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BayesianRobustLinearTransceiverDesignforDual-HopAmplify-and-ForwardMIMORelaySystemsChengwenXing,ShaodanMaandYik-ChungWuDepartmentofElectricalandElectronicEngineeringTheUniversityofHongKong,HongKongEmail:{cwxing,sdma,ycwu}@eee.
hku.
hkAbstract—Inthispaper,weaddresstherobustlineartransceiverdesignfordual-hopamplify-and-forward(AF)MIMOrelaysystems,wherebothtransmittersandreceivershaveimperfectchannelstateinformation(CSI).
WiththestatisticsofchannelestimationerrorsinthetwohopsbeingGaussian,wefor-mulatetherobustlinear-minimum-mean-square-error(LMMSE)transceiverdesignproblemusingtheBayesianframework,andderiveaclosed-formsolution.
Simulationresultsshowthattheproposedalgorithmreducesthesensitivityoftherelaysystemtochannelestimationerrors,andperformsbetterthanthealgorithmusingestimatedchannelonly.
I.
INTRODUCTIONRecently,cooperativecommunicationhasgainedsignicantinterest,duetoitsgreatpotentialstoimprovereliability,coverageandcapacityofwirelesslinks[1][2].
Generallyspeaking,therearethreekindsofrelayprotocols,amplify-and-forward(AF),compress-and-forward(CF)anddecode-and-forward(DF).
Amongthethreeschemes,AFisconceptuallythesimplestone,inwhichtherelayjustscalesthesignaltrans-mittedfromthesource,andthentransmitstothedestination.
Duetoitssimplicityandlowimplementationcomplexity,AFstrategyhasreceivedmanyresearchers'attention.
Ontheotherhand,itiswell-knownthatinfullyscatteredenvironments,multiantennasystemsprovidespatialdiversityandmultiplexinggains.
Thiskindofbenetscanbedirectlyintroducedintocooperativecommunicationsviadeploymentofmultipleantennasattransmittersandreceivers.
Thecom-binationofAFandMIMOsystemsbringsgreatpotentialsinperformanceimprovement.
LineartransceiverdesignforAFMIMOrelaysystemshasbeenaddressedin[2],[3],[4]and[5].
ThecapacityscalinglawofMIMOrelaynetworkshasbeendiscussedin[2].
Thelineartransceiverdesignforthexedrelayincellularnet-workshasbeenaddressedin[3].
Jointlinear-minimum-mean-square-error(LMMSE)transceiverdesignforAFMIMOrelaysystemsisconsideredin[4]and[5].
However,alloftheabovementionedworksrequirethechannelstateinformation(CSI)perfectlyknownatthetransmittersandreceivers.
Unfortunately,inpracticalsystems,channelestimationer-rorsareinevitable,whichshouldbetakenintoaccountintransceiverdesign.
Inthispaper,weproposearobustlineartransceiverdesignmethodforAFMIMOrelaysystems.
TheTheRelayTheDestinationTheSourcesNRNRMDMFGsrHrdHssFig.
1.
Amplify-and-forwardMIMOrelaydiagramchannelestimationerrorsaremodeledasGaussianrandomvariables.
ThestatisticsofthechannelestimationerrorsareincorporatedintothedesignusingtheBayesianframework,andaclosed-formsolutionisobtained.
Simulationresultsshowthattheproposedalgorithmperformsbetterthanthealgorithmusingestimatedchannelonly.
Thefollowingnotationsareusedthroughoutthispaper.
Boldfacelowercaselettersdenotevectors,whileboldfaceuppercaselettersdenotematrices.
ThenotationZHdenotestheHermitianofthematrixZ,andTr(Z)isthetraceofthematrixZ.
ThesymbolIMdenotesanM*Midentitymatrix,while0M,NdenotesanM*Nallzeromatrix.
ThenotationZ12istheHermitiansquarerootofthepositivesemidenitematrixZ,suchthatZ12Z12=ZandZ12isalsoaHermitianmatrix.
II.
PROBLEMFORMULATIONInthispaper,adual-hopamplify-and-forward(AF)cooper-ativecommunicationsystemisconsidered.
Intheconsideredsystem,thereisonesourcewithNSantennas,onerelaywithMRreceiveantennasandNRtransmitantennas,andonedestinationwithMDantennas,asshowninFig.
1.
Atthersthop,thesourcetransmitsdatatotherelay.
Thereceivedsignal,x,attherelayisx=Hsrs+n1(1)wheresisthedatavectortransmittedbythesourcewiththecovariancematrixRs=E{ssH}.
ThematrixHsristheMIMOchannelmatrixbetweenthesourceandtherelay.
Symboln1istheadditiveGaussiannoisewithcovariancematrixRn1.
Attherelay,thereceivedsignalxismultipliedbyaprecodermatrixF,underapowerconstraintTr(FRxFH)≤ThisfulltextpaperwaspeerreviewedatthedirectionofIEEECommunicationsSocietysubjectmatterexpertsforpublicationintheIEEE"GLOBECOM"2009proceedings.
978-1-4244-4148-8/09/$25.
002009PwhereRx=E{xxH}andPisthemaximumtransmitpower.
Thentheresultingsignalistransmittedtothedestina-tion.
Thereceivedsignalatthedestination,y,canbewrittenasy=HrdFHsrs+HrdFn1+n2,(2)whereHrdistheMIMOchannelmatrixbetweentherelayandthedestination,andn2istheadditiveGaussiannoisevectoratthesecondhopwithcovariancematrixRn2.
Inordertoguaranteethetransmitteddatascanberecoveredatthedestination,itisassumedthatMR,NR,andMDaregreaterthanorequaltoNS[4].
Itisassumedthatboththerelayanddestinationhavetheestimatedchannelstateinformation(CSI).
Whenchannelestimationerrorsareconsidered,wehaveHsr=Hsr+ΔHsr,Hrd=Hrd+ΔHrd,(3)wherethesymbolsHsrandHrdaretheestimatedCSI,whileΔHsrandΔHrdarethecorrespondingchannelestimationerrorswhoseelementsarezeromeanGaussianrandomvari-ables.
Ingeneral,theMR*NSmatrixΔHsrcanbewrittenasΔHsr=Σ12srHWΨ12srwheretheelementsoftheMR*NSmatrixHWareindependentandidenticallydistributed(i.
i.
d.
)Gaussianrandomvariableswithzeromeanandunitvariance.
TheMR*MRmatrixΣsrandNS*NSmatrixΨTsraretherowandcolumncovariancematricesofΔHsr,respectively[6].
Itiseasytoseethatvec(ΔHTsr)CN(0MR*NS,ΣsrΨTsr)basedonwhichΔHsrissaidtohaveamatrix-variatecomplexGaussiandistribution,whichcanbewrittenas[7]ΔHsrCNMR,NS(0MR,NS,ΣsrΨTsr),(4)withtheprobabilitydensityfunction(p.
d.
f.
)givenby[8][9]f(ΔHsr)=exp(Tr((ΔHsr0)HΣ1sr(ΔHsr0)Ψ1sr))(π)NSMRdet(Σsr)NSdet(Ψsr)MR.
(5)Similarly,fortheestimationerrorinthesecondhop,wehaveΔHrdCNMD,NR(0MD,NR,ΣrdΨTrd)(6)wheretheMD*MDmatrixΣrdandNR*NRmatrixΨTrdaretherowandcolumncovariancematricesofΔHrd,respectively.
Itisassumedthatthechannelestimationerrors,ΔHsrandΔHrd,areindependent.
Atthedestination,alinearequalizerGisadoptedtodetectthetransmitteddatas.
TheproblemishowtodesignthelinearprecodermatrixFattherelayandthelinearequalizerGatthedestinationtominimizethemeansquareerrors(MSE)ofthereceiveddataatthedestination:MSE(F,G)=E{Tr(Gys)(Gys)H},(7)wheretheexpectationistakenwithrespecttos,ΔHsr,ΔHrd,n1andn2.
III.
ROBUSTTRANSCEIVERDESIGNFORMIMORELAYA.
MSEAveragedoverChannelUncertaintiesSinces,n1andn2areindependent,theMSEexpression(7)canbewrittenasMSE(F,G)=E{(GHrdFHsrINS)s+GHrdFn1+Gn22}=EΔHsr,ΔHrd{Tr((GHrdFHsrI)Rs(GHrdFHsrI)H)}+EΔHrd{Tr(GHrdF)Rn1(GHrdF)H}+Tr(GRn2GH)=EΔHsr,ΔHrd{Tr(GHrdFHsr)Rs(GHrdFHsr)H}+TrGEΔHrd{HrdFRn1FHHHrd}GHTrRs(GHrdFHsr)HTrGHrdFHsrRs+Tr(Rs)+Tr(GRn2GH).
(8)BecauseΔHsrandΔHrdareindependent,thersttermofMSEisEΔHsr,ΔHrd{Tr(GHrdFHsr)Rs(GHrdFHsr)H}=TrGEΔHrdHrdFEΔHsr{HsrRsHHsr}FHHHrdGH.
(9)Fortheinnerexpectation,duetothefactthatthedistributionofΔHsrismatrix-variatecomplexGaussianwithzeromean,thefollowingequationholds[7]EΔHsr{HsrRsHHsr}=EΔHsr{(Hsr+ΔHsr)Rs(Hsr+ΔHsr)H}=Tr(RsΨsr)Σsr+HsrRsHHsrΠ0.
(10)Applying(10)andthecorrespondingresultforΔHrdto(9),thersttermofMSEbecomesTrGEΔHrdHrdFEΔHsr{HsrRsHHsr}FHHHrdGH=Tr(G(Tr(FΠ0FHΨrd)Σrd+HrdFΠ0FHHHrd)GH).
(11)Similarly,thesecondtermofMSEin(8)canbesimpliedasTrGEΔHrd{HrdFRn1FHHHrd}GH=Tr(GTr(FRn1FHΨrd)Σrd+HrdFRn1FHHHrdGH).
(12)Basedon(11)and(12),theMSE(8)equalstoMSE(F,G)=TrG(HrdFRxFHHHrd+K)GHTrRsHHsrFHHHrdGHTrRsGHrdFHsr+Tr(Rs)(13)whereRx=Π0+Rn1(14)K=Tr(F(Π0+Rn1)FHΨrd)Σrd+Rn2.
(15)NoticethatthematrixRxistheautocorrelationmatrixofthereceivesignalxattherelay.
ThisfulltextpaperwaspeerreviewedatthedirectionofIEEECommunicationsSocietysubjectmatterexpertsforpublicationintheIEEE"GLOBECOM"2009proceedings.
978-1-4244-4148-8/09/$25.
002009B.
JointRobustDesignofEqualizerandPrecoderSubjecttothetransmitpowerconstraintattherelay,thejointdesignofequalizerandprecodercanbeexpressedasthefollowingoptimizationproblemminF,GMSE(F,G)s.
t.
Tr(FRxFH)≤P.
(16)SincetheconstraintdoesnotinvolvetheequalizerG,whentheprecodermatrixFisxed,theoptimallinearequalizer,Gopt,satisesthefollowingconditionMSE(F,G)G=0,(17)basedonwhichwehaveGopt=Rs(HrdFHsr)H(HrdFRxFHHHrd+K)1.
(18)Substituting(18)into(13),theMSEatthedestinationequalstoMSE(F)=Tr(Rs)Tr(RsHHsr[FHHHrd(HrdFRxFHHHrd+K)1HrdF]HsrRs).
(19)SinceK12andRx12arebothHermitianmatrices,exploitingthematrixinversionlemma,wehaveFHHHrd(HrdFRxFHHHrd+K)1HrdF=RxH2RxH2FHHHrdKH2(K12HrdFRx12RxH2FHHHrdKH2+IMD)1K12HrdFRx12Rx12=Rx1RxH2(RxH2FHHHrdK1HrdFRx12+IMR)1Rx12.
(20)Putting(20)into(19),anddeningtheconstantpartcTr(Rs)Tr(RsHHsrRx1HsrRs),equation(19)canberewrittenasMSE(F)=Tr(RsHHsrRxH2(RxH2FHHHrdK1HrdFRx12+IMR)1Rx12HsrRs)+c.
(21)From(15),K=Tr(FRxFHΨrd)Σrd+Rn2,soMSE(F)isahighorderfunctionofFandtheproblemofminimizing(21)isverydifculttosolve.
Inordertoproceed,noticethat[10]Tr(FRxFH)λmax(Ψrd)Σrd+Rn2K,(22)whereλmax(Z)denotesthelargesteigenvalueofZ.
SincefortheminimumMSE,thecorrespondingtransmitpowermustbeontheboundary(i.
e.
,Tr(FRxFH)=P),wehavePλmax(Ψrd)Σrd+Rn2K.
AsshowninAp-pendixI,whenKin(21)isreplacedbyitsupper-boundΦPλmax(Ψrd)Σrd+Rn2,theresultantMSEexpression,denotedasMSEU(F)anddenedin(43),isanupper-boundofMSE(F)(i.
e.
,MSEU(F)≥MSE(F)).
Therefore,weproposetodesigntheprecoderFbyminimizingMSEU(F),whichcorrespondstominFTr(Rx12HsrRsRsHHsrRxH2T(RxH2FHHHrdΦ1HrdΘFRx12+IMR)1)s.
t.
Tr(FRxFH)≤P(23)wheretheconstantcisneglected,whichdoesnotaffecttheoptimizationproblem.
NoticethatwhenΨrd∝I,thereplacementinvolvesnoapproximation.
Basedoneigendecompostion,wehaveT=UTΛTUHT,(24)Θ=UΘΛΘUHΘ,(25)wherethematricesUTandUΘconsistoftheeigenvectorsofTandΘ,respectively,whilethediagonalmatricesΛTandΛΘcontainstheeigenvaluesofTandΘ,respectively.
Withoutlossofgenerality,itisassumedthatthediagonalelementsofΛTandΛΘareindecreasingorder.
Substituting(24)and(25)into(23)anddeningFUHΘFRx12UT,(26)theoptimizationproblemcanbewritteninacompactformasminFTrΛT(FHΛΘF+IMR)1s.
t.
Tr(FFH)≤P.
(27)Fortheobjectivefunctionof(27),noticethatTr(ΛTB)≥MRi=1λT,iλB,MRi+1(28)whereBisdenedasB(FHΛΘF+IMR)1,λB,iistheithlargesteigenvalueofB,andthesymbolλT,idenotestheithdiagonalelementofΛT.
In(28),theequalityholdswhenthematrixBisdiagonalwithdiagonalelementsinincreasingorder[11,9.
H.
1.
h].
Therefore,fortheoptimalsolution,(FHΛΘF+IMR)1mustbediagonalwithdiagonalelementsinincreasingorder.
Basedonthediagonalstructure,introducingapermutationmatrixwithdimensionM*MasQM=0010.
.
.
0100M,(29)theobjectivefunctionoftheoptimizationproblem(27)canberewrittenasTrΛT(FHΛΘF+IMR)1=TrΛT(FHΛΘF+IMR)1(30)whereΛT=QMRΛTQMRandΛΘ=QNRΛΘQNRareΛTandΛΘwithdiagonalelementsinreverseorder,andF=ThisfulltextpaperwaspeerreviewedatthedirectionofIEEECommunicationsSocietysubjectmatterexpertsforpublicationintheIEEE"GLOBECOM"2009proceedings.
978-1-4244-4148-8/09/$25.
002009QNRFQMR.
WiththefactthatΛTisadiagonalmatrix,theoptimizationproblem(27)canbereformulatedasminf0(b)=dT{ΛT}d{(FHΛΘF+IMR)1}bs.
t.
Tr(FHF)≤P,(31)wherethesymbold{Z}denotesthevectorformedfromthemaindiagonalofZ.
Noticethatbecauseofthepermutationmatrices,theorderofbisthereversetothatofthemaindiagonalofB(i.
e.
,theelementsofbareindecreasingorder).
TogetherwiththefactthatthediagonalelementsofΛTareinincreasingorder,thefunctionf0(b)isSchur-concave[11,3.
H.
3].
Basedon[12,Theorem1],theoptimalF=QNRFQMRfortheproblem(31)haszeroelementsexceptalongtherightmostmaindiagonal.
DeningN=min(Rank(ΛΘ),MR),theoptimalFhasthefollowingstruc-tureF=diag(f1,fN)0N*(MRN)0(NRN)*N0(NRN)*(MRN).
(32)With(32),theoptimizationproblem(27)canberewrittenasminf2iNi=1λT,iλΘ,if2i+1+MRi=N+1λT,is.
t.
Ni=1f2i≤P(33)whereλΘ,idenotestheithdiagonalelementofΛΘ.
Obvi-ously,thesolutionoftheproblem(33)isthemodiedwater-lling[13],andbasedontheKarush-Kuhn-Tucker(KKT)conditionsof(33),wehave[14]f2i,opt=λT,iμλΘ,i1λΘ,i+i=1,N(34)whereμ>0istheLagrangianmultipliersuchthatNi=1f2i,opt=Pholds.
FromthedenitionofFin(26),(32)and(34),wecanwritetheoptimalFcompactlyasFopt=UΘ,N1√μΛ12ΘΛ12TΛ1Θ+12UHT,NRx12(35)where[(Z)+]i,j=max(0,(Z)i,j).
ThematricesΛΘandΛTaretheprinciplesubmatricesofΛΘandΛTwithdimensionsN*N.
ThematricesUΘ,NandUT,NaretherstNcloumnsofUΘandUT,respectively.
Noticethatwhenthesource-relaylinkisnoiselessandthechannelrealizationisperfectlyknown,equation(35)reducestothepoint-to-pointMIMOrobustLMMSEtransceiver[15].
Ifbothtwochannelsareexactlyknown,(35)isexactlythesolutionin[5].
IV.
NUMERICALEXPERIMENTSInthissection,simulationresultswillbeshowntoverifytheeffectivenessoftheproposedalgorithm.
Inthispaper,thesource,relayanddestinationareallequippedwith3antennas.
Atthesource,itisassumedthatthetransmitpowerTr(Rs)=20dBandthemodulationschemeisQPSK.
Theestimatedchannelmatrices,HsrandHrd,areHsr=0.
27140.
3487i0.
61700.
4784i0.
2315+0.
5103i0.
2354+0.
2462i0.
3534+0.
1253i0.
19640.
7238i1.
18090.
3305i0.
3179+2.
3439i0.
19891.
1954iHrd=0.
90020.
4583i0.
96460.
6782i0.
9360+1.
1348i0.
9969+0.
1589i0.
2910+0.
3071i0.
60350.
4315i0.
67981.
1627i0.
7557+0.
3929i0.
37420.
0623i.
(36)TheestimationerrorcorrelationmatricesareassumedtobeΣsr=1ββ2β1ββ2β1Σrd=1ββ2β1ββ2β1Ψsr=0.
031αα2α1αα2α1Ψrd=0.
041αα2α1αα2α1.
(37)Ineachsimulationrun,channelestimationerrors,ΔHsrandΔHrd,aregeneratedindependently,accordingto(4)and(6),respectively,and1000trialsareaveragedtogiveeachpointinthegures.
Fig.
2showstheMSEofthereceivedsignalatthedesti-nationversusthetransmitpowerattherelayP,forthealgo-rithmusingestimatedchannelmatricesonlyandtheproposedBayesianalgorithm,withdifferentvaluesofβ,whenα=0.
4.
Itcanbeseenthatingeneral,thewholesystemperformancedegradeswhenthecorrelationfactorβincreases.
Thisisduetothefactthatchannelcorrelationsreducethenumberofeffectiveeigenchannels[6].
However,theperformanceoftheproposedalgorithmissignicantlybetterthanthealgorithmusingestimatedchannelmatricesonly,regardlessofthevalueofβ.
Fig.
3showsthecorrespondingresultsfordifferentvaluesofα,whenβ=0.
4.
AsimilarconclusiontothatofFig.
2canbedrawn.
V.
CONCLUSIONSInthispaper,wepresentedthejointdesignoflineartransceiversforAFMIMOrelaysystemsundertheknowledgeofestimatedchannelanderrorcovariancematrices.
Thestatis-ticsofchannelestimationerrorswereincorporatedintothetransceiverdesignusingtheBayesianframework.
Aclosed-formsolutionhasbeenderivedandtwoexistingalgorithmswereshowntobespecialcasesofourframework.
Fromthesimulations,itwasfoundthattheproposedalgorithmreducesthesensitivityoftherelaysystemtochannelestimationerrors,andimprovesthesystemperformancegreatly,comparedtothealgorithmusingestimatedchannelonly.
ThisfulltextpaperwaspeerreviewedatthedirectionofIEEECommunicationsSocietysubjectmatterexpertsforpublicationintheIEEE"GLOBECOM"2009proceedings.
978-1-4244-4148-8/09/$25.
002009051015202530100.
7100.
6100.
5100.
4100.
3100.
2α=0.
4P(dB)MSEAlgorithmbasedonestimatedCSI,β=0.
9AlgorithmbasedonestimatedCSI,β=0.
45AlgorithmbasedonestimatedCSI,β=0Proposedalgorithm,β=0.
9Proposedalgorithm,β=0.
45Proposedalgorithm,β=0β=0.
9β=0.
45β=0Fig.
2.
MSEversustransmitpowerattherelayforthealgorithmbasedonestimatedchannelandtheproposedBayesianalgorithm,withdifferentvaluesofβ,whenα=0.
4051015202530100.
7100.
6100.
5100.
4100.
3100.
2β=0.
4P(dB)MSEAlgorithmbasedonestimatedCSI,α=0.
9AlgorithmbasedonestimatedCSI,α=0.
45AlgorithmbasedonestimatedCSI,α=0Proposedalgorithm,α=0.
9Proposedalgorithm,α=0.
45Proposedalgorithm,α=0α=0.
9α=0.
45α=0Fig.
3.
MSEversustransmitpowerattherelayforthealgorithmbasedonestimatedchannelandtheproposedBayesianalgorithmwithdifferentvaluesofα,whenβ=0.
4APPENDIXIForpositiveHermitianmatrices,MandN,ifMN,thefollowinginequalityholds[10,7.
7.
4]N1M1.
(38)Furthermore,foranymatrixA,theinequalityAHN1AAHM1A(39)alwaysholds[10,7.
7.
3.
a].
Addinganidentitymatrixonbothsidesof(39),theinequalitysigndoesnotchange.
Togetherwith(38),wehave(AHM1A+I)1(AHN1A+I)1.
(40)Withtheresultin(39),foranarbitrarymatrixB,wehaveBH(AHM1A+I)1BBH(AHN1A+I)1B.
(41)PuttingA=HrdFRx12,B=Rx12HsrRs,N=KandM=Tr(FRxFH)λmax(Ψrd)Σrd+Rn2,andtakingthetraceonbothsidesof(41),wehaveMSEU(F)≥MSE(F)(42)whereMSEU(F)isdenedasMSEU(F)=Tr(RsHHsrRxH2(RxH2FHHHrdM1HrdFRx12+IMR)1Rx12HsrRs)+c,(43)andMSE(F)isdenedin(21).
ACKNOWLEDGEMENTThisstudywaspartiallysupportedbyagrantfromtheResearchGrantsCounciloftheHongKongSAR.
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ThisfulltextpaperwaspeerreviewedatthedirectionofIEEECommunicationsSocietysubjectmatterexpertsforpublicationintheIEEE"GLOBECOM"2009proceedings.
978-1-4244-4148-8/09/$25.
002009
hku.
hkAbstract—Inthispaper,weaddresstherobustlineartransceiverdesignfordual-hopamplify-and-forward(AF)MIMOrelaysystems,wherebothtransmittersandreceivershaveimperfectchannelstateinformation(CSI).
WiththestatisticsofchannelestimationerrorsinthetwohopsbeingGaussian,wefor-mulatetherobustlinear-minimum-mean-square-error(LMMSE)transceiverdesignproblemusingtheBayesianframework,andderiveaclosed-formsolution.
Simulationresultsshowthattheproposedalgorithmreducesthesensitivityoftherelaysystemtochannelestimationerrors,andperformsbetterthanthealgorithmusingestimatedchannelonly.
I.
INTRODUCTIONRecently,cooperativecommunicationhasgainedsignicantinterest,duetoitsgreatpotentialstoimprovereliability,coverageandcapacityofwirelesslinks[1][2].
Generallyspeaking,therearethreekindsofrelayprotocols,amplify-and-forward(AF),compress-and-forward(CF)anddecode-and-forward(DF).
Amongthethreeschemes,AFisconceptuallythesimplestone,inwhichtherelayjustscalesthesignaltrans-mittedfromthesource,andthentransmitstothedestination.
Duetoitssimplicityandlowimplementationcomplexity,AFstrategyhasreceivedmanyresearchers'attention.
Ontheotherhand,itiswell-knownthatinfullyscatteredenvironments,multiantennasystemsprovidespatialdiversityandmultiplexinggains.
Thiskindofbenetscanbedirectlyintroducedintocooperativecommunicationsviadeploymentofmultipleantennasattransmittersandreceivers.
Thecom-binationofAFandMIMOsystemsbringsgreatpotentialsinperformanceimprovement.
LineartransceiverdesignforAFMIMOrelaysystemshasbeenaddressedin[2],[3],[4]and[5].
ThecapacityscalinglawofMIMOrelaynetworkshasbeendiscussedin[2].
Thelineartransceiverdesignforthexedrelayincellularnet-workshasbeenaddressedin[3].
Jointlinear-minimum-mean-square-error(LMMSE)transceiverdesignforAFMIMOrelaysystemsisconsideredin[4]and[5].
However,alloftheabovementionedworksrequirethechannelstateinformation(CSI)perfectlyknownatthetransmittersandreceivers.
Unfortunately,inpracticalsystems,channelestimationer-rorsareinevitable,whichshouldbetakenintoaccountintransceiverdesign.
Inthispaper,weproposearobustlineartransceiverdesignmethodforAFMIMOrelaysystems.
TheTheRelayTheDestinationTheSourcesNRNRMDMFGsrHrdHssFig.
1.
Amplify-and-forwardMIMOrelaydiagramchannelestimationerrorsaremodeledasGaussianrandomvariables.
ThestatisticsofthechannelestimationerrorsareincorporatedintothedesignusingtheBayesianframework,andaclosed-formsolutionisobtained.
Simulationresultsshowthattheproposedalgorithmperformsbetterthanthealgorithmusingestimatedchannelonly.
Thefollowingnotationsareusedthroughoutthispaper.
Boldfacelowercaselettersdenotevectors,whileboldfaceuppercaselettersdenotematrices.
ThenotationZHdenotestheHermitianofthematrixZ,andTr(Z)isthetraceofthematrixZ.
ThesymbolIMdenotesanM*Midentitymatrix,while0M,NdenotesanM*Nallzeromatrix.
ThenotationZ12istheHermitiansquarerootofthepositivesemidenitematrixZ,suchthatZ12Z12=ZandZ12isalsoaHermitianmatrix.
II.
PROBLEMFORMULATIONInthispaper,adual-hopamplify-and-forward(AF)cooper-ativecommunicationsystemisconsidered.
Intheconsideredsystem,thereisonesourcewithNSantennas,onerelaywithMRreceiveantennasandNRtransmitantennas,andonedestinationwithMDantennas,asshowninFig.
1.
Atthersthop,thesourcetransmitsdatatotherelay.
Thereceivedsignal,x,attherelayisx=Hsrs+n1(1)wheresisthedatavectortransmittedbythesourcewiththecovariancematrixRs=E{ssH}.
ThematrixHsristheMIMOchannelmatrixbetweenthesourceandtherelay.
Symboln1istheadditiveGaussiannoisewithcovariancematrixRn1.
Attherelay,thereceivedsignalxismultipliedbyaprecodermatrixF,underapowerconstraintTr(FRxFH)≤ThisfulltextpaperwaspeerreviewedatthedirectionofIEEECommunicationsSocietysubjectmatterexpertsforpublicationintheIEEE"GLOBECOM"2009proceedings.
978-1-4244-4148-8/09/$25.
002009PwhereRx=E{xxH}andPisthemaximumtransmitpower.
Thentheresultingsignalistransmittedtothedestina-tion.
Thereceivedsignalatthedestination,y,canbewrittenasy=HrdFHsrs+HrdFn1+n2,(2)whereHrdistheMIMOchannelmatrixbetweentherelayandthedestination,andn2istheadditiveGaussiannoisevectoratthesecondhopwithcovariancematrixRn2.
Inordertoguaranteethetransmitteddatascanberecoveredatthedestination,itisassumedthatMR,NR,andMDaregreaterthanorequaltoNS[4].
Itisassumedthatboththerelayanddestinationhavetheestimatedchannelstateinformation(CSI).
Whenchannelestimationerrorsareconsidered,wehaveHsr=Hsr+ΔHsr,Hrd=Hrd+ΔHrd,(3)wherethesymbolsHsrandHrdaretheestimatedCSI,whileΔHsrandΔHrdarethecorrespondingchannelestimationerrorswhoseelementsarezeromeanGaussianrandomvari-ables.
Ingeneral,theMR*NSmatrixΔHsrcanbewrittenasΔHsr=Σ12srHWΨ12srwheretheelementsoftheMR*NSmatrixHWareindependentandidenticallydistributed(i.
i.
d.
)Gaussianrandomvariableswithzeromeanandunitvariance.
TheMR*MRmatrixΣsrandNS*NSmatrixΨTsraretherowandcolumncovariancematricesofΔHsr,respectively[6].
Itiseasytoseethatvec(ΔHTsr)CN(0MR*NS,ΣsrΨTsr)basedonwhichΔHsrissaidtohaveamatrix-variatecomplexGaussiandistribution,whichcanbewrittenas[7]ΔHsrCNMR,NS(0MR,NS,ΣsrΨTsr),(4)withtheprobabilitydensityfunction(p.
d.
f.
)givenby[8][9]f(ΔHsr)=exp(Tr((ΔHsr0)HΣ1sr(ΔHsr0)Ψ1sr))(π)NSMRdet(Σsr)NSdet(Ψsr)MR.
(5)Similarly,fortheestimationerrorinthesecondhop,wehaveΔHrdCNMD,NR(0MD,NR,ΣrdΨTrd)(6)wheretheMD*MDmatrixΣrdandNR*NRmatrixΨTrdaretherowandcolumncovariancematricesofΔHrd,respectively.
Itisassumedthatthechannelestimationerrors,ΔHsrandΔHrd,areindependent.
Atthedestination,alinearequalizerGisadoptedtodetectthetransmitteddatas.
TheproblemishowtodesignthelinearprecodermatrixFattherelayandthelinearequalizerGatthedestinationtominimizethemeansquareerrors(MSE)ofthereceiveddataatthedestination:MSE(F,G)=E{Tr(Gys)(Gys)H},(7)wheretheexpectationistakenwithrespecttos,ΔHsr,ΔHrd,n1andn2.
III.
ROBUSTTRANSCEIVERDESIGNFORMIMORELAYA.
MSEAveragedoverChannelUncertaintiesSinces,n1andn2areindependent,theMSEexpression(7)canbewrittenasMSE(F,G)=E{(GHrdFHsrINS)s+GHrdFn1+Gn22}=EΔHsr,ΔHrd{Tr((GHrdFHsrI)Rs(GHrdFHsrI)H)}+EΔHrd{Tr(GHrdF)Rn1(GHrdF)H}+Tr(GRn2GH)=EΔHsr,ΔHrd{Tr(GHrdFHsr)Rs(GHrdFHsr)H}+TrGEΔHrd{HrdFRn1FHHHrd}GHTrRs(GHrdFHsr)HTrGHrdFHsrRs+Tr(Rs)+Tr(GRn2GH).
(8)BecauseΔHsrandΔHrdareindependent,thersttermofMSEisEΔHsr,ΔHrd{Tr(GHrdFHsr)Rs(GHrdFHsr)H}=TrGEΔHrdHrdFEΔHsr{HsrRsHHsr}FHHHrdGH.
(9)Fortheinnerexpectation,duetothefactthatthedistributionofΔHsrismatrix-variatecomplexGaussianwithzeromean,thefollowingequationholds[7]EΔHsr{HsrRsHHsr}=EΔHsr{(Hsr+ΔHsr)Rs(Hsr+ΔHsr)H}=Tr(RsΨsr)Σsr+HsrRsHHsrΠ0.
(10)Applying(10)andthecorrespondingresultforΔHrdto(9),thersttermofMSEbecomesTrGEΔHrdHrdFEΔHsr{HsrRsHHsr}FHHHrdGH=Tr(G(Tr(FΠ0FHΨrd)Σrd+HrdFΠ0FHHHrd)GH).
(11)Similarly,thesecondtermofMSEin(8)canbesimpliedasTrGEΔHrd{HrdFRn1FHHHrd}GH=Tr(GTr(FRn1FHΨrd)Σrd+HrdFRn1FHHHrdGH).
(12)Basedon(11)and(12),theMSE(8)equalstoMSE(F,G)=TrG(HrdFRxFHHHrd+K)GHTrRsHHsrFHHHrdGHTrRsGHrdFHsr+Tr(Rs)(13)whereRx=Π0+Rn1(14)K=Tr(F(Π0+Rn1)FHΨrd)Σrd+Rn2.
(15)NoticethatthematrixRxistheautocorrelationmatrixofthereceivesignalxattherelay.
ThisfulltextpaperwaspeerreviewedatthedirectionofIEEECommunicationsSocietysubjectmatterexpertsforpublicationintheIEEE"GLOBECOM"2009proceedings.
978-1-4244-4148-8/09/$25.
002009B.
JointRobustDesignofEqualizerandPrecoderSubjecttothetransmitpowerconstraintattherelay,thejointdesignofequalizerandprecodercanbeexpressedasthefollowingoptimizationproblemminF,GMSE(F,G)s.
t.
Tr(FRxFH)≤P.
(16)SincetheconstraintdoesnotinvolvetheequalizerG,whentheprecodermatrixFisxed,theoptimallinearequalizer,Gopt,satisesthefollowingconditionMSE(F,G)G=0,(17)basedonwhichwehaveGopt=Rs(HrdFHsr)H(HrdFRxFHHHrd+K)1.
(18)Substituting(18)into(13),theMSEatthedestinationequalstoMSE(F)=Tr(Rs)Tr(RsHHsr[FHHHrd(HrdFRxFHHHrd+K)1HrdF]HsrRs).
(19)SinceK12andRx12arebothHermitianmatrices,exploitingthematrixinversionlemma,wehaveFHHHrd(HrdFRxFHHHrd+K)1HrdF=RxH2RxH2FHHHrdKH2(K12HrdFRx12RxH2FHHHrdKH2+IMD)1K12HrdFRx12Rx12=Rx1RxH2(RxH2FHHHrdK1HrdFRx12+IMR)1Rx12.
(20)Putting(20)into(19),anddeningtheconstantpartcTr(Rs)Tr(RsHHsrRx1HsrRs),equation(19)canberewrittenasMSE(F)=Tr(RsHHsrRxH2(RxH2FHHHrdK1HrdFRx12+IMR)1Rx12HsrRs)+c.
(21)From(15),K=Tr(FRxFHΨrd)Σrd+Rn2,soMSE(F)isahighorderfunctionofFandtheproblemofminimizing(21)isverydifculttosolve.
Inordertoproceed,noticethat[10]Tr(FRxFH)λmax(Ψrd)Σrd+Rn2K,(22)whereλmax(Z)denotesthelargesteigenvalueofZ.
SincefortheminimumMSE,thecorrespondingtransmitpowermustbeontheboundary(i.
e.
,Tr(FRxFH)=P),wehavePλmax(Ψrd)Σrd+Rn2K.
AsshowninAp-pendixI,whenKin(21)isreplacedbyitsupper-boundΦPλmax(Ψrd)Σrd+Rn2,theresultantMSEexpression,denotedasMSEU(F)anddenedin(43),isanupper-boundofMSE(F)(i.
e.
,MSEU(F)≥MSE(F)).
Therefore,weproposetodesigntheprecoderFbyminimizingMSEU(F),whichcorrespondstominFTr(Rx12HsrRsRsHHsrRxH2T(RxH2FHHHrdΦ1HrdΘFRx12+IMR)1)s.
t.
Tr(FRxFH)≤P(23)wheretheconstantcisneglected,whichdoesnotaffecttheoptimizationproblem.
NoticethatwhenΨrd∝I,thereplacementinvolvesnoapproximation.
Basedoneigendecompostion,wehaveT=UTΛTUHT,(24)Θ=UΘΛΘUHΘ,(25)wherethematricesUTandUΘconsistoftheeigenvectorsofTandΘ,respectively,whilethediagonalmatricesΛTandΛΘcontainstheeigenvaluesofTandΘ,respectively.
Withoutlossofgenerality,itisassumedthatthediagonalelementsofΛTandΛΘareindecreasingorder.
Substituting(24)and(25)into(23)anddeningFUHΘFRx12UT,(26)theoptimizationproblemcanbewritteninacompactformasminFTrΛT(FHΛΘF+IMR)1s.
t.
Tr(FFH)≤P.
(27)Fortheobjectivefunctionof(27),noticethatTr(ΛTB)≥MRi=1λT,iλB,MRi+1(28)whereBisdenedasB(FHΛΘF+IMR)1,λB,iistheithlargesteigenvalueofB,andthesymbolλT,idenotestheithdiagonalelementofΛT.
In(28),theequalityholdswhenthematrixBisdiagonalwithdiagonalelementsinincreasingorder[11,9.
H.
1.
h].
Therefore,fortheoptimalsolution,(FHΛΘF+IMR)1mustbediagonalwithdiagonalelementsinincreasingorder.
Basedonthediagonalstructure,introducingapermutationmatrixwithdimensionM*MasQM=0010.
.
.
0100M,(29)theobjectivefunctionoftheoptimizationproblem(27)canberewrittenasTrΛT(FHΛΘF+IMR)1=TrΛT(FHΛΘF+IMR)1(30)whereΛT=QMRΛTQMRandΛΘ=QNRΛΘQNRareΛTandΛΘwithdiagonalelementsinreverseorder,andF=ThisfulltextpaperwaspeerreviewedatthedirectionofIEEECommunicationsSocietysubjectmatterexpertsforpublicationintheIEEE"GLOBECOM"2009proceedings.
978-1-4244-4148-8/09/$25.
002009QNRFQMR.
WiththefactthatΛTisadiagonalmatrix,theoptimizationproblem(27)canbereformulatedasminf0(b)=dT{ΛT}d{(FHΛΘF+IMR)1}bs.
t.
Tr(FHF)≤P,(31)wherethesymbold{Z}denotesthevectorformedfromthemaindiagonalofZ.
Noticethatbecauseofthepermutationmatrices,theorderofbisthereversetothatofthemaindiagonalofB(i.
e.
,theelementsofbareindecreasingorder).
TogetherwiththefactthatthediagonalelementsofΛTareinincreasingorder,thefunctionf0(b)isSchur-concave[11,3.
H.
3].
Basedon[12,Theorem1],theoptimalF=QNRFQMRfortheproblem(31)haszeroelementsexceptalongtherightmostmaindiagonal.
DeningN=min(Rank(ΛΘ),MR),theoptimalFhasthefollowingstruc-tureF=diag(f1,fN)0N*(MRN)0(NRN)*N0(NRN)*(MRN).
(32)With(32),theoptimizationproblem(27)canberewrittenasminf2iNi=1λT,iλΘ,if2i+1+MRi=N+1λT,is.
t.
Ni=1f2i≤P(33)whereλΘ,idenotestheithdiagonalelementofΛΘ.
Obvi-ously,thesolutionoftheproblem(33)isthemodiedwater-lling[13],andbasedontheKarush-Kuhn-Tucker(KKT)conditionsof(33),wehave[14]f2i,opt=λT,iμλΘ,i1λΘ,i+i=1,N(34)whereμ>0istheLagrangianmultipliersuchthatNi=1f2i,opt=Pholds.
FromthedenitionofFin(26),(32)and(34),wecanwritetheoptimalFcompactlyasFopt=UΘ,N1√μΛ12ΘΛ12TΛ1Θ+12UHT,NRx12(35)where[(Z)+]i,j=max(0,(Z)i,j).
ThematricesΛΘandΛTaretheprinciplesubmatricesofΛΘandΛTwithdimensionsN*N.
ThematricesUΘ,NandUT,NaretherstNcloumnsofUΘandUT,respectively.
Noticethatwhenthesource-relaylinkisnoiselessandthechannelrealizationisperfectlyknown,equation(35)reducestothepoint-to-pointMIMOrobustLMMSEtransceiver[15].
Ifbothtwochannelsareexactlyknown,(35)isexactlythesolutionin[5].
IV.
NUMERICALEXPERIMENTSInthissection,simulationresultswillbeshowntoverifytheeffectivenessoftheproposedalgorithm.
Inthispaper,thesource,relayanddestinationareallequippedwith3antennas.
Atthesource,itisassumedthatthetransmitpowerTr(Rs)=20dBandthemodulationschemeisQPSK.
Theestimatedchannelmatrices,HsrandHrd,areHsr=0.
27140.
3487i0.
61700.
4784i0.
2315+0.
5103i0.
2354+0.
2462i0.
3534+0.
1253i0.
19640.
7238i1.
18090.
3305i0.
3179+2.
3439i0.
19891.
1954iHrd=0.
90020.
4583i0.
96460.
6782i0.
9360+1.
1348i0.
9969+0.
1589i0.
2910+0.
3071i0.
60350.
4315i0.
67981.
1627i0.
7557+0.
3929i0.
37420.
0623i.
(36)TheestimationerrorcorrelationmatricesareassumedtobeΣsr=1ββ2β1ββ2β1Σrd=1ββ2β1ββ2β1Ψsr=0.
031αα2α1αα2α1Ψrd=0.
041αα2α1αα2α1.
(37)Ineachsimulationrun,channelestimationerrors,ΔHsrandΔHrd,aregeneratedindependently,accordingto(4)and(6),respectively,and1000trialsareaveragedtogiveeachpointinthegures.
Fig.
2showstheMSEofthereceivedsignalatthedesti-nationversusthetransmitpowerattherelayP,forthealgo-rithmusingestimatedchannelmatricesonlyandtheproposedBayesianalgorithm,withdifferentvaluesofβ,whenα=0.
4.
Itcanbeseenthatingeneral,thewholesystemperformancedegradeswhenthecorrelationfactorβincreases.
Thisisduetothefactthatchannelcorrelationsreducethenumberofeffectiveeigenchannels[6].
However,theperformanceoftheproposedalgorithmissignicantlybetterthanthealgorithmusingestimatedchannelmatricesonly,regardlessofthevalueofβ.
Fig.
3showsthecorrespondingresultsfordifferentvaluesofα,whenβ=0.
4.
AsimilarconclusiontothatofFig.
2canbedrawn.
V.
CONCLUSIONSInthispaper,wepresentedthejointdesignoflineartransceiversforAFMIMOrelaysystemsundertheknowledgeofestimatedchannelanderrorcovariancematrices.
Thestatis-ticsofchannelestimationerrorswereincorporatedintothetransceiverdesignusingtheBayesianframework.
Aclosed-formsolutionhasbeenderivedandtwoexistingalgorithmswereshowntobespecialcasesofourframework.
Fromthesimulations,itwasfoundthattheproposedalgorithmreducesthesensitivityoftherelaysystemtochannelestimationerrors,andimprovesthesystemperformancegreatly,comparedtothealgorithmusingestimatedchannelonly.
ThisfulltextpaperwaspeerreviewedatthedirectionofIEEECommunicationsSocietysubjectmatterexpertsforpublicationintheIEEE"GLOBECOM"2009proceedings.
978-1-4244-4148-8/09/$25.
002009051015202530100.
7100.
6100.
5100.
4100.
3100.
2α=0.
4P(dB)MSEAlgorithmbasedonestimatedCSI,β=0.
9AlgorithmbasedonestimatedCSI,β=0.
45AlgorithmbasedonestimatedCSI,β=0Proposedalgorithm,β=0.
9Proposedalgorithm,β=0.
45Proposedalgorithm,β=0β=0.
9β=0.
45β=0Fig.
2.
MSEversustransmitpowerattherelayforthealgorithmbasedonestimatedchannelandtheproposedBayesianalgorithm,withdifferentvaluesofβ,whenα=0.
4051015202530100.
7100.
6100.
5100.
4100.
3100.
2β=0.
4P(dB)MSEAlgorithmbasedonestimatedCSI,α=0.
9AlgorithmbasedonestimatedCSI,α=0.
45AlgorithmbasedonestimatedCSI,α=0Proposedalgorithm,α=0.
9Proposedalgorithm,α=0.
45Proposedalgorithm,α=0α=0.
9α=0.
45α=0Fig.
3.
MSEversustransmitpowerattherelayforthealgorithmbasedonestimatedchannelandtheproposedBayesianalgorithmwithdifferentvaluesofα,whenβ=0.
4APPENDIXIForpositiveHermitianmatrices,MandN,ifMN,thefollowinginequalityholds[10,7.
7.
4]N1M1.
(38)Furthermore,foranymatrixA,theinequalityAHN1AAHM1A(39)alwaysholds[10,7.
7.
3.
a].
Addinganidentitymatrixonbothsidesof(39),theinequalitysigndoesnotchange.
Togetherwith(38),wehave(AHM1A+I)1(AHN1A+I)1.
(40)Withtheresultin(39),foranarbitrarymatrixB,wehaveBH(AHM1A+I)1BBH(AHN1A+I)1B.
(41)PuttingA=HrdFRx12,B=Rx12HsrRs,N=KandM=Tr(FRxFH)λmax(Ψrd)Σrd+Rn2,andtakingthetraceonbothsidesof(41),wehaveMSEU(F)≥MSE(F)(42)whereMSEU(F)isdenedasMSEU(F)=Tr(RsHHsrRxH2(RxH2FHHHrdM1HrdFRx12+IMR)1Rx12HsrRs)+c,(43)andMSE(F)isdenedin(21).
ACKNOWLEDGEMENTThisstudywaspartiallysupportedbyagrantfromtheResearchGrantsCounciloftheHongKongSAR.
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002009
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