Rx1 RxK Rx2 Tx11 Tx1P Tx21 Tx2P TxK1 TxKP Rx3 Tx31 Tx3P MAC1 MAC2 MAC3 MACK Fig1ExampleconnectivitygraphinapartiallyconnectedinterferingMACsmodelLinesrepresentchannelswithnonzerocoefcient ID: 109000
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sufciencyoftheseconditions.WeshowthatinterferencecanbeideallycancelledfromIAinanL-interferingnetworkofarbitrarysize,whilethenumberofantennasperuserremainsbounded.Weadaptthealgorithmfrom[14]forcomputingpre-codingmatricesandreceiverbeamformingvectorsthatrealizeIAinourconsideredscenario.Asanexampleofpotentialapplicationofourresultinthecontextofcellularnetworks,weshowthat,ifallbasestationsareequippedwith7antennas,alluserswith2antennaseach,andeachcellreceivesinterferencefromupto3othercells,then2userspercellcansimultaneouslyachieve1DoFtothebasestationwhilehavingtheout-of-cellinterferencecompletelysuppressedthroughIA.Thisholdsregardlessofthetotalnumberofcellsinthenetwork,andoverconstantMIMOchannels.NotethatthisschemesachieveshalfoftheDoFperuseravailablewithoutinter-cellinterferenceoverthesamechannel.Notealsothatthisresultdoesnotrelyonformingclustersofbasestations,butratheraimsatajointsolutionofIAacrossacompletenetworkofarbitrarynumberofcells.II.PARTIALLYCONNECTEDINTERFERINGMIMOMACSLetusintroducethechannelmodelunderlyingourwork,i.e.thepartiallyconnectedinterferingMIMOMACs(PCI-MIMO-MAC).Weconsidersymmetricsystems,wherebyallinvolvedMACshavethesamedimension,namelyMantennasatthereceiverandNantennasateachofthePtransmitters,andconsideranetworkcomprisedofKsuchMACs(seeFig.1).Fork2f1;:::;Kg,wewritetheM-dimensionalsignalreceivedatthek-thMACasy(k)=PXp=1H(k;k)px(k)p+Xl2I(k)PXp=1H(k;l)px(l)p;(1)wherex(l)pdenotestheN-dimensionalvectorsignalfromthep-thtransmitterinthel-thMAC,andH(k;l)pistheMNmatrixchannelfromthistransmittertothereceiverofthek-thMAC.Thersttermin(1)representsthesuperpositionofsignalsfromusersintheconsideredMAC,whilethesecondtermaccountsfrominterferencefromtheotherMACs.I(k)f1;:::;k�1;k+1;:::;KgdenotesthesetofMACswhichinterferewiththek-thMAC(forthesakeofsimplicity,weassumethateitherallornoneofthetransmittersinagivenMACcanbeheardbythereceiverofthek-thMAC).WealsoletI�1(l)=fkjl2I(k)gforl2f1;:::;Kg,i.e.I�1(l)isthesetofMACswhichareaffectedbyinterferencefromthetransmittersofMACl.IntheexamplepicturedinFig.1,I(1)=f2;3gwhileI�1(1)=f2g.Notethatthismodelcanbeparticularizedinseveralways:BylettingI(k)=f1;:::;k�1;k+1;:::;Kg,oneobtainsthecaseofKfullyconnectedinterferingMACs.BylettingI(k)=fk�1;k+1g8k2f2;:::;K�1g,weobtaintheclassicalWynermodel. Rx1 RxK Rx2 Tx1,1 Tx1,P Tx2,1 Tx2,P TxK,1 TxK,P Rx3 Tx3,1 Tx3,P MAC1 MAC2 MAC3 MACK Fig.1.ExampleconnectivitygraphinapartiallyconnectedinterferingMACsmodel.Linesrepresentchannelswithnon-zerocoefcients.BylettingP=1,weobtainthepartiallyconnectedMIMOIC.Wenowintroduceanotherparticularizationofthismodel,whichwewilluseinthesequel.Denition1(L-interferingMACsnetwork):ThenetworkformedbytheKpartiallyconnectedMIMOMACsasdenedineq.(1)isL-interferingforsomeLK,iff8k2f1;:::;Kg;jI(k)jLandI�1(k)L;(2)wherejjdenotesthecardinalityoperator.Intuitively,Denition1correspondstothecasewherethenumberofMACsinterferingwiththereceiverofanyMACinthenetwork,andthenumberofMACswhosereceiversgetinterferencefromagiventransmitter,areboundedbysomevalueL.Wedeemthismodelmorerealisticthane.g.theWynermodel,sinceitcancorrectlyrepresenttheinterferencesituationencounteredinacellularmodelwithregularcellarrangementandinter-cellinterferencelimitedtoaxedradius,whilethe(single-dimensional)Wynermodelcannotfaithfullyreplicatetheconnectivitysituationencounteredinsucha2-dimensionalnetwork.III.IAOVERTHEPCI-MIMO-MACNETWORKWearenowconcernedwithachievingIAinthePCI-MIMO-MACnetworkintroducedintheprevioussection.A.DenitionAssumethatthesignaltransmittedbyuserpinthel-thMACislinearlyprecodedbytheNDfullcolumnrankmatrixV(l)p,i.e.x(l)p=V(l)ps(l)pwheres(l)pisavectorwithDNcoefcients.WewishallinterferenceatthereceiverofagivenMACkinthenetwork,comingfromtransmittersinotherMACstobe Theorem1:Asufcientconditionforthesystemofequa-tion(3)tobeproperforthecaseofanL-interferingMIMOMACsnetwork(Denition1)isthatPD(N�D)+D0(M�D0)�DD0PL0:(5)Furthermore,(5)isalsoanecessaryconditionif8k2f1;:::;Kg;jI(k)j=L.Notethat(5)isindependentfromthetotalnumberKofMACsinthenetwork.Intuitively,thisisbecausetheconstraintin(2)ensuresthatthenumberofscalarequationsinvolvedinthesystemofequations(3)(oranysubsetthereof)scaleslinearlywithKinsteadofquadraticallyforthefullyconnectedMIMOICwherejI(k)j=K�18k,andsodoesthenumberofvariables.WenowgiveaformalproofofTheorem1.Proof:Inarststep,weprovethat(5)isnecessaryinthecasejI(k)j=L8k.Forthis,weconsiderthetotalnumberofequationsandvariablesinvolvedin(3).Thenumberofdistincttuples(k;l;p)involvedin(3)istriviallyKLP,andeachofthematrixequalitiesrepresentsD0Dscalarequations.ThisyieldsatotalofNe=KLPDD0equations.ThenumberofvariablesintheV(l)pandU(k)matricesmustbecountedwhilepayingattentiontothefactthatmultipleparameterizationsofthesamechoiceofasubspacearepossible,andmustbecountedonlyonce.Asshownin[2],eachV(l)pmustbecountedasD(N�D)variables,whileeachU(k)representsD0(M�D0)variables.Therefore,wehaveNv=KPD(N�D)+KD0(M�D0).Noticenowthatif(5)isnotfullled,wehaveimmediatelythatNvNe,i.e.thesystemisnotproper.Therefore,(5)isnecessary.Wenowprovethesufcientpart.Weneedtocheckthattheinequalitybetweennumberofequationsandvariablesisveriedforallpossiblesubsetsoftheequations.Letusintroducesomeformalism.LetS=f(d0;k;l;p;d)2f1;:::;D0gf1;:::;Kgf1;:::;Kgf1;:::;Pgf1;:::;Dgs:t:l2I(k)g.EachtupleinScorrespondstoonescalarIAequationfromeq.(3).LetASanarbitrarysubsetofS.LetNAvdenotethenumberofvariablesinvolvedinanyoftheequationsdesignatedbyA,andNAe=jAjthenumberofthoseequations.WeneedtoprovethatNAvNAe.Weneedthefollowingdenitions: K=fks:t:9(d0;l;p;d)s:t:(d0;k;l;p;d)2Ag(6) LP=f(l;p)s:t:9(d0;k;d)s:t:(d0;k;l;p;d)2Ag(7) KL=f(k;l)s:t:9(d0;p;d)s:t:(d0;k;l;p;d)2Ag(8) KLP=f(k;l;p)s:t:9(d0;d)s:t:(d0;k;l;p;d)2Ag(9) D(l;p)=fds:t:9(d0;k)s:t:(d0;k;l;p;d)2Ag(10) D0(k)=fd0s:t:9(l;p;d)s:t:(d0;k;l;p;d)2Ag(11)Intuitively, KisthesetofindicescwhichappearinatleastonetupleinA; D(l;p)isthesetofindicesdwhichappearinatleastonetupleinAtogetherwithagiven(l;p);etc.Usingthesedenitions,thenumberofvariablesin-volvedinthebeamformerattransmitterpinMAClis D(l;p)�N� D(l),whilethenumberofvariablesinvolvedintheprojectionlteratreceiverkis D0(k)�M� D0(k).WehavethereforeNAv=Xk2 K D0(k)�M� D0(k)+X(l;p)2 LP D(l;p)�N� D(l;p)(12)Xk2 K D0(k)(M�D0)+X(l;p)2 LP D(l;p)(N�D)(13)sincethecardinalitiesof D(l)and D0(k)areupperboundedrespectivelybyDandD0bydenitionofthesets.Letusnowxk,landp,andconsiderthetuples(d0;k;l;p;d)thatappearinA.Clearlythereareatmost D(l;p) D0(k)suchtuples.Therefore,summingoverallpossible(k;l;p),jAjX(k;l;p)2 KLP D(l;p) D0(k):(14)Since D0(k)D08k,wehavejAjX(k;l;p)2 KLP D(l;p)D0(15)X(l;p)2 LPI�1(l) D(l;p)D0(16)X(l;p)2 LPL D(l;p)D0(17)where(16)stemsfromthefactthat(k;l;p)2 KLPimpliesk2I�1(l),andthatI�1(l)hasatleastasmanyelementsasitsrestrictiontothoseappearinginA.(17)stemsdirectlyfromDenition1.Similarly,startingagainfromeq.(14),jAjX(k;l;p)2 KLPD D0(k)PX(k;l)2 KLD D0(k)(18)PXk2 KjI(k)jD D0(k)LPDXk2 K D0(k):(19)Combining(13),(17)and(19)yieldsNAvjAjM�D0 LPD+N�D LD0:(20)Finally,(5)ensuresthatthesecondtermintheright-handsideof(20)isgreaterorequalto1,yieldingNAvjAj=NAe. A.DiscussionaboutsufciencyAsmentionedabove,havingapropersystemofequations(3)isnotsufcienttoguaranteethatIAaccordingtoDenition2isfeasible.Indeed,takingtheexamplefrom[2]ofthe(33;2)2IC(whichinournotationscorrespondstoK=2,L=1,P=1,M=N=3,D=D0=2),eq.(5)isfullled,whileIAisnotfeasiblesinceitwouldotherwiseviolateageneralboundontheachievableDoF[15],whichwerepeathere:thetotalDoFachievableoveratwo-userICwithrespectivelyM1andM2antennasatthereceiversand