmost nonequivalentunitarygroupcodeswiththeseparameters.Unitarygroupcodeswithmaximum aregiveninTableIforall .Allofthesecodesusetheinitialma-trix(22).AlsoshownforcomparisonatthebottomofthetableareAlamouti’sQPSKcode[1],andthedifferentialversionofthiscodeproposedbyTarokhandJafarkhani[31],whichtakesvaluesin9QAM.For 0.5b/s/Hz,theonlyunitarygroupcodewith isthe cyclicgroupcode,whichisthereforeoptimal.For ,the and cyclicgroupcodesarebothop-timal.The codegiveninTableIisequivalenttothe cyclicgroupcode(seeTableIIIinAppendixB),buthasthead-vantageoftakingvaluesinBPSKratherthanQPSK.Notethatthiscodehasthesamecodematricesas(6)forbinary and thusitisessentiallyAlamouti’sbinarycode.Thedifferentiallyencodedversionofthiscodewasgivenin[31].Tothebestofourknowledge,theobservationsthat(6)isagroupcodeforbi- and ,andthatitisoptimalwithrespectto ,arenew.Thegroupstructuremaybeusefulinsimplifyingtheencodinganddecodingprocedures.For ,thedicycliccodegiveninExample3isoptimal.Sincetheunderlyinggroup isknowninalgebraasthequater-niongroup([14,p.32]),wecallthisthequaternioncode.Notethatthiscodehasthesameminimumproductdistanceasthe code,butachievesa50%higherrate.Whenper-fectCSIisavailableatthereceiver,thequaternioncodeachievesthreequartersoftherateofAlamouti’sQPSKcode,butwitha3-dBhighercodingadvantage.Moreover,whenCSIisabsent,TABLEINITARYROUP=2) Fig.3.Bit-errorprobabilityofthequaternioncode=1) Fig.4.Bit-errorprobabilityoftheoptimal=1)thequaternioncodecanbedifferentiallyencodedanddetectedwithoutexpandingtheconstellation.Fig.3givesaplotofthebit-errorrate(BER)ofthiscodeforonereceiveantenna withandwithoutCSI.AtaBERof ,theperformanceofthe IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.46,NO.7,NOVEMBER2000quaternioncodewithcoherentdetectionisroughly1.1dBbetterthanAlamouti’sQPSKcode,andwithdifferentialdetectionisabout2.0dBbetterthanthecorrespondingcodein[31].For ,thebestcyclicanddicyclicgroupcodeshavethesameminimumdistance.InTableI,wechoosethedicycliccodebecausethesecond-nearestneighborsaresignificantlyfar-therapart.Fig.4showstheBERofthiscodefor ,alongwiththeperformanceofsingle-antennadifferentialQPSK.Notethatthiscodeperformssomewhatbetterthanwouldbeexpectedonthebasisofproductdistancealone.Comparingwith[31,Fig.4]at ,weseethatthe codeis1.4dBworsethanAlamouti’sQPSKcodeforcoherentdemodu-lation,andonly0.5dBworsethanthecorrespondingdifferen-tialcodein[31].Inexchangeforthislossofperformance,thecodeinTableIcanbedifferentiallyencodedwithoutchangingthesignalconstellation,preservestheconstantmodulusprop-ertyoftheconstellation(i.e.,8PSKinsteadof9QAM),andhasasimplerdifferentialencoderanddecoder.Forcoherenttrans-mission,however,theencoderanddecoderofthe aremorecomplexthanAlamouti’scode,andtheconstellationislarger(8PSKversusQPSK).Similarobservationsapplytothe cyclicgroupcode.Finally,for ,the cyclicgroupcodesareop-timalfor and .InTableI,wearbitrarilychoose V.CWehaveconsideredthedesignofspace–timecodingandmodulationtechniquesthatdonotrequirechannelestimatesatthetransmitterorreceiver.Weproposedanewandgeneralap-proachtodifferentialmodulationbasedonunitarygroupcodes,arichclassofspace–timeblockcodeswhichextendSlepian’sideatomultipletransmitantennasandcomplexsignalconstel-lations.Thisapproachcanbeappliedtoanynumberoftransmitantennasandanytargetconstellation.Fortheparticularcase ,wederivedlow-complexitydifferentialreceivers,errorbounds,anddesigncriteriafordifferentialspace–timemodula-tion.Fromtheseresults,itisclearthatdifferentiallyencodedunitarygroupcodescanbedecodedwithorwithoutchanneles-timatesatthereceiver.Thisallowsthereceivertousechannelestimateswhentheyareavailable;however,performancede-gradesby3dBwhenestimatesarenotavailable.Finally,weusedthedesigncriteriatoconstructoptimalunitarygroupcodes .Thesecodescanalsobeusedasspace–timeblockcodeswhenCSIisavailableatthereceiver.Asnotedear-lier,someoftheseresultshavebeenobtainedindependentlybyHochwaldandSweldens[13].Themethodsproposedherearenottheonlywaytoperformdifferentialmodulationwithmultipletransmitantennas.Forex-ample,thedifferentialtransmitdiversityschemesofTarokhandJafarkhani[31],whicharebasedonorthogonalblockdesigns,donotfitwithinourframework.Aspointedoutin[31],how-ever,modulatorsbasedonorthogonalblockdesignsseemtobelimitedto forrealconstellationsandto forcomplexconstellations.Theapproachpresentedhereis,tothebestofourThiscanbeexplainedbyobservingthateachcodematrixinthecodehastwonearestneighbors,whereasAlamouti’sQPSKcodehasfour.knowledge,theonlyknownapproachtodesigningdifferentialspace–timemodulationinothersituations.Inasense,thispaperraisesmorequestionsthanitanswers.Multichannelgroupcodesarearichtopicforfurtherinvestiga-tion.Thestructureandalgebraicpropertiesofthesecodeswillbeinvestigatedinacompanionpaper.Theproposedapproachtodifferentialmodulationextendsinanaturalwayto andtosingle-antennasystems.Thisextensionrequiresaddi-tionaltoolsandintroducessomecomplications,however,andsowillbetreatedelsewhere.SincewehavesuggestedDSTMforapplicationslikefrequencyhoppingandfast-fadingchannels,itisnaturaltoexploreextensionsofDSTMthatexploittimeandfrequencydiversityaswellasspacediversity.Inparticular,DSTMseemstoextendinastraightforwardwaytodispersivechannelswhencombinedwithorthogonalfrequency-divisionmultiplexing.Althoughtheapproachpresentedherepermitsthedesignofdifferentialmodulationschemesforanynumberoftransmitantennasandanysignalconstellation,wehaveonlyskimmedthesurfaceintermsoftheactualdesignofcodes.Ourresultssuggest,however,thattheextensiveliteratureongroupcodescanbeleveragedtoprovideawealthofspace–timeblockcodesanddifferentialspace–timemodulationschemes.A.GroupDesignPreliminariesInthisappendix,wecharacterizeallunitarygroupcodeswith and ,andweidentifythosewithmaximumproductdistance .Weassumefamiliaritywithlinearalgebraatthelevelof[27]andgrouptheoryatthelevelof[14,Chs.1and2].Webeginwithsomeusefulresultson unitarymatrices.Considerthematrix If ,thenthedeterminant isaunit-magnitudecomplexnumber.Since itfollowsthat and .Henceany matrixtakestheform (25)where .Wesaythat diagonal in(25),whichwewriteas .Wesay off-diagonal ,andwewrite .Thenonzeroentriesindiagonalandoff-diagonalunitarymatricesalwayshaveunitmagnitude.LemmaA1: beunitary,andlet betheunitarymatrixin(25). ,then isdiagonalifandonlyif isdiagonaloroff-diagonal. isoff-diagonalifandonlyif and . HUGHES:DIFFERENTIALSPACE–TIMEMODULATIONProof:Toprove,notethat (26)where .If ,then isdiagonalifandonlyif ,whichholdsifandonlyif isdiag- oroff-diagonal .Toprove,notethat if isunitary.If isoff-diagonalthen ,whichistrueifandonlyif and . LemmaA2: beunitary.Then hasanondiagonalunitarysquarerootifandonlyif .Moreover,everysuchrootisoftheform (27)where isreal, isanonzerocomplexnumber,and Proof:Supposethat ,where From(25),wehave Foranynonzerocomplexnumber ,notethat implies .Setting and weseethatif isnotdiagonal then implies and .Sincethisfurtherimplies ,anondiagonalrootexistsonlyif .Substi- and into(25),weobtain(27).B.CyclicGroupsSinceweareinterestedonlyincodeswith bytheremarksfollowing(5)itsufficestosearchforunitarygroupcodes thatmaximizethemodifieddistance wherethesecondequalityissimilarto(21).FromSectionII-B, ifandonlyif ,inwhichcase .Inthissection,wecharacterizeallcyclicgroupswith and Agroup ifthereisaunitarymatrix suchthat where istheorder ,i.e.,thesmallestintegersuchthat .Forexample,if , isodd,and ,thematrix TABLEIIM;kROUP generatesacyclicgroupoforder ,whichwecallthe cyclicgroup.Iteasilyshownthattheminimumdistanceofthisgroupis whichispositiveforallodd .Foragiven ,thesmallestdistanceis ,whichisachievedby and ( or ).The cyclicgroupcodeswithmaximum aregiveninTableIIfor – ,alongwiththeconstellation ,assumingtheuseoftheinitialmatrix(22).Thefollowinglemmasshowthateverycyclicgroupcode isequivalenttoan cyclicgroupcode.LemmaA3:For ,everygroupofdiagonalmatrices and isan cyclicgroup,forsome Proof: beagenericelementof .If and havethesame or ,then Thusallofthematricesin differinboth and .Since and arebothpowersof .Sincethereareexactly distinctpowersof ,eachappearsin onceandonlyonceineachdiagonalposition.Let besuchthat .Notethat hasorder ,so Sinceallofthematricesin differin ,itfollowsthat isanoddpowerof .Hence ,forodd ,therebyprovingthelemma. LemmaA4:For ,everycyclicgroupcode with and isequivalenttoan cyclicgroupcode,forsomeodd Proof: beacyclicgroupcodewithgenerator Recallthatamatrix issaidtobe Normalmatricescanbediagonalizedbyunitarytransformations([27,p.311]).Sincetheunitarymatrix isclearlynormal,thereexistsaunitarymatrix suchthat isdiagonal. isequivalentto .Note isagroupofdiagonalmatriceswith .Hence,byLemmaA3, isan cyclicgroup. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.46,NO.7,NOVEMBER2000TABLEIIIROUPODESFOR16(2+1) ThegroupsinTableIIcanoftenberepresentedinsmallerconstellationsbyusinganondiagonalgenerator.Forexample,thegroupgeneratedby isequivalenttoan cyclicgroup,sincetheeigen-valuesof are .Usedwiththeinitialmatrix(22),thisgroupcodetakesvaluesinthe -PSKconstellation.TableIIIgivesthegeneratorsandminimumdistancesofallsuchgroups – .FromTableII,weseethatallofthecyclicgroupsinTableIIIareoptimal.C.DicyclicGroups beanarbitrarygroupandlet denotethemaximumorderofanyelementin .Recallthat alwaysdivides ,and ifandonlyif iscyclic.Inthissection,weconsidergroupswith .Suchgroupscanalwaysbegeneratedbytwoelements.Forexample,if then isagroupwith and .Since and isthewell-knowngroup([3,p.7]).TableIVgivesallofthedicyclicgroupsfor ,alongwiththeirminimumdistancesandconstella-tions,assumingtheinitialmatrix(22).For ,notethat ,sotheresultinggroupiscyclicratherthandicyclic.Intherestofthissection,weshowthateveryunitarygroup with and isequivalenttothedicyclicgroupcode(28).Wefirstrequiretwotechnicallemmas.LemmaA5: besuchthat and If isadiagonalelementoforder and ,then isnotdiagonal, isanonprimitiveelementof ,and isaprimitiveelementof Proof: isdiagonal,then isagroupofdiagonalma-triceswith .ByLemmaA3, mustbecyclicandthere-forecontainsanelementoforder ,whichcontradictstheas-sumptionthat isthemaximumorder.Hence cannotbedi-agonal,whichproves.Since allelementsin beeitherin orthecoset .Since ,itfollows andhence .If isprimitivein ,then hasorder ,acontradiction.Thus isanon-primitiveelementin ,proving.Finally,notethat TABLEIVROUPODESFOR asubgroupof withindex .Sinceallsubgroupsofindex arenormal([14,p.45]), isanormalsubgroupof .Hence forall ([14,p.41]),where .Inparticular, ,which isprimitivein ,proving LemmaA6: beanoddintegersuchthat ,where .Thenforevery suchthat ,thereexistsaninteger suchthat exceptfor and Proof: denotethegreatestcommondivisorof and .Fromelementarynumbertheory([22,p.102]),the hasexactly solutionsin For ,notethat alwayssatisfies(29).Wethereforerestrictattentionto Supposefirstthat .From[22,p.102],thecon-gruence(29)hasasolution ifandonlyif divides .Itfollowsthatthereisasolutionforevery (30)ifandonlyif divides ,orequivalently,ifandonlyif divides .Since and and arebothpowersof ,nogreaterthan .Moreover,since and areconsecutiveevenintegers,onlyoneisdivisibleby ;hence or .Itfollowsthat isapowerof notgreaterthan ,whichalwaysdivides .Thustoevery (30)thereisan thatsatisfies(29),whichprovesthelemmafor .If ,wehave andhencetheonlynonzeronumberin(30)is .Since and clearlyadmitsnosolution in(29),theproofis LemmaA7:Everygroupcode with and isequivalenttothedicyclicgroupcode(28).More-over,thereisnogroupcodewith and Proof: beanarbitrarygroupcodewith and ,andlet beanyel-ementoforder .Sincethesubgroup iscyclicandhaspositiveminimumdistance,byLemmaA4itisequivalenttoa cyclicgroup.Wecanthereforeassume forsomeodd .Let beany HUGHES:DIFFERENTIALSPACE–TIMEMODULATIONmatrixin .Since ,everyelementin mustbeeitherin orin ;hence Supposefirstthat ,inwhichcase .ByLemma isanonprimitiveelementof ,andhence forsomeinteger .ByLemmasA2andA5, isanondiagonalmatrixoftheform(27),where .Using(5)and(27),wecanexplicitlycalculatethedistancebetweenthegroupmatrices and forall Sincethisvanishesfor ,itfollowsthat whichisacontradiction.We,therefore,concludethat Inparticular,for istheonlypossibility.Hence,thereexistsnogroupwith and Nowsuppose and .ByLemmaA5, isnondi-agonaland liesin andisthereforediagonal.Since ,itfollowsfromLemmaA1 isoff-diagonal.LemmaA5furtherassertsthat isanonprimitiveelementof .ByLemmaA2,itfollowsthat isoftheform (31)where and .For ,theonlyelementsin oftheform arethematrices ,where issuchthat .Hence forsome thatsatisfies .Given ,wecanuse(5)and(31)tocalculatethedistance Notethatifthereexistsan suchthat ,then vanishesandhence .Since isodd,LemmaA6showsthat forall and ,withthepossibleexceptionof and Wehavenowprovedthat,if has ,thentheonlypossiblevaluesof and are and .Itfollowsthat and .Substituting into(31),weobtain .Definingtheunitary ,weobservethat isequivalentto ,where and Since isthedicyclicgroup(28),whichhaspositiveminimumdistance,weconcludethateverygroupwith and isequivalenttothedicyclicgroupcode(28),therebycompletingtheproof. D.OtherExtensionsandOptimalGroups andlet beanyunitarygroupcodewith ,andmaximumelementorder .InSectionsBandCofthisappendix,weshowedthat isequivalentto cyclicgroupcodeif ,and isequivalenttothedicyclicgroupcode(28)if .Inthissection,weshowthattherearenootherpossibilities.Inparticular,weprove .Since divides ,thisimplies or ;hence iseithercyclicordicyclic.LemmaA8: issuchthat and ,then contains andthisistheonlyelementoforder Proof:Toshowthat istheonlypossibleelementof ,let besuchthat and .Itfollowsthattheeigenvaluesof satisfy .Since implies ,botheigenvaluesare .FromtheproofofLemmaA4, isanormalmatrix,whichcanbediagonalizedbyaunitarymatrix .Hence .Toshow contains ,let beanyelementwithorder .Since divides iseven.Setting ,wehave ,andhence . LemmaA9: issuchthat and ,then Proof:SincetheresultfollowsfromLemmaA8for wecanassume .Supposethat and .Since isaprimepower,thefirstSylowTheorem([14,p.94])assertsthateverypropersubgroup isnormalinsomelargersubgroup oforder .Let where isanyelementoforder .Then isagroup andorder .FromLemmaA7,itfollowsthat isequivalenttothedicyclicgroup(28).Wecan,therefore, ,where and .Let bethesubgroupofdiagonalmatricesin .Since contains ,wehave .Conversely,byLemmaA3, isacyclicgroupand,therefore,itsorderisboundedbythemaximumelementorder: .Itfollowsthat whichshowsthateverydiagonalmatrixin isin .Foranyoff-diagonalmatrix ,notethat isdiagonalandthereforecontainedin .Thuseveryoff-diagonalmatrixin iscontainedin .Weconcludethat consistsofallofthediagonalandoff-diagonalmatricesin .Since ,itfollowsfromtheSylowTheoremthat isnormalinsomelargersubgroup .Let beanyelementin .Since isnormalin mustbe andisthereforeeitherdiagonaloroff-diagonal.If isdiagonalthen,byLemmaA1 and iseitherdiagonaloroff-diagonal.However,thisimplies ,whichcontradictsourchoiceof .Hence, beoff-diagonal.ByLemmaA1,thisispossibleonlyif ,whichrequires .For ,LemmaA8impliesthatallelementsof haveorder ,except .Moreover,everymatrixoforder isasquarerootof .ByLemmaA2,eachnondiagonalrootof isoftheform (32)where isreal, iscomplex,and .Since if ,weseethatthediagonalentriesof and bothhavezerorealpartonlyif isoff-diagonal .Thisimplies ,anothercontradiction.Since IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.46,NO.7,NOVEMBER2000bothpossibilitiesleadtocontradictions,weconcludethat . WenowcombinetheresultsoftheAp-pendix–DtoprovetheoptimalityoftheunitarygroupcodesgiveninSectionIV.RecallfromAppendix-Bthat ifandonlyif ,inwhichcasetheminimumproductdistanceisgivenby For ,thereisonlyonegroupwith :the cyclicgroupinTableII.For ,thereexistsnodicyclicgroup(cf.LemmaA7),sothetwocyclicgroupsinTableIIarebothoptimal.Wepreferthe groupinTableIII,however,whichisequivalenttothe cyclicgroupandtakesvaluesinBPSK.For ,the dicyclicgroupinTableIVisoptimal.For ,thebestcyclicanddicyclicgroupshavethesameminimumdistance .Sincethesecond-nearestneighborsareatadistanceof inthedicyclicgroupand inthe and cyclicgroups,wechoosethedicyclicgroup.For ,thefourcyclicgroupsinTableIIareoptimal,ofwhichthe grouphasaparticularlysimple 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