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Beamforming for correlated broadcast channels with quantized channel state information

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Beamforming for correlated broadcast channels with quantized channel state information
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   E MFORMING FORCORREL TED RO DC STCH NNELSWITHQU NTIZEDCH NNEL ST TE INFORM TION RubendeFrancisco*,ClaudeSimon t, Dirk T M Slock*, and Geer Leus t *EurecomInstitute,2229,RoutedesCretes,06560ValbonneSophia-Antipolis,France tTU Delft,Fac.EEMCS,Mekelweg4,2628CDDelft,TheNetherlands 161 ABSTRACT Thedesign of channelquantizationcodebooksforcorrelatedbroadcastchannelswithlimitedfeedbackisaddressed.Adesigncriterionthateffectivelyexploitsthecellstatisticsisproposed,basedonminimizingtheaveragesum-ratedistortioninasystemwithjointlinearbeamformingandmultiuserscheduling.Theproposedaveragedistortionfunctionisoptimizedbygeneratingaset of quantizationcodebooksthroughrandomtrials,keepingthecodebookthatyieldsthelowestdistortion.Comparisonswithlimitedfeedbackapproachesrelyingonrandomcodebooksareprovided,highlightingtheimportance of matchingthecodebookdesigntothecellstatistics.Numericalresultsshowaperformancegaininscenarioswithnon-uniformuserdistributions.Further,weproposeaschemethatexploitsthelimitedchannelknowledgeatthebasestationtoreducethecomputationalcomplexity of determiningthebeamformingvectorsand of findingtheoptimaluserset. 1. INTRODUCTION Multiple-inputmultiple-output(MIMO)systemscansignificantlyincreasethespectralefficiencybyexploitingthespatialdegrees of freedomcreatedbymultipleantennas[1].IntheMIMObroadcastchannel,ithasrecentlybeenproven [2] thatthesumcapacityisachievedbydirtypapercoding(DPC)[3].However,theapplicability of DPCislimitedduetoitscomputationalcomplexityandtheneedforfullchannelstateinformation(CSI)atthetransmitter.Asalowcomplexityalternative,downlinktechniquesbasedonSpaceDivisionMultipleAccess(SDMA)havebeenproposedthatachievethesameasymptoticsumrateasthat of DPC,e.g.,zero-forcingbeamforming[4].Ontheotherhand,whilehavingfullCSIatthereceivercanbeassumed,thisassumptionisnotreasonableatthetransmitterside.Severallimitedfeedbackapproacheshavebeenconsideredinpoint-to-pointsystems[5,6],whereeachusersendstothetransmittertheindex of aquantizedversion of itschannelvectorfromacodebook.AnextensionforMIMObroadcastchannelsismadein[7],inwhicheachmobilefeedsbackafinitenumber of bitsregardingitschannelrealizationatthebeginning of eachblockbasedonacodebook.CodebookdesignsforMIMObroadcastchannelswithlimitedfeedbackfollowingeneralsimpledesigncriteria,withthepurpose of simplifyingcodebookgenerationandsystemanalysis.OpportunisticSDMA(OSDMA)hasbeenproposedin [8] asanSDMAextension of opportunisticbeamforming[9],inwhichfeedbackfrom Theresearch of the authors at Eurecom was supported in part by theEurecomInstitute, andby thenational RNRT projectOpus.Theresearchof the authors atTV Delft was supported in part byNWO-STW under the VIOlprogram(DTC.6577). 978-1-4244-2046-9/08/ 25.00©2008IEEE theuserstothebasestation(BS)isconveyedintheform of abeamformingvectorindexandanindividualsignal-to-interference-plusnoiseratio(SINR).Anextension of OSDMAisproposedin[10],coined as OSDMAwithlimitedfeedback(LF-OSDMA),inwhichthetransmittercountsonacodebookcontaininganarbitrarynumber of unitarybases.Inthisapproach,theusersquantizethechanneldirection(channelshape)totheclosestcodewordinthecodebook,feedingbackthequantizationindexandtheexpectedSINR.Multiuserschedulingisperformedbasedontheavailablefeedback,usingasbeamformingmatrixtheunitarybasisinthecodebookthatmaximizesthesystemsumrate.OtherschemesforMIMObroadcastchannelsproposetousesimpleRandomVectorQuantization(RVQ)[11]forquantizingtheuservectorchannels,such as theapproachdescribedin[7].Asimplegeometricalframeworkforcodebookdesignisproposedin[6],whichdividestheunitsphereinquantizationcellswithequalsurfacearea.Thisframeworkisusedforchanneldirectionquantizationin[12],wherefeedbacktothebasestationconsists of aquantizationindexalongwithachannelqualityindicatorforuserselection.Thesecodebookdesignsdonottakeintoaccounteitherspatialcorrelationsoruserdistributionspresentinthesystem,whichcouldyieldbetterquantizationcodebooksandinturnbettersum-rateperformance.Thegains of adaptivecellsectorizationhavebeenstudiedin[13]inthecontext of CDMAnetworksandsingleantennacommunications,withtheaim of minimizingthetotaltransmitpowerintheuplink of asystemwithnon-uniformuserdistributionoverthecell.Thissituationisanalogoustoasystemwithmultipletransmitantennasinwhichbeamformingisperformed,adaptingitsbeamstounevenuserdistributions.Inascenariowithlimitedfeedbackavailable,adaptation of quantizationcodebookscanbeperformedinsteadinordertoimprovethesystemperformance.In[14],anapproachforexploitinglongtermchannelstateinformationinthedownlink of multiuserMIMOsystemsisproposed.Aflat-fadingmultipathchannelmodelisassumed,withnoline of sight(NLOS)betweenthebasestationanduserterminals.Eachusercanbereachedthroughafinitenumber of multipathcomponentswithacertainmeanangle of departure(AoD)fromtheantennabroadsideandanglespread.Themeananglesofdeparturearefixedandthusnousermobilityisconsidered.Inthispaper,wehighlighttheimportance of cellstatisticsforcodebookdesigninMIMObroadcastchannelswithlimitedfeedback.Theaveragesumratedistortioninasystemwithjointlinearbeamformingandmultiuserschedulingisminimized,exploitingtheinformationonthemacroscopicnature of theunderlyingchannel.Anon-geometricalstochasticchannelmodelisconsidered,inwhicheachusercanbereachedindifferentspatialdirectionsandwithdifferentanglespread.Basedonthismodel,comparisonswithlimitedfeedbackapproachesrelyingonrandomcodebooksareprovidedinordertoillustratetheimportance of matchingthecodebookdesign SPAWC2008 Authorized licensed use limited to: Technische Universiteit Delft. Downloaded on November 12, 2008 at 04:25 from IEEE Xplore. Restrictions apply.  tothecellstatistics.Asshownthroughnumericalsimulations,theproposedapproachprovidesconsiderableperfonnancegainsinscenarioswithnon-unifonnuserdistributions.Sincethebasestationhasnoaccesstoperfectchannelstateinfonnation,thefollowingSINRestimateiscomputedfortheuserset Sand k-th user 2.2.UserSelection2.SYSTEMDESCRIPTION Note,thatthisfunctionalsotakesthenonn of thechannelintoaccountunlikethequantizationfunctionsusedforpurechanneldirectionquantization. We considerabroadcastchannelconsisting of M antennasatthebasestationand K single-antennausersinasinglecellscenario.Let 8 denoteanarbitraryset of userswithcardinality   = M. Giventheuserset S scheduledfortransmission,thesignalreceivedatthe k-th userterminalisgivenby (5)  4 Thesignalsfromthebasestation(BS)arriveateachusertenninal(UT)throughafinitenumber of L paths,whichhavedifferentAoDswithrespecttotheantennaarraybroadsidebutarriveatthereceiverwiththesamedelay.TheAoDforthek-thuserand l-th pathcanbeexpressed as Bkl = Ok + ilBkl where Ok isthemeanAoDforuser k and ilBkl istheangleoffsetforthe l-th multipathcomponent.ThemultipathcomponentshavecomplexGaussiandistributedgains  Ykl withzeromeanandunitvariance.Thechannel of user k isgivenby 1 L hk = IT:L:  Ykla(BkL) yL l l where a( Bkl) arethesteeringvectors.Anomnidirectionalunifonnlineararrayisconsidered(ULA)althoughtheproposedtechniquecanbenefitfromanyarrayconfiguration.Thesteeringvectors a( Bkl) of aULAaregivenby 3.CHANNEL MODEL A 2 SINR = i k LiES,i=/:k I hk w il 2 + (J2 where W k denotesthebeamfonningvectorforuser k. 3.1.UserVectorChannels Inthissectionwepresentthemodelconsideredbothfortheuservectorchannelsandthecellstatistics.Anon-geometricalstochasticchannelisassumed,inwhichthechannelphysicalparametersaredescribedbyprobabilitydensityfunctionsassuminganunderlyinggeometry.Thechannelmodelweproposetouseismainlybasedontheworkin[17],extendedtomultiuserscenarios. We consideranoutdoorenvironmentwithNLOSbetweentransmitterandreceivers,inwhichlocalscatterers,thatarerandomlydistributedaroundeachmobileuser,produceaclusteringeffect.Themultipathcomponents(MPC)arriveinclustersinbothspaceandtime.Forthesake of simplicity,weconsiderflatfadingandhenceallpathsareassumedtoarriveatzerodelay.Furthennore,weassumethateachuserseesMPCsincomingfromsurroundingscatterersthataregroupedintoonecluster.Eachuserisreachedwithadifferentmeanangle of departure(AoD) Ok. TheAoDsassociatedtothemultipathcomponentsaredistributedaroundthemeanaccordingtoacertainpowerangularspectrum(PAS),whichdependsonthespatialdistribution of scatterers.Inpractice,weonlyconsidertheazimuthdirections(angle of propagationwithrespecttotheantennaarraybroadside)sincetheelevationanglespreadisgenerallysmallcomparedtotheazimuthalangle.Differentprobabilitydensityfunctions(PDF)areconsideredintheliterature,suchasGaussian,unifonnorLaplacian [18]. Asummary of themodelparametersisgiveninTable 1. (2)(1) hk = argmin Ilhk - c11 2. cEC Yk = hkWkSk + :L: hkWiSi + nk iES,i=j:k where hk E C 1X1H, Wk E C MX1 , Sk and nk arethechannelvector,thebeamfonningvector,thetransmittedsignal,andtheadditivewhiteGaussiannoiseatreceiver k, respectively.Thefirsttennintheaboveequationistheusefulsignal,whilethesecondtenncorrespondstotheinterferencebytheotherusers. We assumethatthevariance of thetransmittedsignal Sk isnonnalizedtooneand nk isindependentandidenticallydistributed(i.i.d.)circularlysymmetriccomplexGaussianwithzeromeanandvariance (J2. Thechannelisassumedtobeperfectlyknownattheuserside.TheCSIistransmittedtothebasestationoverafeedbacklinkthatislimitedto B bitspertransmission.Hence,theCSIhastobequantizedbeforeitisfedbackusingacodebook C with N entries. We assumethroughoutthepaperthat N = 2 B. Thus,itispossibletofeedbackeveryelement of thecodebook,andthedatarateonthefeedbacklinkisfullyexploited.Strategiesthatexploitthetimecorrelation of thechanneltouselargercodebookswith N > 2 B arepresentedin [15,16]. Theuserchannelsaremappedtotheclosestcodewordin C, as describedby 2.1. Linear Beamforming Thebeamfonningvectorsarecomputedonthebasis of thematrix H, whoserowsarethequantizeduserchannels h k, k E S. Differentlinearbeamfonningtechniquesmaybeconsidered.Commonlyappliedlow-complexitylinearbeamfonningtechniquesaretransmitmatchedfiltering  TxMF andzero-forcing  ZF beamfonning [4]. Transmitmatchedfilteringusesthenonnalizedcolumns of fIR as beamfonningvectors.Zero-forcingbeamfonningusesthenonnalizedcolumns of thepseUdo-inverse of fI. We consideroptimalschedulingthroughoutthepaper,i.e.,wedonotconsiderfairnessissuesbetweentheusers.Let Q betheset of allpossibleusersubsets of cardinality M withdisjointindicesin {I, ... ,K}. Theset of usersscheduledfortransmissionateachtimeslotcorrespondstotheonethatmaximizestheestimatedsumrateoverallpossibleusersets (7)   = arg max log2(1 + SINRk). SEQ L.J kES (3)  6 where A isthewavelength,and d istheantennaspacingattheBS.Thedistribution of theanglesaroundthemeanAoDisassumedtohaveadouble-sidedLaplacianPDF,givenby f ilBkl) =   exp( -I V2 ilB kl/(JO I y 2 Jo 162 Authorized licensed use limited to: Technische Universiteit Delft. Downloaded on November 12, 2008 at 04:25 from IEEE Xplore. Restrictions apply.  o Fig. 1. Broadcastchannelmodelwithuserterminals(UT)surroundedbylocalscatterersgroupedinclusters,locatedindifferentmeanangles of departure(AoDs)withrespecttouniformlineararray(ULA)broadside. -90 m components -180/18090 quantization)forchanneldirectioninformation(CDI)andchannelqualityinformation(CQI).Sincetheamount of feedbackislimited,atradeoffarisesbetweentheamount of bitsusedforCDIquantization,whichhasanimpactonthemultiplexinggain,andtheamount of bitsusedforCQIquantization,whichhasanimpactonthemultiuserdiversitygainachievedfromuserselection.Inthiswork,weconsiderjointquantization of CDIandCQIinformation.Channelquantizationisdonedirectlyovertheuservectorchannelsratherthanquantizingthenormandchanneldirectionseparately,thusprovidingbettergranularity.Hence,sincetheproposedchannelquantizationisadaptedtothecellstatistics,includingtheaverageSNRconditionsandnumber of activeusers,thetradeoffbetweenmultiplexinggainandmultiuserdiversityisimplicitlyoptimized.Theproposedapproachconsists of designingachannelquantizationcodebookvalidforallusersinthecellbyminimizingtheaveragesum-ratedistortion of thescheduledusers.Sinceschedulingandbeamformingareperformedjointlyateachtimeslot,thedistortionmeasureneedstoaccountforbothjointly.Hence,differentlinearbeamformingtechniqueswillresultindifferentoptimizedcodebooks.Thiscriterionyieldsquantizationcodebooksthatarestatisticallymatchedtotheusersthatmaximizetheestimatedsumrate,whichareselectedasdescribedin(3). The quantizationcodebookisoptimizedduringaninitialtrainingperiod,afterwhichthecodebookisfixedandbroadcastedtotheusers. 4.1.DesignCriterion Thecodebook of N codewords,isfoundbysolvingtheoptimizationproblem SR(1i) = max log2(1 + SINRk).  10 seqセ kES Thefirsttermintheequationabovecorrespondstothemaximumsumratethatcan be achievedwiththechosenlinearbeamformingtechniqueandperfectchannelstateinformation,givenbyThebeamformingvectorsandtheusersetobtainedinthecase of perfectchannelstateinformationareingeneraldifferentthantheonesobtainedonthebasis of quantizedchannelinformationforagiventimeslot.Thesecondtermin(9)correspondstotheactualsumrateachievedbythesystem.Thebeamformingvectorsarecomputedonthebasis of thequantizedchannelsandtheusersscheduledfortransmissionareselected as describedin(3).Hence,theachievedsumrateisgivenbywhere d(1i, it isthedistortionmeasurebetweenthesetcontainingtheunquantizeduserchannels 1i = {hI, ... , hK} andthesetcontainingthequantizeduserchannels  t = {hI, ... , hK}. Thedistortionmeasureusedthroughoutthepaperisthesumratelossduetothechannelquantization.Theresultingcodebookdependsonthenumber of scheduledusers M fortransmission,thenumber of activeusers K inthecell,theusedbeamformingtechnique,andthechannelstatistics.Thedistortionmeasurecanbedescribedas (9)(8)(11) c = argminE[d(1i, it ] c d(1i, it = SR(1i) - SR(it). SR(1t) = L log2(1 + SINRk). kES* 4.CODEBOOKDESIGN We presentinthissectionthedesign of theuserchannelcodebook.Comparedtoexistingdesignapproaches[5]werelyonapureMonteCarlobasedapproach.Thisapproachallowsawiderrange of distortionfunctionsthanthecommonlyusedgeneralizedLloydalgorithm,anditalsoallowstoexploitthecellstatistics.Asdiscussedin [20], mosttechniquesrelyingonlimitedchannelstateinformationconsiderseparatefeedbackbits(andthusseparateMostpapersbasedontheabovementionedstochasticmodelsassumethatmeanAoDsareuniformlydistributedoveralldirections.Inindoorscenarios,therelativeclusterAoDisindeeduniformlydistributedover [0,211 ], asithasbeenseenfromchannelmeasurements [19], sincethelocation of clustercentersisuniformlydistributedoverthecell.However,asnotedin [19], thisisnotrealisticinoutdoorscenarioswherethebasestationiselevatedandthemobilestationsareoftensurrounded by localscatterers.Inthesecases,themeanAoDisverydependentonthemacroscopiccharacteristics of eachparticularscenario:topology,userdistribution,mobilitypattern,distribution of scatterers,etc.Hence,themeanAoDsforallusers,  k donotneedtobeuniformlydistributedovertheinterval [0, 211 ] Inourmodel,theyareconsideredtobeuniformlydistributedoveranarbitraryrange of angles Ui [Omini , Omaxi] Agraphicalrepresentation of thebroadcastchannelmodelisdepictedinFig. 1. 3.2.SpatialCellStatistics where CJ } istheangularstandarddeviation, CJ } = vi E[I セ Undertheassumption of usingaULAatthebasestation,thecrosscorrelationcoefficients of eachuser'svectorchannelcan be computedinclosedformgiventhePAS,asshownin [19]. 163 Authorized licensed use limited to: Technische Universiteit Delft. Downloaded on November 12, 2008 at 04:25 from IEEE Xplore. Restrictions apply.  Notethat,asopposedtotheestimatedSINRvaluesemployedforuserselection,theaboveequationcomputestheeffectiveSINRexperienced by each of theusersinthescheduledset S . 4.2.CodebookDesign We areusingaMonteCarlobasedcodebookdesignalgorithmtogeneratethechannelquantizationcodebooks.Theability of thisalgorithmtoworkwitharbitrarydistortionfunctionsmakesitaprimecandidatetosolve(8).TheMonteCarlocodebookdesignalgorithmgeneratesrandomcodebookshavingthesamedistributionasthechannel.Foreveryone of theserandomcodebookstheaveragedistortionisestimatedbyaveragingoveralargenumber of channelrealizations.Finally,thecodebookwiththelowestaveragedistortioniskept.Thiscodebookminimizesthelongtermsampleaveragedistortion,andthus,providesagoodsolutionto(8).Analternativeprocedureconsists of usingthegeneralizedLloydalgorithm [21] toiterativelyfindtheoptimizingcodebookandpartitioncells.However,theMonteCarlocodebookdesignavoidsconvergencetolocalminimaexhibitedbyLloyd'salgorithm,andthusprovidesabetterperformance if thenumber of triedcodebooksissufficientlyhigh.AcodebookdesignthatismoresimilartotheMonteCarlobasedcodebookdesignisrandomcoding [22]. However,randomcoding just uses N randomchannelrealizationsascodebook,anddoesnotallowtooptimizeanarbitrarydistortionfunction. 4.3.PracticalConsiderations Theproposedtechniqueforcodebookdesignisexpectedtoperformbetterinscenarioswithstrongspatialcorrelations.Differentlinearbeamformingtechniqueswillyielddifferentperformances,sincequantizationerrorsaffectthemdifferently.Forinstance,whileTxMFandZFbeamformingexhibitsimilarbehaviorforagivenerrorvariance,optimizedunitarybeamformingprovestobeveryrobust [23]. Sincethestatistics of thebest M usersgovernthedesign,thequantizationcodebooksmayfavorcertainspatiallocationsordirectionsthatprovidegoodsumrates,favoringtheusersinthoseparticularlocations.Inasystemwithlowmobilityandslowvariations,thissituationmayleadtoafairnessissue.Thisbehaviormaybeaccentuatedwhenincorporatingshadowingandpathlosstothechannelmodel.Thiseffectcan be attenuatedbyperformingproportionalfairscheduling(PFS),whichwouldyieldanaveragedistortionfunctionbasedonaweightedsumrate,penalizingtheusersthathavealreadybeenscheduled.Instead of simplygeneratingthequantizationcodebooksduringatrainingperiod,thebasestationmayslowlyadaptthecodebooktochangesintheenvironment:changesintrafficandmobilitypatterns,changes of scatterers,etc.Eachtimeauserentersthesystemorincasethereisacodebookupdate,thebasestationwouldsendtheupdatedcodebooktotheusers,whichingeneralchangesfromcelltocell.Inaddition,similarlytotheworkpresentedin [16] forsingle-userMIMOcommunications,theamount of feedbackcan be reducedbyexploitingtemporalcorrelationsinthesystem. 5.LOW-COMPLEXITYBEAMFORMING AND SCHEDULING Thelimitedchannelknowledgeatthetransmittersidedeterioratestheachievableperformance of thesystem,butcanalsobeexploitedtoreducethecomputationalloadforbeamformingandscheduling.Thequantization of theuserchannelscreatesequivalenceclassesbetweentheusers.Theuserswhosechannelsarequantizedtothesameentryinthecodebookaremembers of thesameequivalenceclass.Thus,thebasestationonlyknowswhichclassauserbelongsto,butitcannotdistinguishbetweentheusersinthesameclass.Itisthussufficienttodothebeamformingandtheschedulingonlybasedontherepresentative of theclass,i.e.,thecodeword,instead of basedonalltheusersintheclass. We denoteasetthatconsists of M representatives of differentclassesasaclassset.Thenumber of classsetstobeconsideredforbeamformingandscheduling Nes = N M issmallerthanthenumber of usersets Nus =   forpracticalsystemparameterswith N « K. Oncetheoptimalclasssetisdetermined,acorrespondingusersetcan be selected by choosingforeveryclassintheclasssetacorrespondinguser.Theuserinsideaspecificclasscanbeselectedrandomly or usingafairnessconstraint.Thecomplexity of determiningthebeamformingvectorsandtheclasssetscanbefurtherreducedusingalookuptablethatstoresforalltheclasssetsthecorrespondingsumrateestimatesandthebeamformingvectors.Weassumethatthislookuptableissortedbasedontheestimatedsumrate of theclasssets,wherethefirstentrycontainstheclasssetwiththehighestestimatedsumrate.Afterthebasestationreceivedthefeedbackfromalltheusers,itchecks if ithasamatchinguserforeveryentryinthefirstclassset. If not,thenthebasestationdoesthesamecheckforthefollowingclasssetsinthelookuptableuntilitfindsaclasssetthathasforeveryclassintheclasssetanactiveuser.Theadvantage of usingprecalculatedbeamformingvectorsstoredinalook-uptableisthatcomputationalmorecomplexbeamformingschemescan be used,e.g.,DirtyPaperCoding.NotethatTxMFisaspecialcasesincetherethedifferentbeamformingvectorsareindependent of theotherusersthatarescheduledfortransmission.Hence,thestorage of only N beamformingvectorsissufficient.Thestoragerequirementsforthelookuptablecanbereducedbystoringonlythemostprobableclasssets.Theprobabilitythatthefirstclasssetisselectedincreaseswiththenumber of usersinthecell.Fortheeventthatnoclasssetinthelookuptableisselected,ausersetcanstill be calculatedusingalow-complexityscheme,e.g.,TxMFbeamforming. 6. SIMULATIONRESULTS We comparetheperformance of linearbeamformingwithquantizedCSIfeedbacktoLF-OSDMA.Theusedlinearbeamformingstrategiesare ZF andTxMF.Weassumea2-GHzsystemwithanantennaspacingatthebasestation of d = 0.4'x セ 15em.Eachuserchannelismodelledwith L = 10multipathcomponents.ThemeanAoD of thedifferentusersisuniformlydistributedovertheinterval[60°,120°],andtheangularspreadisfixed to O ) = 30°.Weassumesingle-antennausersandabasestationwith M = 2antennas.Thedatarateonthefeedbacklinkislimitedto3bits/transmission.Inordertomakeafaircomparisonbetweentheschemes,theSINRfeedback of theLF-OSDMAalgorithmisalsoquantized.Thus,theLF-OSDMAalgorithmhastosharetheavailable3bitsbetweentheCDI,i.e.,theindex of thepreferredbeamformingvector,andtheCQI,i.e.,theSINR of thepreferredbeamformingvector. We simulatetheperformance of allpossibleCDIICQIbitallocations,andfinallyselecttheallocationthatresultsinthehighestsumrate.ThecodebooktoquantizethescalarCQIisdesignedwiththegeneralizedLloydalgorithm [22], usingthemeansquareerrorasdistortionfunction.Theperformance of thedifferentrandomcodebooks,i.e.,theirresultingaveragesumrate,isestimatedthroughaveragingover 164 Authorized licensed use limited to: Technische Universiteit Delft. Downloaded on November 12, 2008 at 04:25 from IEEE Xplore. Restrictions apply.  6.5..----------.------r------ --T----..... ----- 8.REFERENCES [1]H.Bolcskei,D.Gesbert, C. B.Papadias,andA.-J.van der Veen,Eds., Space-TimeWirelessSystems:From Array ProcessingtoMIMOCommunications. Cam-bridgeUniversityPress,2006.[2]H.Weingarten, Y. 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Erceg et al.,  TGn channelmodels, IEEE 802.11-03/940r4, May2004.[20]M.Kountouris, R. de Francisco,D.Gesbert,D. T. Slock,and T. Salzer, Multiuserdiversity-multiplexingtradeoffin MIMO broadcastchannelswithlimitedfeedback, in Proc.40thIEEE AsilomarCon onSignals,Systems, and Computers, PacificGrove,CA,USA,Oct.2006.[21] Y. Linde,A.Buzo,andR.M.Gray, Analgorithmforvectorquantizerdesign, IEEETrans.Commun., vol.28, no. , pp. 84-95, Jan.1980.[22]A.Gershoand R. M.Gray, VectorQuantization and SignalCompressing. KluwerAcademicPublishers,1995.[23] R. de FranciscoandD. T. Slock, Aniterativeoptimizationmethodforunitarybeamformingin MIMO broadcastchannels, in Proc. of 45thAllerton Con onCommun.,Control and Comput., Monticello,IL,USA,Sept.2007. 25 20 8 6 1520 101214 NumberofUsers 10  e LF-OSDMA withquant.SINR ... .. . .. .I   Matchedfiltering -a- Zero-forcing 1.5 L ......I....  L __  ]]ZZZZiZZ]]]]]]セ]]]]]]]]Z j o 2.5   . 3.5 CD iii セ 4 en SNR[dB] theinstantaneoussumrate of 10000 channelrealizations.Fig.2depictstheperformancefordifferentnumbers of userswithafixedSNR of 10dB.Weseethat ZF andTxMFwithquantizedCSIoutperformLF-OSDMAwithquantizedSINRfeedback.ThesameresultcanbeseeninFig.3fordifferentSNRvaluesand K = 10users. We seehowthesumrate of thedifferentschemessaturatesathighSNR,wheretheperformanceislimitedbythequantizationerror. 7.CONCLUSION セ セ 3.5 en 5.5 4.5 Theproblem of designingchannelquantizationcodebooksforcorrelatedbroadcastchannelswithlimitedfeedbackhasbeenaddressedforsystemswherejointlinearbeamformingandmultiuserschedulingisperformed.Thenumericalresultsprovidedhaveshownthebenefits of usingquantizationcodebooksoptimizedaccordingtothecellstatistics.Thegeneratedcodebooksperformwellinscenarioswithreducedangularspreadandeffectiverange of meanangles of departure.Thismakestheproposedapproachparticularlyinterestinginoutdoorsystemswithspatialcorrelationandnonuniformuserdistribution. Fig. 3.SumrateforthecorrelatedchannelmodelfordifferentSNR's.  M = 2, K = 10) Fig. 2.Sumrateforthecorrelatedchannelmodelfordifferentnumbers of users.  M = 2,SNR=10dB) 165 Authorized licensed use limited to: Technische Universiteit Delft. Downloaded on November 12, 2008 at 04:25 from IEEE Xplore. Restrictions apply.
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