capacity of multi-antenna Gaussian channels

1CommunicationTheory]CapacityofMulti-antennaGaussianChannels*EMRETELATARLueentTechnologiesBellLaboratories,600MountainAvenue,MurrayHill,NJ07974,[e-mail]tipletransmittingandorreceivingantennasforsingleusercommunicationsovertheadditiveGaussianchannelwithandwithoutfading.Wederiveformulasforthecapacitiesanderrorexponentsofsuchchannels,anddescribecomputationalprocedurestoevaluatesuchformulas.Weshowthatthepotentialgainsofsuchmulti-antennasystemsoversingle-antennasystemsisratherlargeunderindependenceassumptionsforthefadesandnoisesatdifferentreceivingantennas.1INTRODUCTIONWewillconsiderasingleuserGaussianchannelwithmultipletransmittingandorreceivingantennas.Wewilldenotethenumberoftransmittingantennasbytandthenumberofreceivingantennasbyr.WewillexclusivelydealwithalinearmodelinwhichthereceivedvectoryECrdependsonthetransmittedvectorxE([Iviay=Hx+n(1)whereHisarxtcomplexmatrixandniszero-meancomplexGaussiannoisewithindependent,equalvariancerealandimaginaryparts.WeassumeE[nnt]=Ir,thatis,thenoisescorruptingthedifferentreceiversareindepen-dent.ThetransmitterisconstrainedinitstotalpowertoP,E[xtx]5P.Equivalently,sincextx=tr(xxt),andexpec-tationandtracecommute,Thissecondformofthepowerconstraintwillprovemoreusefulintheupcomingdiscussion.WewillconsiderseveralscenariosforthematrixH:1.Hisdeterministic.2.Hisarandommatrix(forwhichweshallusethenotationH),chosenaccordingtoaprobabilitydistri-bution,andeachuseofthechannelcorrespondstoanindependentrealizationofH.3.Hisarandommatrix,butisfixedonceitischosen.'InvitedpaperThemainfocusofthispaperinonthelasttwoofthesecases.Thefirstcaseisincludedsoastoexposethetech-niquesusedinthelatercasesinamorefamiliarcontext.InthecaseswhenHisrandom,wewillassumethatitsentriesformani.i.d.Gaussiancollectionwithzero-mean,independentrealandimaginaryparts,eachwithvariance1/2.Equivalently,eachentryofHhasuniformphaseandRayleighmagnitude.ThischoicemodelsaRayIeighfadingenvironmentwithenoughseparationwithinthereceivingantennasandthetransmittingantennassuchthatthefadesforeachtransmitting-receivingantennapairareindepen-dent.Inallcases,wewillassumethattherealizationofHisknowntothereceiver,or,equivalently,thechannelout-putconsistsofthepair(y,H),andthedisfributionofHisknownatthetransmitter.2PRELIMINARIESAcomplexrandomvectorxEC?issaidtobeGaussianiftherealrandomvectordER2consistingofitsrealandimaginaryparts,2=[!lit(')],3m(x)isGaussian.Thus,tospecifythedistributionofacomplexGaussianrandomvectorx,itisnecessarytospecifytheexpectationandcovarianceof3,namely,E[2]EEl2andE[(d-E[2])(2-E[IZ])']ER2nx2n.WewillsaythatacomplexGaussianrandomvectorxiscircularlysymmetricifthecovarianceofthecorrespondingdhasthestructureVol.10,No.6,November-December1999585E.TelatarforsomeHermitiannon-negativedefiniteQErc.NotethattherealpartofanHermitianmatrixissymmetricandtheimaginarypartofanHermitianmatrixisanti-symmetricandthusthematrixappearingin(3)isrealandsymmetric.InthiscaseZ[(X-E[x])(x-!€[XI)+]=Q,andthus,acircularlysymmetriccomplexGaussianran-domvectorxisspecifiedbyprescribing'€[XIandE[(x-qXl)(x-mI,t].ForanyzECandAE[PxmdefineLemma1.Themappingsz-+2=[;:',:;]andA+A=Rc(A)-lim(A)havethefollowingproperties:Proof:Theproperties(4a),(4b)and(4c)areimmediate.(4d)followsfrom(4a)andthefactthat1,=12,.(4e)fol-lowsfromdet(A)=det([A:]A[*I-il[I)=det([laA):*I)=det(A)det(A)',(4f),(4g)and(4h)areimmediate.Corollary1.(IE(TXisunitaoifa?zdonly$0EPZnxZnisorthonormal.Corollary2./fQElPxisnon-negativedefinitethensoisQp211xZnTheprobabilitydensity(withrespecttothestandardLebesguemeasureonC')ofacircularlysymmetriccom-plexGaussianwithmeanpandcovarianceQisgivenbyY~,Q(~)=det(nQ)-'/'exp(-(f-fi)tQ-'(2-fi))=det(rQ)-'exp(-(x-p)'Q-'(x-p))wherethesecondequalityfollowsfrom(4d)-(4h).Thedif-ferentialentropyofacomplexGaussianxwithcovarianceQisgivenbyH(7Q)=!-?%a[-logYQ(x)l=logdet(nQ)+(loge)'E[xtQ-'x]=logdet(nQ)+(Ioge)tr('€[xxt]Q-')=logdet(nQ)+(loge)tr(1)=logdet(nee).Forus,theimportanceofthecircularlysymmetriccomplexGaussiansisduetothefollowinglemma:circularlysym-metriccomplexGaussiansareentropymaximizers.Lemma2.SupposethecomplexrandomvectorxEiciszero-meanandsatisfies!E[xx']=Q,i.e.,E[x~x*.]J=Qij,15i,j511.Thentheentropyofxsatisfies%(x)5logdet(neQ)witheqriafityifandoniyifxisctcircirlariysymmetriccomplexGaussianwith'€[xxt]=Q.Proof:Letpbeanydensityfunctionsatisfying!E,[xix;]=Qi,,15i,j5n.Letyp(x)=det(nQ)-lexp(-xtQ-'x).Observethat'EyQ[~ixf]=Qij,andthatlogye(x)isalin-earcombinationofthetermsxi$.ThusqQ[log7~(x)]=Ep[logy~(XI].Then,@(PI-=-2P[10gp(x)]f!-?%~[l~gTQ(~)lL0,withequalityonlyifp=7p.Thus@(p)5H(yp).0Lemma3.lfxEC'isacircularlysymmetriccomplexGaussianthensoisy=AxforanyAEPxn.Proof:Wemayassumexiszero-mean.LetQ=~[xxt].Thenyiszero-mean,3=A%,andqyjq=A55BBtjAf=$AQ$-=ZK',.whereK=AQA'.0Lemma4.lfxandyareindependentcirciilarlysymmetriccoii~plexGairssians,thetiz=xi-yisacirc~rlnrlysymtnet-riccornplL..rGaussian.Pmlf:LetA=E[xxt]andB=E[yy].Then'€[%?I=gcwithC=A+5.0CamcityofMulti-antennaGaussianChannels3THEGAUSSIANCHANNELWITHFIXEDTRANSFERFUNCTIONWewillstartbyremindingourselve