reconstructsoftmod

HindawiPublishingCorporationEURASIPJournalonWirelessCommunicationsandNetworkingVolume2008,ArticleID390489,15pagesdoi:10.
1155/2008/390489ResearchArticleDistributedIterativeMultiuserDetectionthroughBaseStationCooperationShahidKhattak,WolfgangRave,andGerhardFettweisVodafoneChairMobileCommunicationsSystems,TechnischeUniversit¨atDresden,01062Dresden,GermanyCorrespondenceshouldbeaddressedtoShahidKhattak,khattak@ifn.
et.
tu-dresden.
deReceived1August2007;Revised18December2007;Accepted13February2008RecommendedbyHuaiyuDaiThispaperdealswithmultiuserdetectionthroughbasestationcooperationinanuplink,interference-limited,highfrequencyreusescenario.
Distributediterativedetection(DID)isaninterferencemitigationtechniqueinwhichthebasestationsatdierentgeographicallocationsexchangedetecteddataiterativelywhileperformingseparatedetectionanddecodingoftheirreceiveddatastreams.
ThispaperexplorespossibleDIDreceivestrategiesandproposestoexchangebetweenbasestationsonlytheprocessedinformationfortheirassociatedmobileterminals.
Theresultingbackhaultracisconsiderablylowerthanthatofexistingcooperativemultiuserdetectionstrategies.
Single-antennainterferencecancellationtechniquesareemployedtogeneratelocalestimatesofthedominantinterferersateachbasestation,whicharethencombinedwiththeirindependentreceivedcopiesfromotherbasestations,resultinginmoreeectiveinterferencesuppression.
Sincehardinformationbitsorquantizedlog-likelihoodratios(LLRs)aretransferred,weinvestigatetheeectofquantizationoftheLLRvalueswiththeobjectiveoffurtherreducingthebackhaultrac.
Ourndingsshowthatschemesbasedonnonuniformquantizationofthe"softbits"allowforreducingthebackhaulto1–2exchangedbits/codedbit.
Copyright2008ShahidKhattaketal.
ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
1.
INTRODUCTIONAnevergrowingdemandfornewbroadbandmultimediaservicesemphasizestheneedforhigherspectraleciencyinfuturewirelesssystems.
Ahigher-frequencyreuseisthereforeproposed,resultingintheinterferencefromcochannelusersoutsidethecellstodominate,therebyformingasinglemostimportantfactorlimitingthesystemperformance.
Thisinterferencecomingfromoutsidethecellboundariesiscommonlyreferredtoasothercellinterference(OCI).
OCIhasbeentreatedin[1],whereitwassuggestedthatadvancedreceiverandtransmittertechniquescanbeemployedintheuplinkanddownlinkofacellularsystem,respectively.
Giventhatthemobileterminals(MTs)arelow-cost,low-powerindependententities,andarenotexpectedtocooperatetoperformtransmitorreceivebeamforming,theyareassumedtobeassimpleaspossiblewithmostofthecomplexprocessingofacellularsystemmovedtothebasestations(BSs).
Inthispaper,werestrictourselvestoadvancedreceivertechniquesforuplinkcommunication.
Dierentadvancedreceivertechniques,suggestedintheliteraturefortheuplink,givetradeosbetweencomplexityandperformance.
Optimummaximumlikelihooddetection(MLD)[2,3]isprohibitivelycomplexformultiple-inputmultiple-output(MIMO)scenariosemployinghigher-ordermodulation.
Linearreceivers[4–7]aresimpler,butlesseectiveindecouplingtheincomingmultiplexeddatastreams,andoerlowspatialdiversityforfull-ranksystems.
Iterativereceivers[8–10]withsoftdecisionfeedbackoerthebestcompromisebetweencomplexityandperformance,andtheyhavebeenuniversallyadoptedasastrategyofchoice.
OneprincipallineofthoughttoaddresstheOCIproblemwasinitiatedbyWyner'streatmentofbasestationcooperationinasimpleandanalyticallytractablemodelofcellularsystems[11].
Inthismodel,cellsarearrangedineitheraninnitelineararrayorinsometwo-dimensionalpattern,withinterferenceoriginatingonlyfromtheimme-diateneighboringcells(havingacommonedge).
Alltheprocessingisperformedatasinglecentralpoint.
Subsequentworkontheinformationtheoreticcapacityofthecentralizedprocessingsystemsconcludedthattheachievablerateper2EURASIPJournalonWirelessCommunicationsandNetworkingusersignicantlyexceedsthatofaconventionalcellularsystem[12,13].
Recently,decentralizeddetectionusingthebeliefpropa-gationalgorithmforasimpleone-dimensionalWynermodelwasproposedin[14].
ThebeliefpropagationalgorithmeectivelyexchangestheestimatesforallsignalsreceivedateachBS,byalternatelyexchanginglikelihoodvaluesandextrinsicinformation.
Thisideawasextendedto2Dcellularsystemsin[15–17],wherethelimitscomparedtoMAPdecodingwerestudied,showingthegreatpotentialofBScooperationwithdecentralizedprocessing(atleastforregularsituations).
Unfortunately,forastarnetwork(commonlyusedtoday)interconnectingtheBSs,thisresultsinahugebackhaultrac.
Anotherapproachtoconvertsituationswherecochannelusersinterfereeachotherwithcomparablystrongsignalsintoanadvantageforahigh-frequencyreusecellularsystemwasproposedin[18]:dierentBSscooperatebysendingquantizedbasebandsignalstoasinglecentralpointforjointdetectionanddecoding.
Suchadistributedantennasystem(DAS)notonlyreducestheaggregatetransmittedpower,butalsoresultsinmuchimprovedreceivedSINR[19].
Usingappropriatereceivestrategies,botharrayanddiversitygainsareobtained,resultinginasubstantialincreaseinsystemcapacity[20,21].
TheDASscheme,however,islessattractivefornetworkoperatorsduetothelargeamountofbackhaulitrequiresandthecooperativeschedulingnecessarybetweentheadjacentDASunitsinordertoavoidinterference.
Here,backhaulisdenedastheadditionalcommunicationlinkbetweendierentcooperatingentities.
Althoughtheband-widthofwiredlinksusedforbackhaulcanbeveryhigh,theyareusuallyownedbyathirdparty,makingitattractiveforthecellularsystemoperatorstoreducethebackhaulinordertominimizeoperatingcosts.
TheinuenceoflimitedbackhauloncapacityinDAShasbeeninvestigatedin[22,23].
Similarlyasinthementionedworks,weareinterestedinasymmetricmultiuserdetectionscenarios.
Weassumethattheresourcemanagementofthecellularnetworkcandetect(e.
g.
,viasignalstrengthindicators)groupsofMTsthatarestronglyreceivedatseveralbasestations.
However,incontrastto[15,17]andrelatedwork,ourmaininterestisnotthenetworkwideoptimuminformationexchange,butratheritsdecentralizedimplementation.
Tothisend,theconceptofdistributediterativedetection(DID)wasintroducedin[24,25]:eachbasestationinitiallyperformssingle-userdetectionforthestrongestMT,treatingthesignalsreceivedfromallothermobileterminalsasnoise.
TheinformationthatbecomesavailableatthedecoderoutputisthensenttoneighboringBSwhilemutuallyreceivingdatafromitsownneighborsinordertoreconstructandcanceltheinterferenceofitsownreceivedsignal.
Single-userdetectionisthenappliedtothisinterference-reducedsignalbyapplyingparallelinterferencecancellation[26].
Furtherimprovementscanbeachievedbyrepeatedapplicationofthisprocedure.
Thequestionswetrytoanswerhereareasfollows.
(i)Howmuchimprovementcanwegetwithrespecttoconventionalsingle-userdetectionindierentscenarios(varyingstrengthoftheusercouplingthroughthechannel)(ii)Whichadditionalgainispossibleifwereplacethesingle-userdetectionstepinthe0thiterationwithsingle-antennainterferencecancellation(SAIC)whichisimplementedasjointmaximumlikelihooddetection(JMLD)inthesymboldetectoractingasthereceiverfrontend(iii)Whatisareasonabletradeobetweentheamountofinformationexchangeandimprovementbeyondsingle-userdetectionOr,statedotherwise,whathappensunderconstraintsforthemaximumavailabledatarateoverthebackhaullinksbetweenbasestationsandassociatedniteprecisioneectsduetoquantizationTheorganizationoftheremainderofthispaperisasfollows.
Section2presentsthesystemmodel,wherethecouplingamongusers/cellsandthechannelmodelaredescribed.
Section3discussesindetailvariouscomponentsofdistributediterativereceivers.
InSection4dierentdecen-tralizeddetectionstrategiesarecompared.
InSection5weexaminetheeectofquantizationofreliabilityinformation.
Wecomparevariousquantizationstrategiesintermsofinformationlossandnecessarybackhaultrac.
NumericalresultsarepresentedinSection6beforeconclusionsaredrawn.
NotationThroughoutthepaper,complexbasebandnotationisused.
Vectorsarewritteninboldface.
AsetiswrittenindoublestrokefontsuchasIanditscardinalityisdenotedby|I|.
TheexpectedvalueandtheestimatesofaquantitysuchassaredenotedasE{s}ands,respectively.
Randomvariablesarewrittenasuppercaselettersandtheirrealizationwithlower-caseletters.
Aposterioriprobabilities(APPs)willbeexpressedaslog-likelihoodratios(L-values).
Asuperscriptdenotestheorigin(orreceivermodule),whereitisgenerated.
WedistinguishLd1,Ld2,andLextwhichareAPPsgeneratedatthedetectorandthedecoderofagivenBSorexternallytoit.
2.
TRANSMISSIONMODELWeconsideranidealizedsynchronoussingle-carrier(narrowband)cellularnetworkintheuplinkdirection.
NisthenumberofreceiveantennasandMisthenumberoftransmitantennascorrespondingtothenumberofBSsandcochannelMTs,respectively.
Ablockofinformationbitsumfromuserantennamisencodedandbit-interleavedleadingtothesequencexmoflengthK,wherem=1···M.
Thissequenceisdividedintogroupsofqbitseach,whicharethenmappedtoavectorofoutputsymbolsforusermofsizeKs=K/qaccordingtosm=[sm,1,.
.
.
,sm,Ks]=map(xm).
EachsymbolisrandomlydrawnfromacomplexalphabetAofsizeQ=2qwithE{sm,k}=0andE{|sm,k|2}=σ2sform=1···M.
AblockofKssymbolvectorss[k]=[s1,k,s2,k,.
.
.
,sM,k]T(correspondingtoonerespectivecodeword)istransmittedsynchronouslybyallMusers.
AtanyBSl,acorrespondingblockofsymbolsrl[k]isreceived,wheretheindexkisrelatedtotimeorsubcarrierindices(1≤k≤Ks):rl[k]=gl[k]·s[k]+n[k],1≤k≤Ks.
(1)ShahidKhattaketal.
3WithnwedenotetheadditivezeromeancomplexGaussiannoisewithvarianceσ2n=E{n2}.
Foreaseofnotation,weomitthetimeindexkinthefollowing,becausethedetectoroperatesoneachreceivesymbolrlseparately.
Therowvectorglistheelementwiseproductgl,m=hl,m√ρl,mofweightedchannelcoecientshl,mofMco-channelsseenatthelthBS.
Thechannelcoecientvectorhl,obtainedasthecurrentrealizationofachannelmodel(thechannelispassiveontheaverage,i.
e.
,E{|hl,m|2}=1),isassumedtobeknownperfectly.
Thecouplingcoecientsρl,mreectdierentuserpositions(pathlosses)withrespecttobasestationl.
ThesewillbeabstractedinthefollowingbytwocouplingcoecientsρiandρjwhichcharacterizetheBSinteractionwithstrongandweakinterferers.
Equation(1)canthereforebewrittenintermsofthedesiredsignal(denotedwiththeindexd)andweakandstronginterferences:rl=gldsd+i∈Ilglisi+j∈Ilgljsj+n=hldsd+ρii∈Ilhlisistronginterference+ρjj∈Ilhljsjweakinterference+n,(2)whereρld=1.
Wenotethatthisisofcourseavariantofthetwo-dimensionalWynermodel.
WithIlwedenotethesetofindicesofallstronglyreceivedinterferersatBSlwithcardinality|Il|=ml1,wheremlistotalnumberofstronglyreceivedsignalsatBSl.
Additionally,Ilisthecomplementarysetforallweaklyreceivedinterferers:|Il∪Il|=M1.
Notethatthereceivedsignal-to-noiseratio(SNR)isdenedastheratioofreceivedsignalpoweratthenearestBSandthenoisepower.
Specically,theSNRatthelthBScanbewrittenasSNR=E{hldsd2}/E{n2}=σ2s/σ2n.
Theconsideredsynchronousmodelisadmittedlysome-whatoptimisticandwasrecentlycriticizedduetotheimpos-sibilitytocompensatedierentdelaystodierentmobiles(positions)simultaneously[27].
However,thereasontoignoresynchronizationerrorsistwofold.
First,itallowstostudythepossibleimprovementthroughbasestationcooperationwithoutotherdisturbingeectstoobtainbounds(thedegradationfromnonidealsynchronizationshouldthereafterbeincludedasasecondstep).
Second,forOFDMtransmissionorfrequencydomainequalizationthatweenvisageinordertoobtainparallelatchannelsenablingseparateJMLDoneachsubcarrier,wearguethatitispossibletokeeptheinterferenceduetotimingandfrequencysynchronizationerrorsatacceptablelevels.
IncreaseddelayspreadsofmoredistantMTshavetobehandledbyanappropriatelyadjustedguardintervalinthecooperatingregion.
Timingdierencesbetweenmobilesleadtophaseshiftsinthechanneltransferfunction,whicharetakenintoaccountwiththechannelestimate.
ConcerningfrequencyosetsduetovariationsamongoscillatorsandDopplereects,onehastoevaluatetheintercarrierinterferenceinducedbyrelativeshiftsofthesubcarrierspectraofdierentusers.
Roughlyestimatingthiswiththesinc2(f/fsub)functionofthepowerspectraldensityforadjacentsubcarriers,theSINRshouldstillbeCell1Cell2Cell4Cell3d44d14d33d13d11d12d22BSMTρ11=d11d11γ=1ρThρ12=d22d22γ≈1ρThStrongsignalsρ13=d33d13γ6dB,theSAIC-DIDschemesonlystartconvergingatanSNRhigherthan5dB.
ForatypicalcellularsetupusingdirectionalBSantennaswithdown-tilt,mlnormallystaysbetween2and4for4-QAM,resultingintheFERwaterfalltobelocatedaround5dB.
6.
2.
SAIC-DIDwithunquantizedLLRexchangeToseehowtheperformanceofareceivestrategyscaleswiththesizeofthenetwork,Figure11depictsa2*2cellnetworkincomparisontoa3*3cellnetworkfordierentvaluesofthenormalizedpathlossρi.
ThenumberofdominantreceivedsignalsateachBSisxedto4.
Forthesolidcurves,thesetIlisdenedaccordingto(32),withthemodulooperationensuringthatsymmetryconditionsareincorporated;thatis,eachMTisreceivedby4BSs,whileeachBSreceives4MTs.
Interestingly,theperformancefora2*2100101102FER202468101214Eb/N0(dB)ρi=0dBρi=3dBρi=6dBρi=10dBIsolatedρi=randomρj=0(∞dB)ml=43*3setup,1/2pccc(mem2),4-QAM,IIDRayleighchannelFigure8:Eectofpathlossofthedominantinterfererρi,SAIC-DID.
Forthedashedcurvelabeledas"random",eachelementofthepath-lossvector0≤ρl,m≤1,l/=m,israndomlygeneratedwithuniformdistribution.
100101102FER05101520Eb/N0(dB)0iteration1iteration2iteration5iterations10iterationsIsolatedρi=0.
25(6dB)ρj=0(∞dB)ml=43*3,DID,1/2pccc(mem2),4-QAM,IIDRayleighchannelFigure9:IterativebehaviorofSAIC-DIDexchangingsoftAPPvalues.
cellnetworkwithgreatermutual-couplingisonlyslightlyworsethanina3*3cellsetup.
Themutual-couplingina3*3cellsetupcanbeincreasedbysymmetricallyplacingthedominantinterferersoneithersideoftheleadingdiagonal.
Theresultingdierenceinperformancebetweenthesetupsoftwosizesisfurtherreduced(dashedlines).
Thissuggeststhatforagivennumberofdominantinterferersmlandcouplingρi,theperformancedependsonthesizesofthecyclesthatareformedbyexchanginginformationamongtheBSs.
ShahidKhattaketal.
11100101102103FER12345mlρi=10dBρi=6dBρi=3dBSNR=5dBρj=0(∞dB)3*3,DID,1/2pccc(mem2),4-QAM,IIDRayleighchannelFigure10:FERforSAIC-DID,plottedasfunctionofthenumberofdominantcochannelsignalsmlatSNR=5dB.
100101102103FER202468101214Eb/N0(dB)2*2-ρi=6dB2*2-ρi=3dB2*2-ρi=0dB3*3-ρi=6dB3*3-ρi=3dB3*3-ρi=0dBρj=0(∞dB)ml=41/2pccc(mem2),4-QAM,IIDRayleighchannelFigure11:SAIC-DIDperformancecomparisonfor2*2and3*3cellssetup.
EachMTisreceivedstronglyat4BSs,whileeachBSreceivessignalsfrom4MTs.
Thetwocurvesfor3*3cellsetupgivetheboundsfordierentpossiblecombinationsofcouplingswithinthesetup.
Figure12showstheperformanceofSAIC-DIDfor4-QAMand16-QAMmodulations,employinga2*2cellularsetupwithonlyasingledominantinterferer,ml=2,andvaryingthecouplingstrength.
Whiletheperformanceof4-QAMdegradesonlymarginallyforρi=0dBattheFERof102,thelossoftheperformancefor16-QAMisalreadymorethan3dB.
Thisindicatesthatwithadditionalimpairments,strongcochannelinterferersarediculttohandlefor16-QAMmodulation.
100101102103FER024681012Eb/N0(dB)Isolatedρi=6dBρi=3dBρi=0dBρj=0(∞dB)ml=24Tx-4Rx,DID,1/2pccc(mem2),IIDRayleighchannel4-QAM16-QAMFigure12:Eectofpathlossofthedominantinterfererρifordierentmodulationorders.
EachBSseesjusttwodominantsignalsml=2.
100101102103FER111213141516Eb/N0(dB)UnquantizedLLRR=2,LLR(opt)R=8,LLR(opt)R=2,soft-bit(opt)R=4,soft-bit(opt)R=8,soft-bit(opt)R=8,soft-bit(opt)ρi=1(0dB)ρj=0(∞dB)ml=44Tx-4Rx,DID,1/2pccc(mem2),4-QAMIIDRayleighchannelFigure13:EectofquantizationoftheexchangeddecoderLLRvalues,whereρi=0dB.
Curvelabeledwith"+"exchangesonlythosebitsthathavechangedsignsbetweeniterations,andadaptivelysetsthenumberofquantizationintervalsduringeachiterationtoreducebackhaul.
6.
3.
QuantizationofL-valuesandbackhaultrafcTheperformanceoftheproposedschemeforthetwodierentquantizationstrategies,optimalquantizationinthesoft-bitandLLRdomains,andfordierentnumbersofquantizationbitsispresentedinFigure13.
Thenormalizedpathlossρi=1(0dB)ischosensuchthatanylossofqualityoftheestimateshasapronouncedeectonsystemperformance.
Asalreadypredicted,quantizationinthesoft-bitdomainisclearlysuperiortothatintheLLRdomain.
Forsoft-bitdomainquantization,exchanginghardbitswill12EURASIPJournalonWirelessCommunicationsandNetworkingresultinaperformancelossofonedBwhichisreducedtoalmostonequarterofadBfor2-bitquantization(R=4).
Anyfurtherincreaseinquantizationbitswillbringlimitedgains.
Forthedashedcurvelabeledwithaplussign("+")onlythosebitsthathavechangedsignsbetweeniterationsareexchanged,andthenumberofquantizationintervalsRissetadaptivelyduringeachiterationtosavebackhaulcapacity.
ThemaximumnumberofreconstructionlevelsisRmax=8.
Itisillustratedthatdespitealargeimprovementinbackhaul,theperformancedegradesonlymarginally.
Asalreadymentioned,alldecodedinformationbitsareonlyexchangedduringtherstiterationtominimizethebackhaul,whileinthelateriterationsonlythosebitsthathavechangedsignsareexchangedafterapplyingsomelosslesscompression,forexample,run-lengthencoding[38]orvectorquantizationtechniques[39].
Figure14showsthattheaveragebackhaultracduringdierentiterationsisplottedasafunctionofSNRforahardinformationbitexchange.
Intheoperatingregionofinterest(Eb/N0>15dB),thereisnegligibletracafter3iterations.
Thetotalbackhaulinthisoperatingregionliesbetween100%and150%ofthetotalnumberofinformationbitsreceived,whichisasubstantialgainoverDASbackhaultracrequirement[19].
Itmustbementionedthatanyadditionaloverhead,requiredforthecompressiontechnique(suchasrun-length)andusedforexchangingafractionoftheestimates,wasnottakenintoaccount.
6.
4.
SensitivitytoadditionalinterferenceFinallyFigure15showsthedegradationintheperformanceofthereceiverinthepresenceofadditionalweakinterferers.
Asanexample,a(2*2)cellularsystemisconsideredwiththreeinterferers.
Itisassumedthattwointerferersarestronglyreceived(ml=3)withthenormalizedpathlossρi=1(0dB),whilethethirdoneisaweakinterfererwhosenormalizedpathlossρjcanbevaried.
Asillustrated,theperformancedeterioratessharplyifρj>10dB.
Thisisduetothefactthattheproductconstellationofthethreestrongerstreamsisquitedenselypopulatedandanysmalladditionalnoisemayresultinalargechangeinthedemapperoutputestimates,therebymakingthedecoderlesseective.
Astobeexpected,theschemesbecomemoresensitivetothisadditionalnoiseafterquantization.
WithcomparisontoFigure11(2*2,0dBcurve),onecanconcludethatitismorebenecialfortheconsideredscenariotojointlydetectallfourincomingsignalsifthenormalizedpathlossfortheweakinterfererexceeds10dB.
7.
CONCLUSIONSANDFUTUREWORKOutercellinterferenceinfuturecellularnetworkscanbesuppressedthroughbasestationcooperation.
Wepresentedanalternativestrategytothedistributedantennasystem(DAS)formitigatingOCIwhichwetermedasdistributediterativedetection(DID).
Aninterestingfeatureofthisapproachisthefactthatnospecialcentralizedprocessingunitsisneeded.
Inaddition,weexploreditsimplementation102101100101Normalizedbackhaul(%)13141516171819Eb/N0(dB)1stiteration2nditeration3rditeration4thiteration5thiterationρi=1(0dB)ρj=0(∞dB)R=2ml=44Tx-4Rx,DID,1/2pccc(mem2),4-QAM,IIDRayleighchannelFigure14:Backhaultracnormalizedwithrespecttototalinfor-mationbits.
Single-bitquantizationofLLRvaluesisperformed.
Onlythosebitsthathavechangedsignsbetweeniterationsareexchanged(ρi=0dB).
100101102FER024681012141618Eb/N0(dB)R=4-ρj=∞dBR=4-ρj=20dBR=4-ρj=10dBR=4-ρj=6dBR=2-ρj=∞dBR=2-ρj=20dBR=2-ρj=10dBR=2-ρj=6dBρi=1(0dB)ml=34Tx-4Rx,1/2pccc(mem2),4-QAM,IIDRayleighchannelFigure15:FERforSAIC-DIDinthepresenceofaweakinterferer.
ρjrepresentsthepathlossoftheweakinterferer.
withreducedbackhaultracbyperformingjointmaximumlikelihooddetectionforthedesireduserandthedominantinterferers.
Weproposetoexchangenonuniformlyquantizedsoftbitstominimizethebackhaultrac.
Interestingly,thequantizationofreliabilityinformationdoesnotresultinapronouncedperformancelossandsometimesevenhardbitscanbeexchangedwithoutunduedegradation.
Tominimizebackhaulitisfurtherproposedthatonlythosebitsthathavechangedsignsbetweeniterationsbeexchanged.
TheresultisaconsiderablereductioninbackhaultracbetweenbaseShahidKhattaketal.
13stations.
Theschemeislimitedby(undetected)backgroundinterference.
Anextensionofthisworkcouldaddressthequestionunderwhichconditionsreliabilityinformationformorethanonestreamshouldbeexchangedtoobtaindiversityandarraygainandwhenthisdoesnotpay.
Thisshouldprovidesomefurtherinsightintothetradeobetweencapacityincreaseandaordablecomplexity.
APPENDIXOPTIMUMQUANTIZATIONOFTHEL-VALUEDENSITYTooptimizethereconstruction(quantization)levelsrianddecisionlevelsdiforagivendensityp(x),wehavetoiterativelycomputetheintegralsupdatingthereconstructionlevelsgiventhecurrentdecisionlevelsdi(see(29)).
ConsiderrstthebimodalGaussiandensityofL-valuesgivenin(25).
Theintegralstobeevaluatedbecome(withμL=σ2L/2)di+1diexpxμL22σ2L+expx+μL22σ2Ldx=σLπ2erfxμL√2σL+erfx+μL√2σLdi+1di,(A.
1)anddi+1dixexpxμL22σ2L+xexpx+μL22σ2Ldx=σ2LexpxμL22σ2L+expx+μL22σ2Ldi+1di+μLσLπ2erfxμL√2σL+erfx+μL√2σLdi+1di.
(A.
2)TheoptimumpositivequantizationlevelsaredisplayedinFigure16(thenegativelevelsareobtainedbyinversionduetosymmetry).
Astobeexpected,foronequantizationbit,thelevelequalsthemeanmoreorlessexactly.
Withadditionalbits,thelevelsareplacedonbothsidesaroundthemean.
Similarintegralshavetobeevaluatedtoquantizenonuniformlyinthe"soft-bit"domain.
Hereonlyoneintegralcanbecarriedout:di+1dip∧(λ)dλ=12erf2tanh1(λ)μL√2σLd+1di+12erf2tanh1(λ)+μL√2σLd+1di(A.
3)withp∧(λ)givenby(26).
Theotherintegraldi+1diλp∧(λ)dλhastobeevaluatedbynumericalintegration.
ThederivedoptimumquantizationlevelsconvertedbacktotheLLRdomainwithL=2tanh1(λ)areshowninFigure17.
051015202530ri,opt0510152025303540σ2LR=2R=4R=8Reconstructionlevelsof1–3bitquantizers(Lloyd-MaxalgorithminL-valuedomain)Figure16:OptimumnonuniformquantizationlevelsobtainedbyoptimizationintheL-valuedomain.
01234567ri,opt0510152025303540σ2LR=2R=4R=8Reconstructionlevelsof1–3bitquantizers(Lloyd-Maxin'softbit'domain)Figure17:Optimumnonuniformquantizationlevelsobtainedbyoptimizationinthe"soft-bit"domain.
0.
250.
20.
150.
10.
050MutualinformationlossΔI0102030405060σ2LSoftbitquantizationLLRquantizationR=2R=4R=8Figure18:MutualinformationlossΔI(X;L)for1–3quantizationbitsasafunctionofthevarianceoftheL-values.
14EURASIPJournalonWirelessCommunicationsandNetworkingWeobservethatnowtheoptimizedlevelsshowsomesaturationwithincreasingmean/varianceoftheL-valuedensity,becausetheincreaseinreliabilityisnotimportant.
RatheritpaysmoretodistinguishL-valuesofintermediatemagnitude,say,roughlyintherange2≤l≤6.
Forpracticalevaluation,itismoreconvenienttodeter-minethenecessaryquantizerresolutionaccordingtothevarianceoftheL-values.
Wethereforeprovideaplotcorre-spondingtoFigure5withσ2LastheabscissainFigure18.
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