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2008 Interference alignment and the degrees of freedom of the K user channel- TIT.pdf

2008 Interference alignment and…

上传者: 眼泪若素 2013-12-25 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《2008 Interference alignment and the degrees of freedom of the K user channel- TITpdf》,可适用于高等教育领域,主题内容包含IEEETRANSACTIONSONINFORMATIONTHEORY,VOL,NO,AUGUSTInterferenceAlignmentandD符等。

IEEETRANSACTIONSONINFORMATIONTHEORY,VOL,NO,AUGUSTInterferenceAlignmentandDegreesofFreedomoftheKUserInterferenceChannelViveckRCadambe,StudentMember,IEEE,andSyedAliJafar,Member,IEEEAbstractForthefullyconnectedKuserwirelessinterferencechannelwherethechannelcoefficientsaretimevaryingandaredrawnfromacontinuousdistribution,thesumcapacityischaracterizedasC(SNR)=Klog(SNR)o(log(SNR))Thus,theKusertimevaryinginterferencechannelalmostsurelyhasK=degreesoffreedomAchievabilityisbasedontheideaofinterferencealignmentExamplesarealsoprovidedoffullyconnectedKuserinterferencechannelswithconstant(nottimevarying)coefficientswherethecapacityisexactlyachievedbyinterferencealignmentatallSNRvaluesIndexTermsCapacity,degreesoffreedom,interferencealignment,interferencechannel,multipleinput–multipleoutput(MIMO),multiplexingIINTRODUCTIONINFORMATIONtheoristshavepursuedcapacitycharacterizationsofinterferencechannelsforoverthreedecades–TheseeffortshaveproducedanextensivearrayofinterestingresultsthatshedlightonvariousaspectsoftheproblemRecently,aspecialcaseoftheHan–KobayashischemeisshownintoachievethecapacityofthetwouserinterferencechannelwithinonebitReferencealsoprovidesageneralizeddegreesoffreedomcharacterizationthatidentifiesdifferentoperationalregimesforthetwouserinterferencechannelForoptimalwirelessnetworkdesign,thenaturalquestioniswhethertheinsightsfromthetwouserinterferencechannelgeneralizetointerferencechannelscenarioswithmorethantwousersUnfortunately,formorethantwousers,evendegreesoffreedomcharacterizationsarenotknownAtacoarselevel,someoftheinterferencemanagementapproachesusedinpracticeandtheirinformationtheoreticbasismaybesummarizedasfollows:•Decode:Ifinterferenceisstrong,thentheinterferingsignalcanbedecodedalongwiththedesiredsignalthetradeoffisthatwhiledecodingtheinterferencemayimprovetheratesforthedesiredsignal,thedecodabilityoftheinterferingsignalslimitstheotherusers’ratesManuscriptreceivedJuly,revisedMarch,ThisworkwassupportedinpartbytheNationalScienceFoundationunderCAREERGrantandbyDARPAunderITMANETGrantUTAThematerialinthispaperwaspresentedinpartatthethAnnualAllertonConferenceonCommunications,Control,andComputing,Monticello,IL,SeptemberTheauthorsarewiththeCenterofPervasiveCommunicationsandComputing(CPCC),DepartmentofElectricalEngineeringandComputerScience,UniversityofCaliforniaIrvine,Irvine,CAUSA(email:vcadambeuciedusyeduciedu)CommunicatedbyPViswanath,AssociateEditorforCommunicationsColorversionofFigureinthispaperisavailableonlineathttp:ieeexploreieeeorgDigitalObjectIdentifierTITWhilelesscommoninpracticeduetothecomplexityofmultiuserdetection,thisapproachissupportedbythecapacityresultsonthe“verystronginterference”,and“stronginterference”,scenariosinthecontextofthetwouserinterferencechannelTheextensionof“stronginterference”resultstomorethantwousersisnotstraightforwardingeneral•TreatasNoise:Ifinterferenceisweak,thentheinterferingsignalistreatedasnoiseandsingleuserencodingdecodingsufficesThisapproachhasbeenusedinpracticeforalongtime,eg,forfrequencyreuseincellularsystemsHowever,informationtheoreticvalidationforthisapproachhasonlyrecentlybeenobtainedthroughseveralconcurrentworks,,,Whiletreatingweakinterferenceasnoisemaybenaturalfromanengineeringstandpoint,itissomewhatsurprisingfromaninformationtheoreticperspectivethatintroducingstructureintotheinterferencesignalsisnotusefulinthisregimeThisresulthasbeenestablishedformorethantwousersaswell•Orthogonalize:Ifthestrengthofinterferenceiscomparabletothedesiredsignal,theninterferenceisavoidedbyorthogonalizingthechannelaccessThisisthebasisfortime(frequency)divisionmediumaccessschemesthatavoidinterferencebetweencoexistingwirelesssystemsbydividingspectruminacakecuttingfashionInformationtheoreticvalidationforthisapproachcomesfromthecapacityprelog(degreesoffreedom)characterizationsConsideringonlysingleantennanodes,thesingleuserAWGNchannelcapacityintheabsenceofinterferencemaybeexpressedasSNRSNRsothatintheabsenceofinterferencetheGaussianchannelhasdegreeoffreedomThesumcapacity(peruser)ofthetwouserinterferencechannelisknowntobeSNRSNRsothateachusergetsonlyhalfthedegreesoffreedomItisconjecturedinthatthesumcapacity(peruser)fortheuserinterferencechannelisSNRSNROrthogonalaccessschemescanbeusedtodividethedegreeoffreedomamongtheuserssuchthateachusergetsafractionandthesumofthesefractionsisequaltoWerefertothisapproachasthe“cakecutting”approachInthispaper,weexploretheregimeidentifiedwiththe“orthogonalize”approachabove,wherealldesiredandinterferingsignalsareofcomparablestrengthWeshowthat,forabroadclassofwirelessnetworks,evenwhentherearemorethantwoIfthecapacitycanbeexpressedasC(SNR)=dlog(SNR)o(log(SNR))thenwesaythechannelhasddegreesoffreedom(alsoknownasthecapacityprelogorthemultiplexinggain)$IEEEIEEETRANSACTIONSONINFORMATIONTHEORY,VOL,NO,AUGUSTinterferingusers,thesumcapacity(peruser)isSNRSNRie,everyonegetshalfthecakeThekeytothisresultisanachievableschemecalledinterferencealignmentthatisespeciallyrelevanttotheinterferencechannelwithmorethantwousersWebeginwiththesystemmodelIISYSTEMMODELConsidertheuserinterferencechannel,comprisedoftransmittersandreceiversEachnodeisequippedwithonlyoneantenna(multipleantennanodesareconsideredlaterinthispaper)Thechanneloutputatthethreceiveroverthethtimeslotisdescribedasfollows:where,istheuserindex,isthetimeslotindex,istheoutputsignalofthethreceiver,istheinputsignalofthethtransmitter,isthechannelfadecoefficientfromtransmittertoreceiveroverthethtimeslotandistheadditivewhiteGaussiannoise(AWGN)termatthethreceiverWeassumeallnoisetermsareindependentidenticallydistributed(iid)zeromeancomplexGaussianwithunitvarianceToavoiddegeneratechannelconditions(eg,allchannelcoefficientsareequalorchannelcoefficientsareequaltozeroorinfinity)weassumethatthechannelcoefficientvaluesaredrawniidfromacontinuousdistributionandtheabsolutevalueofallthechannelcoefficientsisboundedbetweenanonzerominimumvalueandafinitemaximumvalue,Weassumethatchannelknowledgeiscausalandgloballyavailable,ie,attimesloteachnodeknowsallchannelcoefficientsRemark:ForthepurposeofthisworkthereisnofundamentaldistinctionbetweentimeandfrequencydimensionsThechanneluseindexinthemodeldescribedabovecouldequivalentlybeusedtodescribetimeslots,frequencyslotsoratimefrequencytupleifcodingisperformedinbothtimeandfrequencyThevaryingnatureofthechannelcoefficientsfromonechannelusetoanotheris,however,animportantassumptionWealsodefinetheterm“constant”channel,asthecasewhereallchannelcoeffcientsarefixedWeassumethattransmittershaveindependentmessagesintendedforreceivers,respectivelyThetotalpoweracrossalltransmittersisassumedtobeequaltoWeindicatethesizeofthemessagesetbyForcodewordsspanningchanneluses,theratesareachievableiftheprobabilityoferrorforallmessagescanbesimultaneouslymadearbitrarilysmallbychoosinganappropriatelylargeThecapacityregionoftheuserinterferencechannelisthesetofallachievableratetuplesADegreesofFreedomSimilartothedegreesoffreedomregiondefinitionforthemultipleinput–multipleoutput(MIMO)channelinwedefinethedegreesoffreedomregionfortheuserinterferencechannelasfollows:()IIIOVERVIEWOFMAINRESULTSThemaininsightofferedinthispaperishowtheideaofinterferencealignmentcanbeappliedtotheuserinterferencechanneltorestrictallinterferenceateveryreceivertoapproximatelyhalfofthereceivedsignalspace,leavingtheotherhalfinterferencefreeforthedesiredsignalWepresentatoyexampletoillustratethiskeyconceptAInterferenceAlignmentToyExampleConsidertheconstantuserinterferencechanneldefinedby()whereatthethchanneluse,arethethreceiver’soutputsymbolandzeromean,unitvariance,complexcircularlysymmetricadditivewhiteGaussiannoise(respectively)andisthethtransmitter’sinputsymbolAlldirectchannelcoefficientsareequaltowhileallcrosschannel(carryinginterference)coefficientsareequaltoThechannelcoefficientsarefixedforallchannelusesAllsymbolsarecomplexandalltransmittedsignalsaresubjecttoapowerconstraint,sothatIntheabsenceofinterference,anyusercanachieveacapacityandtheoptimalinputdistributioniscircularlysymmetriccomplexGaussianWithalluserspresenttheoptimal(sumcapacityachieving)schemeisasfollowsEachtransmittersacrificeshalfthesignalspaceandonlysendsarealGaussiansignalwithpowerEachreceiverdiscardstheimaginarypartofthereceivedsignalthatcontainsalltheinterferenceandisabletodecodethedesiredsignalfreefrominterferenceatarate,wherethefactorofshowsupinthedenominatorbecauseonlythe“real”partoftheadditivenoise(whichhaspower)isrelevantThus,thesumratewithinterferencealignmentisInterestingly,thesumcapacityofthischannelisalso,whichmeansthatforthissymmetricchannelinterferencealignmentiscapacityoptimalatanySNRTheconverseargumentisasfollowsConsideranytwousers,sayusersandandeliminateallotherusersThiscannothurttheusersbeingconsideredConsideranyreliablecodingschemeforthistwouserinterferencechannelBecausethecodingschemeisreliablebyassumption,usercansuccessfullydecodehismessageandsubtractitoutfromthereceivedsignalNowhecanaddbackaphaseshiftedversionofhissignaltoCADAMBEANDJAFAR:INTERFERENCEALIGNMENTANDDEGREESOFFREEDOMOFTHEUSERINTERFERENCECHANNELreconstructanewreceivedsignalthatisstatisticallyequivalenttothereceivedsignalofreceiverThisimpliesthatreceivercandecodebothmessagesThus,thesumrateachievedbyusersandcannotbemorethanthesumcapacityofthetwousermultipleaccesschanneltoreceiverButthismultipleaccesschannel(MAC)hassumcapacitySimilarly,consideringanytwouserswefindthattheirsumrateisboundedabovebyAddingalltheseboundstogether,wefindthattheouterboundonthesumrateofallusersintheinterferencechannelisSincethisisachievablewithinterferencealignment,itisalsothecapacityofthisuserinterferencechannelThisistrueatanySNRvalueOneparticularlyinterestingaspectofthisexampleisthatwhilethecapacityachievingschemeusesGaussianinputs,theyarenotcircularlysymmetricGaussiansThisisremarkablebecauseforGaussianpointtopoint(MIMO),multipleaccess,broadcastchannelswithcomplexchannelcoefficients,theinputs(eveniftheyarecorrelatedandhavedifferentpowers)areindividually(elementwise)circularlysymmetricGaussianBOtherExamplesInterferencealignmentexamplessimilartotheonespresentedabovecanalsobeconstructedinotherdimensionssuchasspace(beamformingacrossmultipleantennas),time(eitherthroughpropagationdelaysorthroughcodingacrosstimevaryingchannels),frequency(eitherthroughdopplershiftsorbycodingacrossmultiplecarrierswithfrequencyselectivecoefficients)andcodes(throughlatticeormultilevelcodesthataligninterferencewithinsignallevels)AppendixIprovidesasimpleexampleofinterferencealignmentwheneachchannelhasadelayassociatedwithitAsanotherexample,considertwoparallelinterferencechannels(forexampleovertwoorthogonalcarriers)Onthefirstcarriersupposeallchannelcoefficientsareequalto,whileonthesecondcarriersupposealldesiredchannelsareequaltooneandtheinterferingchannelcoefficientsareequaltoThenitiseasilyseenthatbyspreadingthesignaloverthetwocarrierswiththespreadingcodeallinterferenceisalignedThisexampleispresentedintoestablishtheresultthatparallelinterferencechannelsareinseparable,ie,jointcodingacrossparallelchannelsisnecessarytoachievecapacity(unlikeGaussianmultipleaccessandbroadcastchannelswhereseparatecodingwithoptimalpowerallocationacrosscarrierssufficestoachievecapacity)Interferencealignmentisachievedthroughlatticecodesinthecontextofmanytooneandonetomanyinterferencechannelsinandforcertainfullyconnectedinterferencechannelsin,whichalsodrawsaninterestinganalogybetweenthepropagtiondelayexampleprovidedinAppendixIandthealignmentofsignallevelsthroughmultilevelcodesQuitesimply,amultiplicationofthetransmittedsignalwiththechannelcoefficient(say)leadstoadecimalpointshiftofthearyrepresentation(ie,thebaserepresentation)ofthetransmittedsignalvaluewhichissimilartoapropagationdelayintimeTheenablingpremiseforinterferencealignmentinalltheprecedingexamplesistherelativityofalignmentie,thealignmentofsignalvectorspacesisrelativetotheobserver(thereceiver)TwotransmittersmayappeartobeaccessingthechannelsimultaneouslytoonereceiverwhiletheyappeartobeorthogonaltoanotherreceiverSinceeachreceiverhasadifferentview,thereexistscenarioswhereeachreceiver,fromitsownperspective,appearstobeprivilegedrelativetoothersThegoalofinterferencealignmentistocreatesuchscenariosinawirelessnetworkSpecifically,interferencealignmentreferstoaconstructionofsignalsinsuchamannerthattheycastoverlappingshadowsatthereceiverswheretheyconstituteinterferencewhiletheyremaindistinguishableatthereceiverswheretheyaredesiredTheideaofinterferencealignmentevolvedoutofthedegreesoffreedominvestigationsonthetwouserMIMOchannel,,andthecompoundbroadcastchannelThetwouserchannelisacommunicationsystemwithtwotransmitters,tworeceivers,andfourindependentmessages,onefromeachtransmittertoeachreceiverTakingadvantageoftheMACandthebroadcastchannel(BC)componentscontainedwithinthechannel,MaddahAli,Motahari,andKhandaniproposedanelegantcodingscheme(theMMKscheme)inforthetwouserMIMOchannelTheMMKschemenaturallycombinessuccessivedecodinganddirtypapercoding,theoptimalschemesfortheconstituentMACandBCInterestingly,theMMKschemeachievesdegreesoffreedomonthetwouserchannelwhenallnodesareequippedwithantennasThekeytothisresultistheimplicitinterferencealignmentthatisfacilitatedbytheiterativeoptimizationoftransmitprecodingandreceivecombiningvectorsThefirstexplicitinterferencealignmentschemeispresentedinwhereitisshownthatdirtypapercodingandsuccessivedecodingarenotrequiredtoachievethemaximumdegreesoffreedomonthetwouserMIMOchannelTheachievabilityofdegreesoffreedomandtheconverseareestablishedinInterferencealignmentisusedin,toobtaininnerboundsonthedegreesoffreedomregionoftheMIMOchannelInterferencealignmentisalsoakeyingredientofthedegreesoffreedomcharacterizationofthecompoundbroadcastchannelinCDegreesofFreedomoftheUserInterferenceChannelInthispaperweestablishthattheusertimevaryinginterferencechanneldefinedinSectionIIhasdegreesoffreedomEquivalently,athighSNR,everyuseris(simultaneouslyandalmostsurely)abletoachievereliablecommunicationatratesapproachingonehalfofthecapacitythathecouldachieveintheabsenceofallinterferenceAninterestingimplicationofthisresultisthattimevaryinginterferencenetworksarenotfundamentallyinterferencelimitedTheresulthasthesameflavorasthetoyexamplespresentedearlierinthissectionInbothcasestheconclusionisthateveryonegetshalfthecakeWhilethetoyexamplesrepresentcontrivedscenarioswherethechannelparametersarecarefullyselectedtofacilitateinterferencealignment,thedegreesoffreedomresultisforchannelswhosecoefficientsarerandom,ie,selectedbynatureThereisapenaltyinvolvedwithrandomchannelcoefficients,butthepenaltyisSNR,ie,itbecomesanegligiblefractionoftheusers’ratesathighSNRIndeed,weexpectthattheratepenaltywillincreasewiththenumberofusers,sothatitwilltakehigherandhigherSNRtoapproachhalfofeachuser’scapacityasthenumberofusersIEEETRANSACTIONSONINFORMATIONTHEORY,VOL,NO,AUGUSTFigInterferencealignmentonthethreeuserinterferencechanneltoachieve=degreesoffreedomincreasesThedegreesoffreedomperspectiveistoocoarsetocapturethispenaltyandthereforedoesnotrevealthiscompetitionamongusersInthissense,thepicturepresentedbythedegreesoffreedomresultisoptimisticThedegreesoffreedomfortheconstantinterferencechannel(withtheexceptionofcertainMIMOscenarios)remainsanopenproblemformorethantwousersTheinterferencealignmentschemesusedinthispaperarebasedonbeamformingovermultiplesymbolextensionsofthetimevaryingchannelTheseschemesdonotexactlyachievetheouterboundonthedegreesoffreedomforafinitesymbolextensionInstead,byusinglongersymbolextensionsweareabletoapproacharbitrarilyclosetotheouterboundIntuitively,thiscanbeunderstoodasfollowsInordertoachieveexactlydegreesoffreedom(peruser)overafinitesymbolextension,everyreceivermustbeabletopartitionitsobservedsignalspaceintotwosubspacesofequalsize,oneofwhichismeantforthedesiredsignalsandtheotheristhe“wastebasket”foralltheinterferencetermsMoreover,thevectorspacescorrespondingtotheinterferencecontributedbyallundesiredtransmittersmustexactlyalignateveryreceiverwithinthewastebasketwhichhasthesamesizeaseachoftheinterferencesignalsItturnsoutthisproblemisoverconstrainedanddoesnotadmitasolutionWecircumventthisproblembyallowingsomeoverflowspace(afewextrasymbols)forinterferencetermsthatdonotalignperfectlyFortunately,wefindthatthesizeoftheoverflowspacebecomesanegligiblefractionofthetotalnumberofdimensionsasweincreasethesizeofthesignalspaceThus,foranyitispossibletoaligninterferencetotheextentthattheachieveddegreesoffreedomarewithinanfractionoftheouterboundThetradeoffisthatthesmallerthevalueof,thelargerthenumberofsymbols(timeslots)neededtorecoverafractionoftheouterboundvaluepersymbolAsanexample

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