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IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.5,JULY1999Space±TimeBlockCodesfromOrthogonalDesignsVahidTarokh,Member,IEEE,HamidJafarkhani,andA.R.Calderbank,Fellow,IEEEAbstractÐWeintroducespace±timeblockcoding,anewpar-adigmforcommunicationoverRayleighfadingchannelsusingmultipletransmitantennas.Dataisencodedusingaspace±timeblockcodeandtheencodeddataissplitintostreamswhicharesimultaneouslytransmittedusingtransmitantennas.Thereceivedsignalateachreceiveantennaisalinearsuperposition TAROKHetal.:SPACE±TIMEBLOCKCODESFROMORTHOGONALDESIGNSTheoutlineofthepaperisasfollows.InSectionII,wede-scribeamathematicalmodelformultiple-antennatransmissionoverawirelesschannel.Wereviewthediversitycriterionforcodedesigninthismodelasestablishedin[10].InSectionIII,werevieworthogonaldesignsanddescribetheirapplicationtowirelesscommunicationsystemsemployingmultipletransmitantennas.Itwillbeprovedthattheschemeprovidesmaximumpossiblespatialdiversityorderandallowsaremarkablysimpledecodingstrategybasedonlyonlinearprocessing.InSectionIV,wegeneralizetheconceptoftheorthogonaldesignsanddevelopatheoryofgeneralizedorthogonaldesigns.Usingthismathematicaltheory,weconstructcodingschemesforanyarbitrarynumberoftransmitantennas.Theseschemesachievethefulldiversityorderthatcanbeprovidedbythetransmitandreceiveantennas.Moreover,theyhaveverysimplemaximum-likelihooddecodingalgorithmsbasedonlyonlinearprocessingatthereceiver.Theyprovidethemaximumpossibletransmis-sionrateusingtotallyrealconstellationsasestablishedinthetheoryofspace±timecoding[10].InSectionV,wedenecomplexorthogonaldesignsandstudytheirproperties.WewillrecovertheschemeproposedbyAlamouti[1]asaspecialcase,thoughitwillbeprovedthatgeneralizationtomorethantwotransmitantennasisnotpossible.Wethendevelopatheorycomplexgeneralizedorthogonaldesigns.Thesedesignsexistforanynumberoftransmitantennasandagainhaveremarkablysimplemaximum-likelihooddecodingalgorithmsbasedonlyonlinearprocessingatthereceiver.Theyprovidefullspatialdiversityand ofthemaximumpossiblerate(asestablishedpreviouslyinthetheoryofspace±timecoding)usingcomplexconstellations.Forcomplexconstellationsandforthespeciccasesoftwo,three,andfourtransmitantennas,thesediversityschemesareimprovedtoprovide,respectively ,and ofmaximumpossibletransmissionrate.SectionVIpresentsourconclusionsandnalremarks.Forthereaderwhoisinterestedonlyinthecodecon-structionbutisnotconcernedwiththedetails,weprovideasummaryofthematerialatthebeginningofeachsubsection.II.TODELANDTHEInthissection,wemodelamultiple-antennawirelesscom-municationsystemundertheassumptionthatfadingisquasi-staticand at.Wereviewthediversitycriterionforcodedesignassumingthismodel.Thisdiversitycriterioniscrucialforourstudiesofspace±timeblockcodes.Weconsiderawirelesscommunicationsystemwherethebasestationisequippedwith andtheremoteisequipped antennas.Ateachtimeslot ,signals aretransmittedsimultaneouslyfromthe transmitantennas.Thecoefcient isthepathgainfromtransmitantenna receiveantenna .ThepathgainsaremodeledassamplesofindependentcomplexGaussianrandomvariableswithvariance perrealdimension.Thewirelesschannelisassumedtobequasi-staticsothatthepathgainsareconstantoveraframeof andvaryfromoneframetoanother.Attime thesignal receivedatantenna isgivenby (1)where areindependentsamplesofazero-meancomplexGaussianrandomvariablewithvariance SNR percom-plexdimension.Theaverageenergyofthesymbolstransmittedfromeachantennaisnormalizedtobe Assumingperfectchannelstateinformationisavailable,thereceivercomputesthedecisionmetric overallcodewords anddecidesinfavorofthecodewordthatminimizesthissum.Givenperfectchannelstateinformationatthereceiver,wemayapproximatetheprobabilitythatthereceiverdecideserroneouslyinfavorofasignal assumingthat wastransmitted.(Fordetailssee[6],[10].)Thisanalysisleadstothefollowingdiversitycriterion.·DiversityCriterionForRayleighSpace±TimeCode:Inordertoachievethemaximumdiversity ,thematrix ...... ............... hastobefullrankforanypairofdistinctcodewords and .If hasminimumrank overthesetofpairsofdistinctcodewords,thenadiversityof isachieved.SubsequentanalysisandsimulationshaveshownthatcodesdesignedusingtheabovecriterioncontinuetoperformwellinRicianenvironmentsintheabsenceofperfectchannelstateinformationandunderavarietyofmobilityconditionsandenvironmentaleffects[11].III.ORTHOGONALESIGNSASODESFORInthissection,weconsidertheapplicationofrealorthog-onaldesigns(SectionIII-A)tocodingformultiple-antennawirelesscommunicationsystems.Unfortunately,thesedesignsonlyexistinasmallnumberofdimensions.EncodingusingorthogonaldesignsisshowntobetrivialinSectionIII-B.Maximum-likelihooddecodingisshowntobeachievedbydecouplingofthesignalstransmittedfromdifferentantennasandisprovedtobebasedonlyonlinearprocessingatthereceiver(SectionIII-C).Thepossibilityoflinearprocessingatthetransmitter,leadstotheconceptoflinearprocessingorthogonaldesignsdevelopedinSectionIII-D.Wethenproveanormalizationresult(Theorem3.4.1)whichallowsusto IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.5,JULY1999focusonaspecicclassoflinearprocessingorthogonaldesigns.Tostudythesetofdimensionsforwhichlinearprocessingorthogonaldesignsexist,weneedabriefreviewoftheHurwitz±RadontheorywhichisprovidedinSectionIII-E.Usingthistheory,weprovethatallowinglinearprocessingatthetransmitteronlyincreasesthehardwarecomplexityatthetransmitteranddoesnotexpandthesetofdimensionsforwhicharealorthogonaldesignexists.Areaderwhoisonlyinterestedincodeconstructionandapplicationsofspace±timeblockcodesmaychoosetofocusattentiononSectionsIII-A,III-B,andIII-CaswellasTheo-rem3.5.1,Denition3.5.2,andLemma3.5.1.A.RealOrthogonalDesignsArealorthogonaldesignofsize isan matrixwithentriestheindeterminates TheexistenceproblemfororthogonaldesignsisknownastheHurwitz±Radonprobleminthemathematicsliterature[5],andwascompletelysettledbyRadoninanothercontextatthebeginningofthiscentury.Infact,anorthogonaldesignexistsifandonlyif or Givenanorthogonaldesign ,onecannegatecertaincolumnsof toarriveatanotherorthogonaldesignwherealltheentriesoftherstrowhavepositivesigns.Bypermutingthecolumns,wecanmakesurethattherstrowof is .Thuswemayassumewithoutlossofgenerality hasthisproperty.Examplesoforthogonaldesignsarethe design (3)the design andthe design Thematrices(3)and(4)canbeidentied,respectively,withcomplexnumber andthequaternionicnumber B.TheCodingSchemeInthissection,weapplyorthogonaldesignstoconstructspace±timeblockcodesthatachievediversity.Weassumethattransmissionatthebasebandemploysarealsignalconstella- with elements.Wefocusonprovidingadiversityorderof .Corollary3.3.1of[10]impliesthatthemaximumtransmissionrateis bitspersecondperhertz(bits/s/Hz).Weprovidethistransmissionrateusingan design.Attimeslot1, bitsarriveattheencoderandselectconstellationsignals .Setting for ,wearriveatamatrix withentries .Ateachtimeslot theentries aretransmittedsimultaneouslyfromtransmitantennas Clearly,therateoftransmissionis bits/s/Hz.Wenowdemonstratethatthediversityorderofsuchaspace±timeblockcodeis Theorem3.2.1:Thediversityorderoftheabovecodingschemeis Proof:Therankcriterionrequiresthatthematrix benonsingularforanytwodistinctcodesequences .Clearly, where isthematrixconstructedfrom byreplacing with forall .Thedeterminantoftheorthogonalmatrix iseasilyseentobe where isthetransposeof .Hence whichisnonzero.Itfollowsthat isnonsingularandthemaximumdiversity isachieved. C.TheDecodingAlgorithmNext,weconsiderthedecodingalgorithm.Clearly,therowsof areallpermutationsoftherstrowof possiblydifferentsigns.Let denotethepermutationscorrespondingtotheserowsandlet denotethesignof inthe throwof .Then meansthat isuptoasignchangethe thelementof .Sincethecolumns arepairwise-orthogonal,itturnsoutthatminimizingthemetricof(2)amountstominimizing (6)where (7) TAROKHetal.:SPACE±TIMEBLOCKCODESFROMORTHOGONALDESIGNSandwhere denotesthecomplexconjugateof Thevalueof onlydependsonthecodesymbol ,thereceivedsymbols ,thepathcoefcients ,andthestructureoftheorthogonaldesign .Itfollowsthatminimiz-ingthesumgivenin(6)amountstominimizing(7)forall .Thusthemaximum-likelihooddetectionruleistoformthedecisionvariables forall anddecideinfavorof amongalltheconstellationsymbols if Thisisaverysimpledecodingstrategythatprovidesdiversity.D.LinearProcessingOrthogonalDesignsTherearetwoattractionsinprovidingtransmitdiversityviaorthogonaldesigns.·Thereisnolossinbandwidth,inthesensethatorthogonaldesignsprovidethemaximumpossibletransmissionrateatfulldiversity.·Thereisanextremelysimplemaximum-likelihooddecod-ingalgorithmwhichonlyuseslinearcombiningatthereceiver.Thesimplicityofthealgorithmcomesfromtheorthogonalityofthecolumnsoftheorthogonaldesign.Theabovepropertiesarepreservedevenifweallowlinearprocessingatthetransmitter.Therefore,werelaxthedeni-tionoforthogonaldesignstoallowlinearprocessingatthetransmitter.Signalstransmittedfromdifferentantennaswillnowbelinearcombinationsofconstellationsymbols.Denition3.4.1:Alinearprocessingorthogonaldesignin isan matrix suchthat:·Theentriesof arereallinearcombinationsofvariables .· ,where isadiagonalmatrixwith diagonalelementoftheform withthecoefcients allstrictlypositiveItiseasytoshowthattransmissionusingalinearprocessingorthogonaldesignprovidesfulldiversityandasimplieddecodingalgorithmasabove.Thenexttheoremshowsthatwemay,withnolossofgenerality,constrainthematrix Denition3.4.1tobeascaledidentitymatrix.Theorem3.4.1:Alinearprocessingorthogonaldesign invariables existsifandonlyifthereexistsalinearprocessingorthogonaldesign suchthat Proof: bealinearprocessingorthogonaldesign,andlet wherethematrices arediagonalandfull-rank(sincethecoefcients arestrictlypositive).Thenitfollowsthat (9) (10)and isafull-rankdiagonalmatrixwithpositivediagonalentries.Let denotethediagonalmatrixhavingthepropertythat .Wedene Thenthematrices satisfythefollowingproperties: (11) Itfollowsthat isalinearprocessingorthogonalarrayhavingtheproperty Inviewoftheabovetheorem,wemay,withoutanylossofgenerality,assumethatalinearprocessingorthogonaldesign satises E.TheHurwitz±RadonTheoryInthissection,wedeneaHurwitz±Radonfamilyofmatrices.Thesematricesencodetheinteractionsbetweenvariablesinanorthogonaldesign.Denition3.5.1:Asetof realmatrices iscalledasize Hurwitz±Radonfamilyofmatricesif and WenextrecallthefollowingtheoremofRadon[8].Theorem3.5.1: ,where isoddand with and .AnyHurwitz±Radonfamilyof matricescontainsstrictlylessthan matrices.Furthermore .AHurwitz±Radonfamily matricesexistsifandonlyif or Denition3.5.2: bea matrixandlet beanyarbitrarymatrix.Thetensorproduct isthematrixgivenby ........................... (13) IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.5,JULY1999Denition3.5.3:Amatrixiscalledanintegermatrixifallofitsentriesareintheset TheproofofthenextLemmaisdirectlytakenfrom[5]andweincludeitforcompleteness.Lemma3.5.1:Forany thereexistsaHurwitz±Radonfamilyofmatricesofsize whosemembersareintegerProof:Theproofisbyexplicitconstruction.Let denotetheidentitymatrixofsize .Werstnoticethatif with odd,then .Moreover,givenafamily Hurwitz±Radonintegermatrices ofsize ,theset isaHurwitz±Radonfamilyof integermatricesofsize .Inlightofthisobservation,itsufcestoprovethelemmafor .Tothisend (14) (15)and (16)Let and Then Weobservethat isaHurwitz±Radonintegerfamilyofsize isaHurwitz±Radonintegerfamilyofsize and isanintegerHurwitz±Radonfamilyofsize Thereadermayeasilyverifythatif isanintegerHurwitz±Radonfamilyof matrices,then isanintegerHurwitz±Radonfamilyof integermatrices If,inaddition, isanintegerHur-witz±Radonfamilyof matrices,then isanintegerHurwitz±Radonfamilyof integermatrices Weproceedbyinduction.For ,wealreadycon-structedanintegerHurwitz±Radonfamilyofsize withentriesintheset .Now(17)givesthetransition to .Byusing(18)andletting , ,wegetthetransitionfrom to .Similarly,with , and , ,wegetthetransitionfrom to andto . Thenexttheoremshowsthatrelaxingthedenitionoforthogonaldesignstoallowlinearprocessingatthetransmitterdoesnotexpandthesetofdimensions forwhichthereexistsanorthogonaldesignofsize Theorem3.5.2:Alinearprocessingorthogonaldesignof existsifandonlyif and Proof: denotealinearprocessingorthogonalde-sign.Sincetheentriesof arelinearcombinationsofvariables wecanwriterow of as ,where isanappropriatereal-valued matrixand .Orthogonalityof translatesintothefol-lowingsetofmatrixequalities: (19) (20)where istheidentitymatrix.WenowconstructaHur-witz±Radonsetofmatricesfromtheoriginaldesign.Let for .Then andwehave (21) (22) Theseequationsimplythat isafamilyofmatrices.BytheHurwitz±RadonTheo-rem(3.5.1),wecanconcludethat and or . Inparticular,wehavethefollowingspecialcase.Corollary3.5.1:Anorthogonaldesignofsize existsifandonlyif and Proof:ImmediatefromTheorem3.5.2. Tosummarize,relaxingthedenitionoforthogonaldesigns,byallowinglinearprocessingatthetransmitter,failstoprovidenewtransmissionschemesandonlyaddstothehardwarecomplexityatthetransmitter.IV.GRTHOGONALThepreviousresultsshowthelimitationsofprovidingtransmitdiversitythroughlinearprocessingorthogonalde-signsbasedonsquarematrices.Sincethesimplemaximum-likelihooddecodingalgorithmdescribedaboveisachievedbecauseoforthogonalityofcolumnsofthedesignmatrix,wemaygeneralizethedenitionoflinearprocessingorthogonaldesigns.Notonlydoesthiscreatenewandsimpletransmis-sionschemesforanynumberoftransmitantennas,butalsogeneralizestheHurwitz±Radontheorytononsquarematrices. TAROKHetal.:SPACE±TIMEBLOCKCODESFROMORTHOGONALDESIGNSInthissection,weintroducegeneralizedrealorthogonalandposethefundamentalquestionofgeneralizedorthogonaldesigntheory.Theanswertothisfundamentalquestionprovidesuswithtransmissionschemesthatareinsomesenseoptimalintermsofthedecodingdelay.Wethensettlethefundamentalquestionofgeneralizedorthogonaldesigntheoryforfull-rateorthogonaldesigns(inasensetobedenedinthesequel)andconstructfull-ratetransmissionschemesforanynumberoftransmitantennas.Areaderwhoisinterestedonlyincodeconstructionandapplicationsofspace±timeblockcodesisadvisedtogothroughtheresultsofthissection.A.ConstructionandBasicPropertiesDenition4.1.1:Ageneralizedorthogonaldesign ofsize isa matrixwithentries suchthat where isadiagonalmatrixwithdiagonal oftheform andcoefcients arestrictlypositiveintegers.Therateof is ThefollowingtheoremisanalogoustoTheorem3.4.1Theorem4.1.1: generalizedorthogonaldesign invariables existsifandonlyifthereexistsageneralizedorthogonaldesign inthesamevariablesandofthesamesizesuchthat Inviewoftheabovetheorem,withoutanylossofgenerality,weassumethatany generalizedorthogonaldesign invariables satises Transmissionusingageneralizedorthogonaldesignisdis-cussednext.Weconsiderarealconstellation ofsize Throughputof canbeachievedasdescribedinSec-tionIII-A.Attimeslot1, bitsarriveattheencoderandselectconstellationsymbols .Theencoderpop-ulatesthematrixbysetting ,andattime thesignals aretransmittedsimultaneouslyfrom .Thus bitsaresentduringeach transmissions.Itcanbeproved,asinTheorem3.1,thatthediversityorderis .Itshouldbementionedthattherateofageneralizedorthogonaldesignisdifferentfromthethroughputoftheassociatedcode.Tomotivatethedenitionoftherate,wenotethatthetheoryofspace±timecodingprovesthatforadiversityorderof ,itispossibletotransmit bitspertimeslotandthisisbestpossible(see[10,Corollary3.3.1]).Therefore,therate ofthiscodingschemeisdenedtobe whichisequalto Thegoalofthissectionistoconstructhigh-ratelinearprocessingorthogonaldesignswithlowdecodingcomplexityandfulldiversityorder.Wemust,however,takethememoryrequirementsintoaccount.Thismeansthatgiven and ,wemustattempttominimize Denition4.1.2:Foragiven wedene betheminimumnumber suchthatthereexistsa generalizedorthogonaldesignwithrateatleast .Ifnosuchorthogonaldesignexists,wedene generalizedorthogonaldesignattainingthevalue calleddelay-optimal.Thevalueof isthefundamentalquestionofgen-eralizedorthogonaldesigntheory.Themostinterestingpartofthisquestionisthecomputationof sincethegeneralizedorthogonaldesignsoffullratearebandwidth-efcient.Toaddressthisquestion,wewillneedthefollowingConstructionI: and InLemma3.5.1,weexplicitlyconstructedafamilyofinteger matriceswith members .Let andconsiderthe matrix whose columnis for .TheHurwitz±Radonconditionsimplythat isageneralizedorthogonaldesignoffullrate.Theorem4.1.2:Thevalue isthesmallestnumber suchthat Proof: beanumbersuchthat .Let andapplyConstructionItoarriveat generalizedorthogonaldesignoffullrate.Bydenition, ,andhence Next,weconsideranygeneralizedorthogonaldesign ofsize in variables(rateone)where Thecolumnsof arelinearcombinationsofthevariables .The thcolumncanbewrittenas somereal-valued matrix .Sincethecolumnsof orthogonalwehave (25) Thismeansthatthematrices areaHurwitz±Radonfamilyofsize .Thus and ,and .Combiningthisresultwithinequality(24)concludestheproof. Corollary4.1.1:Forany Proof:TheprooffollowsimmediatelyfromTheo-rem4.1.2. Corollary4.1.2:Thevalue ,wheretheminimizationistakenovertheset and Inparticular, , ,and for Proof: .Werstclaimthat isapoweroftwo.Tothisend,supposethat where isanoddnumber.Then .But .Thiscontradictsthefactthat andprovestheclaim.Thus forsome .Anapplicationoftheexplicitformulafor giveninTheorem3.5.1completestheproof. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.5,JULY1999Itfollowsthatorthogonaldesignsaredelayoptimalfor and WehaveexplicitlyconstructedaHurwitz±Radonfamilyofmatricesofsize with memberssuchthatallthematricesinthefamilyhaveentriesintheset .GivensuchafamilyofHurwitz±Radonmatricesofsize wecanapplyConstructionItoprovidea orthogonaldesignwithfullrate.Thisfull-rategeneralizedorthogonaldesignhasentriesoftheform Thisisthemethodusedtoprovethefollowingtheoremwhichcompletestheconstructionofdelay-optimalgeneralizedorthogonaldesignsofrateonefor transmitantennas.Theorem4.1.3:Theorthogonaldesigns (27) (28) (29)and aredelay-optimaldesignswithrateone.Proof:Theorthogonaldesignsconstructedaboveachievethevalue for . V.GRTHOGONALESIGNSASPACEThesimpletransmitdiversityschemesdescribedaboveassumearealsignalconstellation.Itisnaturaltoaskforextensionsoftheseschemestocomplexsignalconstellations.HencethenotionofcomplexorthogonaldesignsisintroducedinSectionV-A.WerecovertheAlamoutischemeasa complexorthogonaldesignsinSectionV-B.Motivatedbythepossibilityoflinearprocessingatthetransmitter,wedenecomplexlinearprocessingorthogonaldesignsinSectionV-C,butweshallprovethatcomplexlinearprocessingorthogonaldesignsonlyexistintwodimensions.ThismeansthattheAlamoutiSchemeisinsomesenseunique.However,wewouldliketohavecodingschemesformorethantwotransmitantennasthatemploycomplexconstellations.HencethenotiongeneralizedcomplexorthogonaldesignsisintroducedinSectionV-E.Wethenprovebyexplicitconstructionthat generalizedcomplexorthogonaldesignsexistinanydimension.InSectionV-F,itisshownthatthisisnotthebestratethatcanbeachieved.Specically,examplesofrate generalizedcomplexlinearprocessingorthogonaldesignsindimensionsthreeandfourareprovided.Areaderwhoisonlyinterestedincodeconstructionandtheapplicationofspace±timeblockcodesmaychoosetoreadSectionV-B,Denition5.4.1,Denition5.5.2,theproofofTheorem5.5.2,Corollary5.5.1,theremarkafterCorollary5.5.1,andSectionV-F.A.ComplexOrthogonalDesignsWedeneacomplexorthogonaldesign ofsize asanorthogonalmatrixwithentriestheindeterminates ,theirconjugates ormultiplesoftheseindeterminatesby where Withoutlossofgenerality,wemayassumethattherstrow is ThemethodofencodingpresentedinSectionIII-Acanbeappliedtoobtainatransmitdiversityschemethatachievesthefulldiversity .Thedecodingmetricagainseparatesintodecodingmetricsfortheindividualsymbols Anexampleofa complexorthogonaldesignisgivenby B.TheAlamoutiSchemeThespace±timeblockcodeproposedbyAlamouti[1]usesthecomplexorthogonaldesign Supposethatthereare signalsintheconstellation.Atthersttimeslot, bitsarriveattheencoderandselecttwocomplexsymbols and .Thesesymbolsaretransmittedsimultaneouslyfromantennasoneandtwo,respectively.Atthesecondtimeslot,signals and aretransmittedsimultaneouslyfromantennasoneandtwo,respectively.Maximum-likelihooddetectionamountstominimizingthedecisionstatistic overallpossiblevaluesof and .Theminimizingvaluesarethereceiverestimatesof and ,respectively.Asinthe TAROKHetal.:SPACE±TIMEBLOCKCODESFROMORTHOGONALDESIGNSprevioussection,thisisequivalenttominimizingthedecision fordetecting andthedecisionstatistic fordecoding .Thisisthesimpledecodingschemedescribedin[1],anditshouldbeclearthataresultanalogoustoTheorem3.2.1canbeestablishedhere.ThusAlamouti'sschemeprovidesfulldiversity using receiveantennas.ThisisalsoestablishedbyAlamouti[1],whoprovedthatthisschemeprovidesthesameperformanceas levelmaximumratiocombining.C.OntheExistenceofComplexOrthogonalDesignsInthissection,weconsidertheexistenceproblemforcomplexorthogonaldesigns.First,weshowthatacomplexorthogonaldesignofsize determinesarealorthogonaldesignofsize ConstructionII:Givenacomplexorthogonaldesign ofsize ,wereplaceeachcomplexvariable bythe realmatrix Inthisway isrepresentedby (35) isrepresentedby andsoforth.Itiseasytoseethatthe matrixformedinthiswayisarealorthogonaldesignofsize Wecannowprovethefollowingtheorem:Theorem5.3.1:Acomplexorthogonaldesign ofsize existsonlyif or Proof:Givenacomplexorthogonaldesignofsize applyConstructionIItoprovidearealorthogonaldesignofsize .Sincerealorthogonaldesignscanonlyexistfor and itfollowsthatcomplexorthogonaldesignsofsize cannotexistunless or . For ,Alamouti'sschemegivesacomplexorthogonaldesign.Wewillprovelaterthatcomplexorthogonaldesignsdonotexistevenforfourtransmitantennas.D.ComplexLinearProcessingOrthogonalDesignsDenition5.4.1:Acomplexlinearprocessingorthogonaldesigninvariables isan matrix ·theentriesof arecomplexlinearcombinationsof andtheirconjugates; ,where isadiagonalmatrixwherealldiagonalentriesarelinearcombinationsof withallstrictlypositiverealcoefcients.TheproofofthefollowingtheoremissimilartothatofTheorem3.4.1.Theorem5.4.1:Acomplexlinearprocessingorthogonal invariables existsifandonlyifthereexistsacomplexlinearprocessingorthogonaldesign suchthat Inviewoftheabovetheorem,withoutanylossofgenerality,weassumethatanycomplexlinearprocessingorthogonal satises Wecannowprovethefollowingtheorem:Theorem5.4.2:Acomplexlinearprocessingorthogonaldesignofsize existsifandonlyif Proof:WeapplyConstructionIItothecomplexlinearprocessingorthogonaldesignofsize toarriveatalinearprocessingorthogonaldesignofsize .Thus or whichimpliesthat or .For Alamouti'smatrixisacomplexlinearprocessingorthogonaldesign.Therefore,itsufcestoprovethatfor linearprocessingorthogonaldesignsdonotexist.TheproofisgivenintheAppendix.Wecannowimmediatelyrecoverthefollowingresult.Corollary5.4.1:Acomplexorthogonaldesignofsize existsifandonlyif Proof:ImmediatefromTheorem5.4.2. Weconcludethatrelaxingthedenitionofcomplexor-thogonaldesignstoallowlinearprocessingwillonlyaddtohardwarecomplexityatthetransmitterandfailstoprovidetransmissionschemesinnewdimensions.E.GeneralizedComplexOrthogonalDesignsWenextdenegeneralizedcomplexorthogonaldesigns.Denition5.5.1: bea matrixwhoseentriesare ortheirproductwith .If isadiagonal IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.5,JULY1999matrixwith thdiagonalelementoftheform andthecoefcients allstrictlypositivenumbers, isreferredtoasageneralizedorthogonaldesignof andrate ThefollowingTheoremisanalogoustoTheorem3.4.1.Theorem5.5.1: complexgeneralizedlinearpro-cessingorthogonaldesign invariables existsifandonlyifthereexistsacomplexgeneralizedlinearprocessingorthogonaldesign inthesamevariablesandofthesamesizesuchthat Inviewoftheabovetheorem,withoutanylossofgenerality,weassumethatany generalizedorthogonaldesign invariables satisestheequality aftertheappropriatenormalization.Transmissionusingacomplexgeneralizedorthogonalde-signissimilartothatofageneralizedorthogonaldesign.Maximum-likelihooddecodingisanalogoustothatofAlam-outi'sschemeandcanbedoneusinglinearprocessingatthereceiver.Thegoalofthissectionistoconstructhigh-ratecomplexgeneralizedlinearprocessingorthogonaldesignswithlowdecodingcomplexitythatachievefulldiversity.Wemust,however,takethememoryrequirementsintoaccount.Thismeansthatgiven and ,wemustattempttominimize Denition5.5.2:Foragiven and ,wedene theminimumnumber forwhichthereexistsacomplexgeneralizedlinearprocessingorthogonaldesignofsize andrateatleast .Ifnosuchorthogonaldesignexists,we Thequestionofthecomputationofthevalueof isthefundamentalquestionofgeneralizedcomplexorthogonaldesigntheory.Toaddressthisquestiontosomeextent,wewillestablishthefollowingTheorem.Theorem5.5.2:Thefollowinginequalitieshold.·i)Forany ,wehave ·ii)For ,wehave Proof:WerstproveParti).If ,thenthereisnothingtobeproved.Thusweassumethat andconsideracomplexgeneralizedlinearprocessingorthogonaldesign ofrateatleastequalto andsize .ByapplyingConstructionII,wearriveata realgeneralizedlinearprocessingorthogonaldesignofrateatleastequalto .Thus ToprovePartii),weconsiderarealorthogonaldesign ofsize andrateatleastequalto invariables where .Weconstructacomplex ofsize .Wereplacethesymbols everywherein bytheirsymbolicconjugates toarriveatanewarray .Thenwedene tobethe arraywiththerow the throwof andtherow the throwof .Itiseasytoseethat acomplexgeneralizedorthogonaldesignofrateatleastequal .Thus . Corollary5.5.1: ,wehave Proof:ItfollowsimmediatelyfromPartii)ofTheorem5.5.2andCorollary4.1.1. Corollary5.5.1provesthereexistsrate plexgeneralizedorthogonaldesigns,andtheproofofPartii)ofTheorem5.5.2givesanexplicitconstructionforthesedesigns.Forinstance,rate codesfortransmissionusingthreeandfourtransmitantennasaregivenby (37)and Thesetransmissionschemesandtheiranalogsforhigher givefulldiversitybutlosehalfofthetheoreticalbandwidthefciency.F.FewSporadicCodesItisnaturaltoaskforhigherratesthan whendesigninggeneralizedcomplexlinearprocessingorthogonaldesignsfortransmissionwith multipleantennas.For ,Alamouti'sschemegivesarateonedesign.For and ,weconstruct generalizedcomplexlinearprocessingorthogonaldesignsgivenby (39) TAROKHetal.:SPACE±TIMEBLOCKCODESFROMORTHOGONALDESIGNS and (40)for .Thesecodesaredesignedusingthetheoryofamicabledesigns[5].Apartfromthesetwodesigns,wedonotknowofanyothergeneralizeddesignsinhigherdimensionswithrategreaterthan .Webelievethattheconstructionofcomplexgeneralizeddesignswithrategreaterthan isdifcultandwehopethatthesetwoexamplesstimulatefurtherwork.VI.CWehavedevelopedthetheoryofspace±timeblockcoding,asimpleandelegantmethodfortransmissionusingmultipletransmitantennasinawirelessRayleigh/Ricianenvironment.Thesecodeshaveaverysimplemaximum-likelihooddecodingalgorithmwhichisonlybasedonlinearprocessing.Moreover,theyexploitthefulldiversitygivenbytransmitandreceiveantennas.ForarbitraryrealconstellationssuchasPAM,wehaveconstructedspace±timeblockcodesthatachievethemaximumpossibletransmissionrateforanynumber transmitantennas.Foranycomplexconstellation,wehaveconstructedspace±timeblockcodesthatachievehalfofthemaximumpossibletransmissionrateforanynumber transmitantennas.Forarbitrarycomplexconstellationsandforthespeciccases and ,wehaveprovidedspace-timeblockcodesthatachieve,respectively,all, ,and ofthemaximumpossibletransmissionrate.Webelievethatthesediscoveriesonlyrepresentthetipoftheiceberg.Theorem:Acomplexorthogonaldesignofsize doesnotProof:Theproofisdividedintosixsteps.StepI:Inthisstep,weprovidenecessaryandsufcientconditionsfora matrixofindeterminatestobeacomplexlinearprocessinggeneralizedorthogonaldesign.Tothisend, beacomplexlinearprocessinggeneralizedorthogonaldesignofsize .Eachentryof isalinearcombination .Itfollowsthat (41)where arecomplex matrices.Since wecanconcludefromtheabovethat Conversely,anysetof complexmatrices satisfyingtheaboveequationsdenesalinearprocessingcomplexorthogonaldesign.StepII:Inthisstep,wewillprovethatgivenacomplexlinearprocessinggeneralizedorthogonaldesign ,wecouldconstructanothercomplexlinearprocessinggeneralizedor-thogonaldesign suchthatforanyrow,oneof and doesnotoccurintheentriesofthatrowof .Inotherwords,forany whereforanyxed either forall or forall .Intheformer(respectively,latter)casewesay (respectively, )doesnotoccurinthe throwof Using(42),werstobservethat Hence Similarly, .Thismeansthatthematrices and areidempotentfor .Since ,thematrices and projectionsontoperpendicularvectorspaces and thusarediagonalizablewithalleigenvaluesintheset .If ,thenexactly (respectively, )ofeigenvaluesof (respectively, )are Next,using(42),weobservethatfor Thusthematrices and commute.Similarly,itfollowsthat isacommutingfamilyofdiagonalizablematrices.Hence,thesematricesaresimultaneouslydiagonalizable.Sincetheeigenvaluesof areintheset ,weconcludethatthereexistsaunitary suchthat where arediagonalmatriceswithdiago-nalentriesintheset .Moreover,because the thentryof iszero(respectively,one)ifandonlyifthe thentryof isone(respectively,zero).Since thenonzeroentriesof appearinthoserows wherethe thelementof iszero.Similarly, IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.45,NO.5,JULY1999impliesthatthenonzeroentriesof appearinthoserows,wherethecorrespondingdiagonalelementof zero.Thusthenonzeroentriesof and indifferentrows. thenitfollowsfromthematrixequationsgiveninStepIthat isacomplexlinearprocessinggeneralizedorthogonaldesignwiththedesiredproperty.StepIII:Wecannowassumewithoutanylossofgen-eralitythat isacomplexlinearprocessinggeneralizedorthogonaldesignwiththepropertiesdescribedinStepII.Inthisstep,weapplyConstructionIIto andstudythepropertiesoftheassociatedreallinearprocessinggeneralizedorthogonaldesign.Byinterchanging with everywhereinthedesignifnecessary,wecanfurtherassumethatonly intherstrowof .WenextapplyConstructionIIto andconstructarealorthogonaldesign ofsize invariables .Thematrix canbewrittenas (43)where arereal matrices.Furthermore,assumingthepropertyestablishedinStepII,wecaneasilyobservebydirectcomputationthat (44)where where isadiagonalmatrixofsize whosediagonalentriesbelongtotheset .Moreover,the thentryof equals .Welet thevectorwhose thcomponentisthe thelementof .The thelementof isequalto (respectively, )if )occursinrow Using(43)and wearriveatthefollowingsetofequations: (45)Let ,thenusing(44)and(45)wehave (46) (47) (48) (49) (50) (51) (52) StepIV:Wenextprovethatthematrices anticommutewith and butcommutewith and .First,weobservethatby(51)and(53) Sincethematrices and areantisymmetric,theaboveequationsprovethat anticommuteswith and .Furthermore,since ,weconcludefrom(46)±(53)thatwhen and whichimpliesthat Since anticommuteswith ,wearriveat Because isorthogonal,itisinvertibleandthuswhen- and ,wehave Theassertionfor noweasilyfollowssince StepV:Recallthat isthevector thcomponentisthe thelementof .Inthisstep,weprovethatanytwovectors and haveHammingdistanceexactlyequaltotwo.Tothisend,since commuteswith for andanticommuteswith and ,wecaneasilyconcludefromthenonsingularityof that ,for and for .ThustheHammingdistanceofanytwodistinctvectors and isneitherzeronorfour.WerstprovethattheHammingdistanceofanytwodistinct and cannotbeone.Tothisend,letussupposethattwodistinctvectors and haveHammingdistanceoneanddifferonlyinthe thposition.Theninthe throwof wehaveeitheroccurrencesof and oroccurrencesof and butnotboth.Inanyotherrowof ,wehaveeitheroccurrencesof and oroccurrencesof and notboth.Itiseasytoseethatthecolumnsof cannotbeorthogonaltoeachother.WenextprovethattheHammingdistanceofanytwodistinctvectors and cannotbethree.Tothisend,letussupposethattwodistinctvectors and haveHamming TAROKHetal.:SPACE±TIMEBLOCKCODESFROMORTHOGONALDESIGNSdistancethree.Since forall concludethat forall .Wecannow andobservethatthevector isdistinctwith and .Moreover,itcoincideswithboth and therstposition.Itfollowsusingasimplecountingargument hasHammingdistanceonewitheither or .Butwejustprovedthatthisisnotpossible.Weconcludethatanytwodistinctvectors and Hammingdistanceexactlyequaltotwo.StepVI:Inthisstep,wewillarriveatacontradictionthatconcludestheproof.Becauseanytwodistinctvectors and haveHammingdistanceexactlyequaltotwo,thematrix whose throw isaHadamardmatrix.Itfollowsthatanytwodistinctcolumnsof alsohaveHammingdistance .Thuswecannowassumewithoutlossofgeneralitythat(afterpossiblerenamingofthevariablesandbyexchangingtheroleofsomevariableswiththeirconjugates) occurinrowone occurinrowtwoof .Therstrowof isthusexpressibleas andthesecondrowof isoftheform forappropriatevectors .Because weobservethat arevectorsofunitlength.Moreover,if thevectors and areorthogonaltoeachother.Sincetherstandsecondrowsof areorthogonal,weobserve isorthogonalto .Thismeansthat containsasetofveorthonormalvectorsincomplexspaceofdimension .Thiscontradictionprovestheresult. TheauthorsthankDr.NeilSloaneforvaluablecommentsanddiscussions.[1]S.M.Alamouti,ªAsimpletransmitterdiversityschemeforwire-lesscommunications,ºIEEEJ.Select.AreasCommun.,vol.16,pp.1451±1458,Oct.1998.[2]N.BalabanandJ.Salz,ªDualdiversitycombiningandequalizationindigitalcellularmobileradio,ºIEEETrans.Veh.Technol.,vol.40,pp.342±354,May1991.[3]G.J.FoschiniJr.,ªLayeredspace-timearchitectureforwirelesscommu-nicationinafadingenvironmentwhenusingmulti-elementantennas,ºBellLabsTech.J.,pp.41±59,Autumn,1996.[4]G.J.FoschiniJr.andM.J.Gans,ªOnlimitsofwirelesscommunicationinafadingenvironmentwhenusingmultipleantennas,ºWirelessPersonalCommun.,Mar.1998.[5]A.V.GeramitaandJ.Seberry,OrthogonalDesigns,QuadraticFormsandHadamardMatricesLectureNotesinPureandAppliedMathemat-,vol.43.NewYorkandBasel:MarcelDekker,1979.[6]J.-C.Guey,M.P.Fitz,M.R.Bell,andW.-Y.Kuo,ªSignaldesignfortransmitterdiversitywirelesscommunicationsystemsoverRayleighfadingchannels,ºinProc.IEEEVTC'96,1996,pp.136±140.[7]Y.KawadaandN.Iwahori,ªOnthestructureandrepresentationsofCliffordAlgebras,ºJ.Math.Soc.Japan,vol.2,pp.34±43,1950.[8]J.Radon,ªLinearescharenorthogonalermatrizen,ºinausdemMathematischenSeminarderHamburgishenUniversit,vol.I.pp.1±14,1922.[9]N.SeshadriandJ.H.Winters,ªTwosignalingschemesforimprovingtheerrorperformanceoffrequency-division-duplex(FDD)transmissionsystemsusingtransmitterantennadiversity,ºInt.J.WirelessInform.,vol.1,no.1,1994.[10]V.Tarokh,N.Seshadri,andA.R.Calderbank,ªSpace±timecodesforhighdataratewirelesscommunication:PerformanceanalysisandcodeIEEETrans.Inform.Theory,vol.44,pp.744±765,Mar.[11]V.Tarokh,A.Naguib,N.Seshadri,andA.R.Calderbank,ªSpace-timecodesforhighdataratewirelesscommunications:Performancecriteriainthepresenceofchannelestimationerrors,mobilityandmultipleIEEETrans.Commun.,vol.47,pp.199±207,Feb.1999.[12]E.Telatar,ªCapacityofmulti-antennaGaussianchannels,ºAT&T-BellLaboratoriesInternalTech.Memo.,June1995.[13]J.Winters,J.Salz,andR.D.Gitlin,ªTheimpactofantennadiversityonthecapacityofwirelesscommunicationsystems,ºIEEETrans.,vol.42.Nos.2/3/4,pp.1740±1751,Feb./Mar./Apr.1994.[14]A.Wittneben,ªBasestationmodulationdiversityfordigitalSIMUL-CAST,ºinProc.IEEEVTC,May1991,pp.848±853. ,ªAnewbandwidthefcienttransmitantennamodulationdiver-sityschemeforlineardigitalmodulation,ºinProc.IEEEICC,1993,pp.1630±1634.