Capacity Limits of MIMO Channels

684IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.21,NO.5,JUNE2003CapacityLimitsofMIMOChannelsAndreaGoldsmith,SeniorMember,IEEE,SyedAliJafar,StudentMember,IEEE,NiharJindal,StudentMember,IEEE,andSriramVishwanath,StudentMember,IEEEInvitedPaperAbstract—WeprovideanoverviewoftheextensiverecentresultsontheShannoncapacityofsingle-userandmultiusermultiple-inputmultiple-output(MIMO)channels.Althoughenormouscapacitygainshavebeenpredictedforsuchchannels,thesepredictionsarebasedonsomewhatunrealisticassumptionsabouttheunderlyingtime-varyingchannelmodelandhowwellitcanbetrackedatthereceiver,aswellasatthetransmitter.MorerealisticassumptionscandramaticallyimpactthepotentialcapacitygainsofMIMOtechniques.Fortime-varyingMIMOchannelstherearemultipleShannontheoreticcapacitydefinitionsand,foreachdefinition,differentcorrelationmodelsandchannelinformationassumptionsthatweconsider.Wefirstprovideacomprehensivesummaryofergodicandcapacityversusoutageresultsforsingle-userMIMOchannels.Theseresultsindicatethatthecapacitygainobtainedfrommultipleantennasheavilydependsontheavailablechannelinformationateitherthereceiverortransmitter,thechannelsignal-to-noiseratio,andthecorrelationbetweenthechannelgainsoneachantennaelement.Wethenfocusattentiononthecapacityregionofthemultiple-accesschannels(MACs)andthelargestknownachievablerateregionforthebroadcastchannel.Incontrasttosingle-userMIMOchannels,capacityresultsforthesemultiuserMIMOchannelsarequitedifficulttoobtain,evenforconstantchannels.WesummarizeresultsfortheMIMObroadcastandMACforchannelsthatareeitherconstantorfadingwithperfectinstantaneousknowledgeoftheantennagainsatbothtransmitter(s)andreceiver(s).WeshowthatthecapacityregionoftheMIMOmultipleaccessandthelargestknownachievablerateregion(calledthedirty-paperregion)fortheMIMObroadcastchannelareintimatelyrelatedviaadualitytransformation.ThistransformationfacilitatesfindingthetransmissionstrategiesthatachieveapointontheboundaryoftheMIMOMACcapacityregionintermsofthetransmissionstrategiesoftheMIMObroadcastdirty-paperregionandvice-versa.Finally,wediscusscapacityresultsformulticellMIMOchannelswithbasestationcooperation.Thebasestationsthenactasaspatiallydiverseantennaarrayandtransmissionstrategiesthatexploitthisstructureexhibitsignificantcapacitygains.ThissectionalsoprovidesabriefdiscussionofsystemlevelissuesassociatedwithMIMOcellular.Openproblemsinthisfieldaboundandarediscussedthroughoutthepaper.IndexTerms—Antennacorrelation,beamforming,broadcastchannels(BCs),channeldistributioninformation(CDI),channelstateinformation(CSI),multicellsystems,multiple-accesschan-nels(MACs),multiple-inputmultiple-output(MIMO)channels,multiusersystems,Shannoncapacity.ManuscriptreceivedNovember8,2002;revisedJanuary31,2003.ThisworkwassupportedinpartbytheOfficeofNavalResearch(ONR)underGrantsN00014-99-1-0578andN00014-02-1-0003.TheworkofS.VishwanathwassupportedbyaStanfordGraduateFellowship.TheauthorsarewiththeDepartmentofElectricalEngineering,StanfordUniversity,Stanford,CA94305USA(e-mail:[e-mail];[e-mail];[e-mail];[e-mail]).DigitalObjectIdentifier10.1109/JSAC.2003.810294I.INTRODUCTIONWIRELESSsystemscontinuetostriveforeverhigherdatarates.Thisgoalisparticularlychallengingforsystemsthatarepower,bandwidth,andcomplexitylimited.However,anotherdomaincanbeexploitedtosignificantlyincreasechannelcapacity:theuseofmultipletransmitandreceiveantennas.PioneeringworkbyWinters[81],Foschini[20],andTelatar[69]ignitedmuchinterestinthisareabypredictingremarkablespectralefficienciesforwirelesssystemswithmultipleantennaswhenthechannelexhibitsrichscat-teringanditsvariationscanbeaccuratelytracked.Thisinitialpromiseofexceptionalspectralefficiencyalmost“forfree”resultedinanexplosionofresearchactivitytocharacterizethetheoreticalandpracticalissuesassociatedwithmultiple-inputmultiple-output(MIMO)wirelesschannelsandtoextendtheseconceptstomultiusersystems.ThistutorialsummarizesthesegmentofthisrecentworkfocusedonthecapacityofMIMOsystemsforbothsingle-usersandmultipleusersunderdifferentassumptionsaboutspatialcorrelationandchannelinformationavailableatthetransmitterandreceiver.ThelargespectralefficienciesassociatedwithMIMOchan-nelsarebasedonthepremisethatarichscatteringenvironmentprovidesindependenttransmissionpathsfromeachtransmitan-tennatoeachreceiveantenna.Therefore,forsingle-usersys-tems,atransmissionandreceptionstrategythatexploitsthisstructureachievescapacityonapproximatelysepa-ratechannels,whereisthenumberoftransmitantennasandisthenumberofreceiveantennas.Thus,capacityscaleslin-earlywithrelativetoasystemwithjustonetransmitandonereceiveantenna.Thiscapacityincreaserequiresascat-teringenvironmentsuchthatthematrixofchannelgainsbe-tweentransmitandreceiveantennapairshasfullrankandin-dependententriesandthatperfectestimatesofthesegainsareavailableatthereceiver.Perfectestimatesofthesegainsatboththetransmitterandreceiverprovidesanincreaseintheconstantmultiplierassociatedwiththelinearscaling.MuchsubsequentworkhasbeenaimedatcharacterizingMIMOchannelcapacityundermo