The Degree of Freedom of MIMO Interference Channel

1TheDegreesofFreedomofMIMOInterferenceChannelswithoutStateInformationatTransmittersYanZhuandDongningGuoDepartmentofElectricalEngineeringandComputerScienceNorthwesternUniversityEvanston,IL60208UnitedStatesAbstractThispaperfullydeterminesthedegree-of-freedom(DoF)regionoftwo-userinterferencechannelswitharbitrarynumberoftransmitandreceiveantennasandisotropicfading,wherethechannelstateinformationisavailabletothereceiversbutnottothetransmitters.Theresultcharacterizesthecapacityregiontothefirstorderofthelogarithmofthesignal-to-noiseratio(SNR)inthehigh-SNRregime.TheDoFregionisachievedusingrandomGaussiancodebooksindependentofthechannelstates.HencetheDoFgainduetobeamformingandinterferencealignmentiscompletelylostinabsenceofchannelstateinformationatthetransmitters(CSIT).IndexTermsCapacityregion,channelstateinformation,degreesoffreedom,interferencechannel,isotropicfading,multipleantennas,multiple-input-multiple-output(MIMO).I.INTRODUCTIONManytypicalscenariosincellularandadhocnetworkscanbewellmodeledusinginterferencechannels.TherehavebeenrecentbreakthroughsinunderstandingthefundamentallimitsofsuchThisworkissupportedbytheNSFundergrantCCF-0644344andDARPAundergrantW911NF-07-1-0028.September1,2010DRAFTarXiv:1008.5196v1[cs.IT]31Aug20102channels,withorwithoutmultipleantennas[1]–[5].Moststudiesofinterferencechannelsassumethatfullchannelstateinformation(CSI)isavailabletoalltransmittersandreceivers.Inpractice,however,thestateofthechannelisusuallymeasuredatthereceivers,anditisoftendifficultforthetransmitterstoacquiretheCSIpreciselyinatimelymanner.Inthispaper,weconsideratwo-usermultiple-inputmultiple-output(MIMO)interferencechannelsubjecttoisotropicfading,whichisindependentovertime,andwhoserealizationispreciselyknowntothereceiversbutnottothetransmitters.ThemodelisdescribedinSectionII.(AnexampleofthechannelmodelisillustratedinFig.1.)Thedegree-of-freedom(DoF)regionoftheMIMOinterferencechanneliscompletelycharacterizedasTheorem1inSectionIII.TheresultindicatesthatwithoutCSIatthetransmitters(CSIT),noadditionalgainintermsofDoFcanbeachievedusingbeamformingorinterferencealignment,whichisincontrastwiththeresultsforthecasewithfullCSIshownin[6].AdetailedproofTheorem1isdevelopedinSectionsIIIandIV.Tx1Tx2Rx2Rx1H11H12H22H21Fig.1.Two-userinterferencechannelwithmultipleantennas.TherehavebeenseveralrelatedworksoninterferencechannelwithoutCSIT,e.g.,[7]–[12].Slowfadingisassumedin[7],[8],andismodeledusingacompoundinterferencechannel.Thecapacityofatwo-userinterferencechannelwithsingleantennaisstudiedin[7],whilethediversity-multiplextradeoffofthesamemodelisstudiedin[8].Morecloselyrelatedtothisworkarestudiesofthecaseoffastfadingin[9]–[12].In[9],Akuiyiboetalderivedanouterboundofcapacityregionfortwo-userMIMOinterferencechannelswithRayleighfading,whichisonlytightforsomespecialcasesevenintermsoftheDoF.TighterouterboundsontheDoFregionhavebeendevelopedbyHuangetalin[10]assumingRayleighfading,andbyVazeandVaranasiin[11]assumingamoregeneralmodel,andandbytheauthorsassumingSeptember1,2010DRAFT3generalisotropicfading[12].1Agapremainsbetweentheinnerandouterboundsin[10]–[12].Aspecificexampleisthecasewherethetwousershaveoneandthreetransmitantennas,andtwoandfourreceiverantennas,respectively,asshowninFig.1.TheDoFpair(1;1)hasbeenshowntobeachievablebutthebestouterboundsin[10]–[12]includesthepair(1;1:5).Thispaperclosesthegapbyshowingthatachievableregionof[12]istheexactDoFregion.Intheaforementionedspecialcase,thepair(1;1:5)isnotachievable.II.CHANNELMODELConsideratwo-userinterferencechannel,whereeachtransmitterhasdedicatedinformationforitsintendedreceiver.SupposeeachtransmittertisequippedwithMtantennasandreceiverrisequippedwithNrantennasforeveryt;r=1;2.Thesignalsreceivedbythetwouserscanbedescribedas2y[i]=H11[i]w[i]+H12[i]x[i]+u1[i](1a)z[i]=H21[i]w[i]+H22[i]x[i]+u2[i](1b)wherei=1;:::;nisthetimeindex,w(M11)andx(M21)denotethetransmitsignals,Hrt(NrMt)denotesthechannelmatrixbetweentransmittertandreceiverr,andur(Nr1)denotesthethermalnoiseatreceiverr,whichconsistsofindependentidenticallydistributed(i.i.d)circular-symmetriccomplex-Gaussian(CSCG)randomvariablesofunitvariance(denotedbyurCN(0;INr)).ThefadingprocessesfHrt[i]gandnoiseprocessesfur[i]g(r;t=1;2)aremutuallyindependentandi.i.d.overtime.Wealsoassumeapowerconstraintforeachuser,i.e.,allcodewords,(w[1];:::;w[n])and(x[1];:::;x[n])satisfy1nnXi=1kw[i]k2and1nnXi=1kx[i]k2:Sincethenoiseprocessesarenormalized,isregardedasthemaximumaveragetransmitsignal-to-noiseratios(SNR)ofbothusers.Furthermore,weassumethateachrealizationof(Hr1;Hr2)1Thefadingmodelsof[11]and[12]overlapbutdonotfullycovereachother.BothmodelsincludeindependentRayleighfadingstudiedin[10]asaspecialcase.2Asaconvention,weuseboldfontstodenoterandomvariables,randomvectorsandrandommatrices,andweusethecorrespondingnormalfontstodenotetheirrealizations.September1,2010DRAFT4isavailableatreceiverronly,whilethetransmittershavenoknowledgeaboutthechannelmatricesexcepttheirstatistics.Definition1:Acomplex-valuedrandommatrixRisisotropicifandonlyifthematricesRandRQareidenticallydistributedfor