A new approach for capacity analysis of large dime

arXiv:cs/0612076v1[cs.IT]15Dec2006SUBMITTEDTOIEEETRANSACTIONSONINFORMATIONTHEORY1ANewApproachforCapacityAnalysisofLargeDimensionalMulti-AntennaChannelsW.Hachem(∗),O.Khorunzhiy,Ph.Loubaton,J.NajimandL.PasturAbstractThispaperadressesthebehaviourofthemutualinformationofcorrelatedMIMORayleighchannelswhenthenumbersoftransmitandreceiveantennasconvergeto+∞atthesamerate.UsinganewandsimpleapproachbasedonPoincar´e-Nashinequalityandonanintegrationbypartsformula,itisrigorouslyestablishedthatthemutualinformationconvergestoaGaussianrandomvariablewhosemeanandvarianceareevaluated.Theseresultsconfirmpreviousevaluationsbasedonthepowerfulbutnonrigorousreplicamethod.Itisbelievedthatthetoolsthatareusedinthispaperaresimple,robust,andofinterestforthecommunicationsengineeringcommunity.IndexTermsCentralLimitTheorem,CorrelatedMIMOChannels,LargeRandomMatrixTheory,MutualInfor-mation,Poincar´e-NashInequality.I.INTRODUCTIONItiswidelyknownthathighspectralefficienciesareattainedwhenmultipleantennasareusedatboththetransmitterandthereceiverofawirelesscommunicationsystem.Indeed,duetothemobilityandThisworkwaspartiallysupportedbythe“FondsNationaldelaScience”viatheACIprogram“NouvellesInterfacesdesMath´ematiques”,projectMALCOMnumber205.W.HachemandJ.NajimarewithCNRSandENST(UMR5141),Paris,France.{hachem,najim}@enst.fr,O.KhorunzhiyiswithEquipe”Probabilit´es-Statistiques”Universit´edeVersailles-Saint-Quentin,[e-mail],P.LoubatoniswithIGMLabInfo,UMR8049,InstitutGaspardMonge,Universit´edeMarneLaVall´ee,[e-mail],L.PasturiswithKharkivNationalUniversity-InstituteforLowTemperaturePhysics-Kharkiv,[e-mail].(*)Correspondingauthor.15December2006DRAFTSUBMITTEDTOIEEETRANSACTIONSONINFORMATIONTHEORY2tothepresenceofalargenumberofreflectedandscatteredsignalpaths,theelementsoftheN×nMultipleInputMultipleOutput(MIMO)channelmatrixwithNantennasatthereceiver’ssiteandnantennasatthetransmitter’sareoftenmodeledasrandomvariables.Assumingarandommodelforthismatrix,Telatarrealizedinthemid-ninetiesthatShannon’scapacityofsuchchannelsincreasesattherateofmin(N,n)forafixedtransmissionpower[1].AresultofthesamenaturecanbefoundintheworkofFoschiniandGans[2].Theauthorsof[1]and[2]assumedthattheelementsofthechannelmatrixGarecentered,independentandidenticallydistributed(i.i.d.)elements.Inthiscontext,awellknownresultinRandomMatrixTheory(RMT)[3]saysthattheeigenvaluedistributionoftheGrammatrixGG∗whereG∗istheHermitianadjointofGconvergestoadeterministicprobabilitydistributionasngoestoinfinityandN/nconvergestoaconstantc0.DenotebyI(ρ)=logdetρnGG∗+IN
thecapacityofchannelGforaSignaltoNoiseRatioatareceiverantennaequaltoρ/n.Oneconsequenceof[3]isthatthecapacitypertransmitantennaI(ρ)/n,beinganintegralofalogfunctionwithrespecttotheempiricaleigenvaluedistributionofGG∗,convergestoaconstant.Thisfactalreadyobservedin[1]sustainstheassertionofthelinearincreaseofcapacitywiththenumberofantennas.Inaddition,thisconvergenceprovestobesufficientlyfast.Asamatteroffact,theasymptoticresultspredictedbytheRMTremainrelevantforsystemswithamoderatenumberofantennas.Thenextstepwastoapplythistheorytochannelmodelsthatincludeacorrelationbetweenpaths(orentriesofG).OneofthemainpurposesofthisgeneralizationistobetterunderstandtheimpactofthesecorrelationsonShannon’smutualinformation.Letusciteinthiscontextthecontributions[4],[5],[6],[7]and[8],alldevotedtothestudyofthemutualinformationinthecasewheretheelementsofchannel’smatrixarecenteredandcorrelatedrandomvariables.In[9],adeterministicequivalentiscomputedunderbroadconditionsforthecapacitybasedonRicechannelsmodeledbynon-centeredmatriceswithindependentbutnotidenticallydistributedrandomvariables.Thelinkbetweenmatriceswithcorrelatedentriesandmatriceswithindependententriesandavarianceprofileisstudiedin[10].Oneofthemostpopularcorrelatedchannelmodelsusedforthesecapacityevaluationsistheso-calledKroneckermodelG=ΨWeΨwhereWisaN×nmatrixwithGaussiancenteredi.i.d.entries,andΨandeΨareN×Nandn×nmatricesthatcapturethepathcorrelationsatthereceiverandatthetransmittersidesrespectively[11],[12].ThismodelhasbeenstudiedbyChuahet.al.in[5].WithsomeassumptionsonmatricesΨandeΨ,theseauthorsshowedthatI(ρ)/nconvergestoadeterministicquantitydefinedasthefixedpointofanintegralequation.Lateron,Tulinoet.al.[8]obtainedthelimitofI(ρ)/nforacorrelationmodelmoregeneralthantheKroneckermodel.BoththeseworksrelyonaresultofGirko15December2006DRAFTSUBMITTEDTOIEEETRANSACTIONSONINFORMATIONTHEORY3describingtheeigenvaluedistributionoftheGrammatrixassociatedwithamatrixwithindependentbutnonnecessarilyidenticallydistributedentries,aclosemodelasweshallseeinamoment.In[7],Moustakaset.al.studiedthemutualinformationfortheKroneckermodelbyusingtheso-calledreplicamethod.TheyfoundanapproximationV(ρ)ofE[I(ρ)]accuratetotheorder1/ninthelargenregime.Usingthissamemethod,theyalsoshowedthatthevarianceofI(ρ)−V(ρ)isoforderoneandwereabletoderivethisvarianceforlargen.Althoughthereplicatechniqueispowerfulandhasawiderangeofapplications,therigorousjustificationofsomeofitspartsremainstobedone.Inthispaper,wep