Design of coded modulation schemes for orthogonal transmit diversity,” presented at the

DesignofCodedModulationSchemesfor

OrthogonalTransmitDiversity

MohammadJaberBorran,MahsaMemarzadeh,andBehnaamAazhang

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Abstract

Inthispaper,weproposeatechniquetodecoupletheproblemsofspatialandtemporaldiversitygainmaximizationinvolvedinthedesignofspace-timecodesinfastRayleighfadingchannelsbyusingorthogonaltransmitdiversitysystems.Wewillalsointroducetheideaofconstellationexpansion(indimensionorsize)todesigncodedmodulationschemeswithmaximumtemporaldiversitygainfororthogonaltransmitsystems.TheChernoffupperboundfortheerrorprobabilityisusedtoshowthatthecodedesigncriterionreducestothemaximizationofHammingandproductdistancesintheexpandedconstellation.Theproposedtechniqueisdemonstratedbydesigningmultilevelandmultipletrelliscodedmodulationschemesfororthogonaltransmitdiversitysystems.Thesecodesareshowntohavebetterperformancecomparedtothecodingschemesdesignedforsingletransmitandreceiveantennas.

Keywords

CodedModulation,OrthogonalTransmitDiversity,MultilevelCoding,MultipleTrellisCodedModulation,Mul-tidimensionalTrellisCodedModulation

I.INTRODUCTION

Itiswellknown[1]thatthedesignofoptimumspace-timecodesformultipletransmitandreceiveantennasystemsoverfastRayleighfadingchannels,withindependentfadingcoefficientsbetweendifferenttrans-mittersandreceivers,isbasedontwocriteria:thedistancecriterion,andtheproductcriterion.Assumethat

and

,representthe

sequencesoftransmittedanderroneouslydecodedsymbols,respectively,where

isthenumberoftransmit

antennasandisthelengthofthecodeblock.In[1],itisshownthatwhencompletechannelstateinforma-tionisavailableatthereceiver,theChernoffupperboundforthepairwiseerrorprobabilitycanbeexpressedas

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throughthecodingscheme(temporaldiversitygain).Thedistinctionbetweenthesetwopotentialsourcesofdiversitygainisnoteasilyseenfromtheabovedistancecriterion.Decouplingtheproblemsofmaximizingthesetwodiversitygains,ifpossible,couldsignificantlysimplifythecodedesignprocedure.Inthatcase,wecouldusetheexistingcodingschemesdesignedforfastfadingchannelsandsingletransmitandreceivedantennas,becausetheseschemesarealreadydesignedtomaximizethetemporaldiversitygain.

Consideringasingleantennaatthereceiver,themaximumachievablespatialdiversitygainisequaltothenumberoftransmitantennas.Assumingacoherencetimeequaltothenumberoftransmitantennasforthechannel,oneofthetransmissionschemesthatcanprovidefullspatialdiversitygainandhasarelativelysimplestructureistheOrthogonalTransmitDiversity(OTD)systemproposedbyAlamouti[2].Moreover,ithasalreadybeenshown[3]thatthissystempreservesthecapacityofmultipletransmitandsinglereceiveantennasystems.ThesearemotivationstoconsidertheOTDsystemasameansofprovidingthespatialdiversitygainandgetthetemporaldiversitygainthroughanouterbandwidthefficientcodedmodulationscheme.Intheremaining,wewillconsiderthecaseoftwotransmitandonereceiveantennas,thoughthegeneralizationofthediscussiontoanynumberoftransmitantennasforwhichanorthogonaltransmissionschemeexists,isstraightforward.

InSectionII,wewillexplainthetechniqueofdecouplingtheproblemsofspatialandtemporaldiversitymaximizationusingtheOTDsystemwithmoredetail.Laterinthesamesectionwewillintroducetheideaofconstellationexpansion(indimensionorsize)todesigncodedmodulationschemesfortheorthogonaltransmitdiversitysystems.ItwillbefurthershownthatthecodedesigncriteriareducetothemaximizationofHammingandproductdistancesintheexpandedconstellation.SimulationresultsfortwobandwidthefficientcodedmodulationschemeswillbedemonstratedinSectionIII,andSectionIVpresentstheconclu-sions.

II.CODEDESIGNCRITERIA

FOR

ORTHOGONALTRANSMITDIVERSITYSYSTEM

Underchannelconditionsdescribedintheprevioussection,itwasmentionedthattheorthogonaltransmitdiversitysystemiscapableofprovidingthemaximumachievablespatialdiversitygain.InFig.1,theOTDsystemisdepictedforthecaseoftwotransmitandsinglereceiveantennas.TheChernoffupperboundforthepairwiseerrorprobabilityofthissystemisderivedin[3]as

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Fig.1.OrthogonalTransmitDiversity(OTD)System

transmitantennas,resultsfromtheorthogonaltransmissionsystem.Hence,applyingtheOTDsystemofFig1,wehavemaximizedthespatialdiversitygainachievablebytwotransmitantennas.Thismeansthattheproblemofdiversitygainmaximizationhassomehowbeendecoupledintotwoproblems:maximizationofthespatialdiversitygain,andthetemporaldiversitygain.TheformercanbeachievedbyusingtheOTDsystemandthelatteristheresultofanoutercodedmodulationschemeoptimizedintermsofitstemporaldiversitygain.

In[4],ithasbeenshownthattheperformanceofcodesinfastfadingchannelswithsingletransmitandreceiveantennasiscontrolledbytwofactors:thecodeminimumHammingdistance(lengthoftheshortesterroreventpath),andtheproductofsquaredsymboldistancesalongtheshortesterroreventpath.Thesetwofactorsdeterminethetemporaldiversitygainandthecodinggainofthecode,respectively.However,from(2)itcanbenoticedthattheoptimumcodefortransmissionovertheOTDsystemisnotsimplyobtainedbythecriteriaof[4],butinsteaditinvolvesmaximizationofnewdistancequantitiesdefinedintermsofpairsofconsecutivesymbols.ThesenewpairwiseHammingandpairwiseproductdistances,denotedbyand

respectively,aredefinedas:

(3)

and

(4)

ThisisthemotivationtointroduceanewcodedesigntechniquefortheOTDsystembasedonexpand-October25,2001

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ingthesignalconstellation.Theexpansioncanbeperformedineitherdimensionorsizeoftheconstel-lation(goingtohigherordersofmodulation).Eachpointinthenewconstellationcanbeconsideredastheconcatenationoftwoconsecutivesignalpointsfromtheoriginalsignalset.Denotingthesequencesoftransmittedanderroneouslydecodedsymbolsinthenewconstellationby

and

,respectively,wewillhave:

Thus,(3)and(4)canberewrittenas

(5)

and

(6)

Hence,thecodedesigncriteriawillbebasedonmaximizingtheminimumsymbolHammingandproductdistancesintheexpandedconstellation.

Thus,toachieveacertaindiversitygainof,onlyaminimumsymbolHammingdistanceof

inthe

expandedconstellationisneeded.ThisdoesnotnecessarilyrequireaminimumHammingdistanceof

intheoldsignalset,asopposedtothedesigntechniquesofcodedmodulationschemesforsingletransmitandreceiveantennasystems[5].Infact,itisclearfromtheChernoffupperboundof(2)thatadoptingtheorthogonaltransmitdiversity,theHammingdistancerequirementsofthecodedsymbolshalves.Thisistheconsequenceoftheinherentspatialdiversitygainoftworesultingfromtheorthogonaltransmissionsystem.ThereductionintheHammingdistancerequirementsallowsustohavelargersubsetsinthesignalsetpartitioning,whichinturnresultsinhigherachievablecoderates.ThiswillbeexplainedwithmoredetailinSectionIII.

III.DESIGN

OF

TRELLISCODEDMODULATIONSCHEMES

FORTHE

OTDSYSTEM

InSectionII,itwasdescribedthatthecodedesigncriteriafortheOTDsysteminfastfadingchannels,isbasedonmaximizationofHammingandproductdistancesintheexpandedconstellation,whereeachpointistheconcatenationoftwopointsfromtheoriginalsignalset.Assumingthattheoriginalsignalsetistwodimensional(2D),onewayofconstructingthenewconstellationistoconsiderthefourdimensional(4D)Cartesianproductoftheoriginal2Dsignalsetbyitself.Anotherwayistoconstructanew2Dconstellation

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ofsize

,where

isthesizeoftheoriginalsignalset.Thecodedesignwilllaterbeperformedforthe

newconstellationadoptingthedesigncriteriaofSectionII.Attheoutputofthecodedmodulationblock,eachencodedsymbolfromthenewconstellationwillbeconsideredasconcatenationoftwosignalpointsfromtheoriginalconstellation,andwillbetransmittedintwoconsecutivesymbolintervalsthroughtheOTDsystem.ItshouldbenoticedthatMultipleTrellisCodedModulation(MTCM)andMultiLevelCoding(MLC)designtechniquessatisfyingthecriteriaofSectionII,havealreadybeenproposedintheliterature([5,6]).Theseareappropriatecodedmodulationschemesforfastfadingchannels,astheycanbedesignedtoachievegooddistancepropertiesrequiredbythecriteriaderivedin[4].

Intheremaining,wewilldemonstratetheabovedesignprocedurethroughtwoexamples,wheretwotrelliscodedmodulationschemeshavebeendesignedfortransmissionthroughtheOTDsysteminafastRayleighfadingchannelwithcoherencetimeoftwosymbolintervals.

A.ConstellationExpansioninDimension:DesignofMultidimensionalMLCforOTD

Supposethatthegoalistodesignacodedmodulationschemewithrate3bits/sec/Hzandtotaldiversitygainof4.Weusea4D256-pointlatticeconstellationwith2Dconstituentscomingfroma16QAMconstel-lation,alongwithatwo-levelcodewhichprovidesminimumHammingdistanceof2.Twoconvolutionalencodersofrates2/3and4/5areusedasthefirstandsecondlevelencoders.The4Dsetpartitioningchain

isusedtopartitionthe256-pointconstellationinto8subsetsofsize32,asexplainedin

[7]and[8].Oneofthe8subsetsischosenbytheoutputsofthefirstlevelencoder.Theoutputsofthesecondlevelencoderarethenusedtochooseoneofthe32pointsinsidethechosensubset.Each4Dpointisthenmappedintoitstwo2Dcoordinatestoproduceasequenceofsignalpointsfroma16QAMconstellation.ThissequenceislatertransmittedthroughtheOTDsystemofFig.1.

TheerrorrateperformanceoftheabovecodeisshowninFig.2(a)andiscomparedtotheuncoded8PSKOTDscheme.Ascanbeseenfromtheerrorratecurves,thecodedschemeshowsmorethan4dBgainovertheuncodedschemeaterrorratesof

andlower.

B.ConstellationExpansioninSize:DesignofMTCMforOTD

Considerdesigningacodewiththeoveralldiversitygainof4andrate1.5bits/s/HzusingQPSKmodula-tion.InordertoachieveaminimumHammingdistanceof2resultinginatotaldiversitygainof4usingtheOTDsystem,itsufficestoconsideranMTCMcodewithmultiplicityof4andperformthesetpartitioningtaskfora2-foldCartesianproductofa16PSKsignalset.Eachpointinthe16PSKsignalsetisconsideredastheconcatenationoftwoconsecutiveQPSKsymbols.Ifthesetpartitioningschemeof[5]isadoptedfor

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10010Uncoded Orthogonal Transmission (R = 3 bits/s/Hz)MLC for Orthogonal Transmission (R = 3 bits/s/Hz)−110−110−2Symbol Error ProbabilitySymbol Error ProbabilitySlope = − 2 10−310−210−310−4Slope = − 4 Uncoded Orthogonal Transmission (R = 1.5 bits/s/Hz) MTCM for Single Transmission (R = 1.5 bits/s/Hz) MTCM for Orthogonal Transmission (R = 1.5 bits/s/Hz)10−468101214SNR per Bit16182010−567891011SNR per Bit1213141516(a)(b)Fig.2.Symbolerrorrateperformanceof(a)MultidimensionalMultilevelTCM,and(b)MultipleTCM

a2-foldCartesianproductof16PSKsymbols,amaximumof16codesetscanbeassignedtoeachsubset.Thismeansthatamaximumof16parallelpathscanbeconsideredforthetrellis.So,ifa4statefullyconnectedtrellisisconsidered,theencoderwouldbecapableofencoding6inputbits.Togetherwiththe4QPSKsymbolsassignedtoeachtransitionofthetrellis,thisresultsinarateof

bits/sec/Hz.

Notethatifwewantedtousethesametrellistodesignacodewithdiversitygainof4forsingletransmitandreceiveantennasystem,themaximumachievableratewouldbe1bit/s/Hz.Thatisbecausethesetpar-titioningofthe4-foldCartesianproductofQPSKsymbols,wouldresultinsubsetswithamaximumsizeoffour[5].Thus,forcomparisonpurpose,an8statefullyconnectedtrellishasbeenusedtodesignanMTCMcodewithrate1.5bits/s/Hzanddiversitygainof4forsingletransmissionscheme.

TheerrorperformancesofthesetwotransmissionschemesareshowninFig.2(b)andhavebeencomparedtotheuncodedcase.ItcanbeseenthattheMTCMcodedesignedfortheOTDsystemoutperformsthesingletransmitandreceiveantennacase,andshowsmorethan5dBgainovertheuncodedschemeaterrorratesof

andlower.

IV.CONCLUSIONS

Inthispaper,weintroducedamethodtodecoupletheproblemsofmaximizationofspatialandtemporaldiversitygainbyusingtheorthogonaltransmitdiversitysystem.Wealsoderivedthecriteriaforthedesignofcodedmodulationschemesfortheorthogonalsystembasedontheexpansionofthesignalconstellation.Itwasshownthatthisexpansioncanbeperformedineitherdimensionorsizeofthesignalset.TheproposeddesigncriteriawerethenappliedtothedesignoftrelliscodedmodulationschemesfortheOTDsystem.The

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newcodingschemeswereshowntohavesignificantlybetterperformancecomparedtothecodesdesignedforsingletransmitandreceiveantennasystemsusingsimulations.

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