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A Matrix-Algebraic Approach to Successive Interference Cancellation in CDMA

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A Matrix-Algebraic Approach to Successive Interference Cancellation in CDMA Lars K. Rasmussen & Teng J. Lim Centre for Wireless Communications 20 Science Park Road #02-34/37 Teletech Park Singapore Science Park II Singapore 117674 E-Mail: cwclkr@leonis.nus.edu.sg E-Mail: cwclimtj@leonis.nus.edu.nus.sg FAX: +65 779 5441 Department of Signals and Systems Chalmers University of Technology S-412 96 Gothenburg Sweden E-Mail: anne@s2.chalmers.se Ann-Louise Johansson Submitted to IEEE Transactions on
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  AMatrix-AlgebraicApproachtoSuccessiveInterference  CancellationinCDMA     LarsK.Rasmussen&TengJ.Lim  CentreforWirelessCommunications 20ScienceParkRoad #02-34/37TeletechPark SingaporeScienceParkII Singapore117674 E-Mail:cwclkr@leonis.nus.edu.sg E-Mail:cwclimtj@leonis.nus.edu.nus.sg FAX:+657795441  Ann-LouiseJohansson  DepartmentofSignalsandSystems ChalmersUniversityofTechnology S-41296Gothenburg Sweden E-Mail:anne@s2.chalmers.se  SubmittedtoIEEETransactionsonCommunications 14May1997,revisedJune1998  Abstract  InthispaperwedescribethelinearSICschemebasedonmatrix-algebra.Weshowthat thelinearSICschemes(single-andmulti-stage)correspondtolinearmatrixlteringthatcan beperformeddirectlyonthereceivedchip-matchedlteredsignalvectorwithoutexplicitly performingtheinterferencecancellation.Thisleadstoananalyticalexpressionforcalculating theresultingbiterrorratewhichisofparticularuseforshort-codesystems.Convergence issuesarediscussedanditisshownthatthesimpleimplementationofthelinearSICprovides similarorbetterperformancethanthedecorrelatoratonlyafewstages.Theconceptof    - convergenceisintroducedtodeterminethenumberofstagesrequiredforpracticalconvergence forbothshortandlongcodes.IthaspreviouslybeenobservedthatthelinearSIChasan optimumnumberofstagesforwhichthebiterrorrateisminimised.Thisbehaviourishere relatedtothemeansquarederrorwhichcanbeusedtoestimatethenumberofstagesrequired tominimisethebiterrorrate.  I.Introduction  Inamobilecommunicationssystemmultipleaccesstothecommonchannelresourcesisvital.In asystembasedonspread-spectrumtransmissiontechniquescodedivisionprovidessimultaneous accessformultipleusers.Byselectingmutuallyorthogonalcodesforallusers,theyeachachieve interferencefreesingle-userperformance.Itishowevernotpossibletomaintainorthogonal spreadingcodesatthereceiverinamobileenvironmentandthusmultiple-accessinterference (MAI)arises.Conventionalsingle-userdetectiontechniquesareseverelyaectedbyMAI,making suchsystemsinterferencelimited1].TraditionalmatchedlterreceiversforCDMAalsorequire strictpowercontrolinordertoalleviatethenear-farproblemwhereahigh-poweredusercreates signicantMAIforlow-poweredusers. 1   L.K.Rasmussen,T.J.Lim,A.-L.Johansson: AMatrix-AlgebraicApproach...  Page2 Moreadvanceddetectionstrategiescanbeadoptedtoimproveperformance.In2]Verd ude- velopedtheoptimalcomplexity-unconstrainedmaximum-likelihood(ML)detectorformultiuser CDMA.Thisdetectorperformsanexhaustivesearchovertheconstrainedspaceofpossiblehy- potheses.Theinherentcomplexityhoweverincreasesexponentiallywiththenumberofusers, renderingtheoptimalMLdetectorimpractical. Forpracticalimplementationparallelandsuccessiveinterferencecancellation(SIC)schemes havebeensubjecttomostattention.Thesetechniquesrelyonsimpleprocessingelementscon- structedaroundthematchedlterconcept.Therststructurebasedontheprincipleofinter- ferencecancellationwasthemulti-stagedetectorin3].Herethecancellationisdecision-directed (i.e.,non-linear)andisdoneinparallel.In4]Dentetal.proposedaserialapproach,asingle- stagenon-linearSICscheme.Thisschemehasbeenanalysedindetailin5].In6],Kawabeet al.suggestedamulti-stagenon-linearSICtechnique.Acloselyrelatedschemewassuggested bySawahashietal.in7].JohanssonandSvenssonhavesuggestedbothsingle-andmulti-stage linearSICdetectorsin8],whileJamalandDahlmanhavecomparedtheperformanceofthe linearandthenon-linearSICapproachesin9]. AnalgebraicapproachtoSICwasinitiallyintroducedin10]andfurtherdevelopedin11]. CloselyrelatedworkbyElders-Bolletal.waspresentedin12]-14]wheretheysuggestlinear detectorsbasedontheapplicationofclassiciterativetechniquesforsolvinglinearsystems.The Gauss-SeideliterationwashereidentiedasSIC.Iterativemethodsforlineardetectordesign havealsoalsobeenproposedbyJunttietal.in15].Theequivalencetointerferencecancellation washowever,notrecognised. InthispaperwedescribethelinearSICschemebasedonmatrix-algebra.Weshowthatthe linearSICschemes(single-andmulti-stage)correspondtolinearmatrixlteringthatcanbeper- formeddirectlyonthereceivedchip-matchedlteredsignalvectorwithoutexplicitlyperforming theinterferencecancellation.Thisleadstoananalyticalexpressionforcalculatingtheresulting biterrorrate(BER)whichisofparticularuseforshort-codesystems.Convergenceissuesare discussedanditisshownthatthesimpleimplementationofthelinearSICprovidessimilaror betterperformancethanthedecorrelatoratonlyafewstages.Theconceptof    -convergenceis introducedtodeterminethenumberofstagesrequiredforpracticalconvergenceforbothshort andlongcodes.IthaspreviouslybeenobservedthatthelinearSIChasanoptimumnumberof stagesforwhichtheBERisminimised.Thisbehaviourishererelatedtothemeansquarederror (MSE)whichcanbeusedtoestimatethenumberofstagesrequiredtominimisetheBER. Thepaperisorganisedasfollows.InSectionIItheuplinkmodelisdescribedandthetech- niquesforSICarebrieysummarised.InSectionIIIweintroduceamatrix-algebraicapproachfor describinglinearSIC,whichallowsfornewinsightintothebehaviouroftheschemes.Theequiv- alentmatrixltersforlinearSICarederivedandconvergenceissuesarediscussedformulti-stage schemes,includingtheconceptof    -convergenceforbothshortandlongcodes.Anexpression fortheMSEoftheschemeisderivedtondthenumberofstagesrequiredtominimisetheMSE whichinmostcasescorrespondstominimisingtheBER.Numericalexamplesarepresentedin SectionIVandconclusionsaredrawninSectionV.Throughoutthispaperscalarsarelower-case, vectorsareboldfacelower-case,andmatricesareboldfaceupper-case.Subscriptingisdropped wherenoambiguitiesarise.Thesymbols(    )  >  ,(    )  H  ,(    )  ?  1  and  kk  arethetransposition,hermi-   L.K.Rasmussen,T.J.Lim,A.-L.Johansson: AMatrix-AlgebraicApproach...  Page3 tian,inversionandEuclideanvector-normoperatorsrespectively,andthedelimiter  fg   y  denes aspaceofdimension  y  .Allvectorsaredenedascolumnvectorswithrowvectorsrepresented bytransposition.  R  denotesthesetofrealnumbers,andthefollowingnotationisusedforthe productofmatrices,  n  2  Y   i  =  n  1  X  i  =  (  X  n  2  X  n  2  ?  1    X  n  1  +1  X  n  1  if  n  1     n  2  I  if  n  1  >n  2  :  (1)  II.SystemModels  Inthissection,themodelfortheuplinkoftheCDMAcommunicationsystemconsideredthrough- outthispaperisbrieydescribed.Theuplinkmodelisbasedonadiscrete-timesymbol- synchronousCDMAsystemassumingsingle-pathchannelsandthepresenceofstationaryad- ditivewhiteGaussiannoise(AWGN)withzeromeanandvariance    2  =  N  0  =  2.Also,thelinear SICschemeisdescribedandtherelevantnotationisintroduced.  A.  UplinkModel  Aspecicuserinthis  K  -usercommunicationsystemtransmitsabinaryinformation-symbol  d  k  2f?   1  ;  1  g   ,bymultiplyingwithaspreadingcode  s  k  2   n  ?  1  p  N  ;  1  p  N  o  N  ,oflength  N  chipsand thentransmittingoveranAWGNchannelusingBPSK  1  .Thespreadingcodestransmittedby eachuserinanygivensymbolintervalareassumedtobesymbol-synchronousandthechannel imposesnophaserotationonthetransmittedsignal.Symbol-synchronismisassumedforclarity. Similarargumentsholdforthesymbol-asynchronouscaseasdemonstratedin14].Eachuseris receivedatauser-specicenergylevel  c  2  k  thatisassumedconstantoveronebit-interval.Note thatwehaveassumedthat  s  >  k  s  k  =1.Theoutputofachip-matchedlteristhenexpressedasa linearcombinationofspreadingcodes,specically,thechipmatchedlteredreceivedvector,  r  ,is acolumnvectoroflength  N  ,encompassingthetransmissionsforallusers.Thereceivedvector  r  ishencedescribedthroughmatrix-algebraas  r  =  SCd  +  n  2   R  N  ;  (2) where  S  =(  s  1  ;  s  2  ;    ;  s  K  )  2     ?   1  p    N ;  1  p    N    N;K  ;  (3)  C  =diag(  c  1  ;c  2  ;    ;c  K  )  2   R  K;K  ;  (4)  d  =(  d  1  ;d  2  ;    ;d  K  )  >  2f?   1  ;  1  g   K  :  (5) Toavoidrankdeciencies,weassumethat  K     N  and  S  hasfullrank,i.e.,thespreadingcodes forallusersarelinearlyindependent.Thesamplednoisecorruptingtheoutputofthechip- matchedlterisindependentineachsamplesincethechannelnoiseisassumedtobewhiteand thechipwaveformsareassumedtofullltheNyquistcriterion(e.g.rectangularchippulses).We  1  Binarydataandchipformatsareassumedforclarity.Allthepresentedconceptsgeneraliseto  m  -aryformats.   L.K.Rasmussen,T.J.Lim,A.-L.Johansson: AMatrix-AlgebraicApproach...  Page4 thereforeobtainanoisevector  n  whereeachsampleisGaussiandistributedwithzeromeanand variance  N  0  =  2. Thereceivedvectoriscontainedinavectorspaceofdimension  N  ,  r  2   R  N  .Itis,however, onlythepartof  r  residinginthesignalspace  2  thatisaectingthedetectordecision.Thesignal spaceisdeterminedbyspan  f   S  g   .If  N  =  K  ,  r  2   span  f   S  g   .Ingeneralhowever,  r  =  r  s  +  r  s    with  r  s  2   span  f   S  g   and  r  s    2   null  ?  S  >    wherenull  ?  S  >    denotesthenullspaceof  S  >  orequivalently,the orthogonalcomplementof  S  .Wecandetermine  r  s  and  r  s    byorthogonalprojections.  r  s  =  S    S  >  S    ?  1  S  >  r  =  P  S  r  (6)  r  s    =(  I  ?   P  S  )  r  =  P  S    r  :  (7) Thiswillbeofsignicanceforconvergencerateconsiderations.  B.  LinearSuccessiveInterferenceCancellation  Successiveinterferencecancellationschemesarebestdescribedbydeninganinterferencecan- cellingunit(ICU)asshowninFigure1.Thisunitisthenusedasabuilding-blockinthe multi-stageSICschemeshowninFigure2.Thesingle-stageschemeisobtainedbyomittingall buttherststage.Itisassumedthatthe  K  usersareorderedaccordingtotheirreceivedsignal power.TheinputresidualsignalvectortoanICUofuser  k  atstage  i  is  e  i;k  .Fortherstuser intherststage  e  1  ;  1  =  r  .ThecontributiontobecancelledintheICUofuser  k  atstage  i  is  s  k  s  >  k  e  i;k  .Ingeometricaltermsthisisaprojectionoftheresidualreceivedvector  e  i;k  ontothe relevantspreadingcode  s  k  .Fortherststage  y  0  ;k  =0forallICUblocks.  III.Matrix-AlgebraicApproaches  Throughoutthissectionthesubscriptingisaccordingtothefollowingconventions.Variables independentofthedetectorstageis,whenneeded,subscriptedwithauserindex,i.e.,  x  k  .The rstsubscriptonvariablesdependentonthedetectorstage,  x  i;k  ,denotesthecurrentstage,the secondsubscripttheuserindex.  A.  LinearSuccessiveInterferenceCancellation  TherstuserintherststageoftheSICschemeisoperatingdirectlyonthereceivedsignal vector  e  1  ;  1  =  r  .Therst-stagelteroutputis  y  1  ;  1  =  s  >  1  e  1  ;  1  =  s  >  1  r  ,leadingtotheresultinginput vectorforthenextunitas  e  1  ;  2  =  e  1  ;  1  ?   s  1  s  >  1  e  1  ;  1  =    I  ?   s  1  s  >  1    r  ;  (8) where  I  isan  N     N  identitymatrix.Thenextstepisthen  y  1  ;  2  =  s  >  2  e  1  ;  2  =  s  >  2  ?  I  ?   s  1  s  >  1    r  ,and  e  1  ;  3  =  e  1  ;  2  ?   s  2  s  >  2  e  1  ;  2  =    I  ?   s  2  s  >  2    I  ?   s  1  s  >  1    r  :  (9)  2  Theirrelevancetheoremallowsfortheportionofthenoisewhichliesoutsideofspan  f  S  g  tobeignored16]. 

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