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Collaborative Decoding: Achieving Cooperative Diversity in Wireless Networks Using Soft-Input Soft-Output Decoders


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COLLABORATIVEDECODING:ACHIEVINGCOOPERATIVEDIVERSITYINWIRELESSNETWORKSUSINGSOFT-INPUTSOFT-OUTPUTDECODERSByARUNAVUDAINAYAGAMADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2006

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Copyright2006byArunAvudainayagam

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Tomyparents,HemaandNayagam,andmywife,Krithi.

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ACKNOWLEDGMENTSNumerousindividualshavedirectlyandindirectlycontributedtomygraduateex-perienceingeneralandthisdissertationinparticular.ThefollowingfewwordscannotfullyconveymygratitudeforallthehelpandguidanceIhavereceivedovertheyears.Anyacknowledgmentshouldstartwithmyadvisor,Dr.JohnShea.Ihavegreatlybenetedfromhisexpertise,valuableinsightsandapproachtotacklingproblems.Hismentoringhasmademestrivetobecomeabetterengineerandresearcher.Ithankhimforallhishelp,guidance,andpatience.IwouldalsoliketothankDr.TanWongformanyfruitfulandinterestingdiscussions.SpecialthanksgotoDr.FangfortakingmeunderhiswingwhenIrstbeganmygraduatestudiesandforencouragingmetocontinueatUFformyPh.D.Imustalsomentionmyfel-lowgraduatestudentswhomademystayatWINGaverymemorableexperience.AbhinavRoogta,HanjoKimandJangwookMoondeserveaspecialmentionforvarioussuccessfulandsomenot-so-successfulbutinterestingnonethelesscollaborations.IamalsogratefultotheOfceofNavalResearchforsponsoringpartofmyresearch.Thisdissertationwouldnothavebeenpossiblewithoutthesupportandencouragementofmyparents,HemaandNayagam.IthankyouforallyoursacricesthathasmouldedmeintothispersonthatIamtoday.Andtoendwithalastbutnotleastcliche;mymostlovingandsincerethanksgotomywife,Krithi.ShewasprobablythemostaffectedbymyscheduleduringthecourseofthisworkandIcannotexpresshowinvaluablehersupporthasbeenovertheyears. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................ iv LISTOFTABLES ................................... vii LISTOFFIGURES ................................... viii ABSTRACT ....................................... x CHAPTERS 1INTRODUCTION ................................ 1 1.1ObjectivesandMainContributions .................... 2 1.2OutlineoftheDissertation ........................ 4 2BACKGROUNDANDRELATEDRESEARCH ................ 6 2.1Information-TheoreticStrategies ..................... 6 2.2RepetitionBasedCooperation ....................... 9 2.3CodedCooperation ............................ 14 3SOFT-INPUTSOFT-OUTPUTDECODING ................... 17 3.1TheLog-MAPandMax-log-MAPAlgorithms .............. 17 3.2TheDensityFunctionofReliabilitiesAssociatedwithaMax-Log-MAPDecoder ................................. 19 3.2.1AHighSNRApproximationtotheDensityFunctionofRelia-bilities .............................. 20 3.2.2OntheCorrelationBetweenOutputErrorEvents ......... 22 3.3AMathematicallyTractableDensityFunction .............. 24 3.4AClosed-FormExpressionfortheBit-Error-RateofSISODecoders .. 25 3.5ExtensiontoBlock-FadingChannels ................... 26 3.6NumericalResults ............................. 28 4CODEDCOOPERATIONTHROUGHCOLLABORATIVEDECODING ... 34 4.1CollaborativeDecodingthroughReliabilityExchange ......... 36 4.1.1CollaborativeDecodingthroughtheReliabilityExchangeoftheLeastReliableBits ....................... 37 4.1.2CollaborativeDecodingthroughtheReliabilityExchangeoftheMostReliableBits ....................... 41 v

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4.2GuidelinesfortheDesignofCollaborativeDecodingSchemes ..... 47 5ONCORRELATEDBITERRORSATTHEOUTPUTOFAMAX-LOG-MAPDECODER ................................ 51 5.1TerminologyandNotation ........................ 52 5.2RevisitingMax-log-MAPDecodingofConvolutionalCodes ...... 52 5.2.1ObtainingtheMLandCompetingPathusingtheBCJRAlgorithm 55 5.2.2OntheUtilityofCompetingPathsintheDesignofCollabora-tiveDecoding .......................... 59 6IMPROVEDLEAST-RELIABLE-BITSCOLLABORATIVEDECODINGFORBANDWIDTH-CONSTRAINEDSYSTEMS ................. 61 6.1CollaborativeDecodingwithConstrainedOverheads .......... 62 6.1.1Constrained-overheadIncrementalMRC ............. 64 6.1.2OverviewofImprovedLeast-ReliableBitsCollaborativeDe-coding .............................. 65 6.2EstimationofRequestSize ........................ 67 6.3EstimationoftheRequestSet ....................... 71 6.4DetailedDescriptionofI-LRBCollaborativeDecoding ......... 72 6.5Results ................................... 74 7CONCLUSIONANDDIRECTIONSFORFUTURERESEARCH ....... 80 7.1Conclusion ................................ 80 7.2DirectionsforFutureResearch ...................... 81 REFERENCES ..................................... 83 BIOGRAPHICALSKETCH .............................. 87 vi

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LISTOFTABLES Table page 3-1Erroreventmultiplicityofthe5;7convolutionalcode ............ 29 4-1OverheadofLRB-1fordifferentnumberofnodes. .............. 39 5-1Notationusedinthischapter .......................... 53 6-1InstantaneousSNRestimationfortrellissectionsbasedontheaverageoftheinstantaneousSNRsoftheparitybitsinthecandidateset ....... 71 vii

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LISTOFFIGURES Figure page 2-1Therelaychannel. ............................... 7 2-2Themultipleaccesschannelwithcooperatingencoders. ........... 8 2-3Thedecode-and-forwardcooperationscheme. ................ 11 2-4Theamplify-and-forwardcooperationscheme. ................ 12 2-5Codedcooperationusingrate-compatiblepuncturedconvolutionalcodes. .. 14 2-6Codedcooperationusingturbocodes. .................... 15 3-1Themeanofreliabilitiesasafunctionofthesignal-to-noiseratiowhenthecorrelationbetweentheoutputerroreventsareignored. .......... 28 3-2Themeanofreliabilitiesasafunctionofsignal-to-noiseratioaftertakingintoaccountthecorrelationbetweenoutputerrorevents. ......... 30 3-3ThePDFofreliabilitiesofthe;78CCfortwodifferentsignal-to-noiseratios. .................................... 31 3-4ThePDFofreliabilitiesofthe;78CCobtainedusingthesimpler,math-ematicallytractableexpressiongivenin 3.18 .............. 32 3-5Theprobabilityofbiterrorformax-log-MAPdecodingofconvolutionalcodesevaluatedusingtheclosed-formapproximationgivenin 3.23 ... 33 3-6Themeanofreliabilitiesofthe;7convolutionalcodesasafunctionofsignal-to-noiseratioofablock-fadingchannel. .............. 33 4-1Systemmodelforcollaborativedecoding. .................. 35 4-2Principleofinteractive/collaborativedecodingwithtwonodes. ....... 36 4-3Performanceoftwocollaborativedecodingschemesinwhichreceiversre-questinformationforasetofleast-reliablebits. .............. 40 4-4Reliabilitydensityfunctionsassociatedwithcorrectlyandincorrectlyde-codedbits. .................................. 42 4-5Performanceoftwocollaborativedecodingschemesinwhichreceiversbroadcastinformationaboutasetofmost-reliablebits. .......... 43 viii

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4-6Bitindicesofreliabilitiesexchangedasafunctionofiteration. ....... 44 4-7Performanceofsuboptimalvariantstwocollaborativedecodingschemesinwhichharddecisionsareexchangedinsteadofsoftinformation. ..... 46 4-8PerformanceoftheMRB-2schemewitheightnodesonablock-fadingchannel. ................................... 48 6-1Thecode-trellisforthe;7convolutionalcodewithexamplesoftheno-tationusedinthischapter. ......................... 73 6-2Probabilityofblockerrorfordifferentnumberofcollaboratingnodeswhentheoverheadconstraintisxedat5%oftheoverheadforMRC. ..... 75 6-3Throughputfordifferentnumberofcollaboratingnodeswhentheoverheadconstraintisxedat5%oftheoverheadforMRC. ............ 76 6-4ProbabilityofblockerroroverheadforCOI-MRCandI-LRBwitheightcooperatingnodes,anddifferentconstraintsontheoverhead. ...... 77 6-5AveragecooperationoverheadforCOI-MRCandI-LRBwitheightcoop-eratingnodes,anddifferentconstraintsontheoverhead. ......... 77 6-6ThroughputforCOI-MRCandI-LRBwitheightcooperatingnodes,anddifferentconstraintsontheoverhead. ................... 78 6-7Averagenumberofiterationspercollaborativedecodingattemptrequiredbyeightreceivers. ............................. 79 ix

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCOLLABORATIVEDECODING:ACHIEVINGCOOPERATIVEDIVERSITYINWIRELESSNETWORKSUSINGSOFT-INPUTSOFT-OUTPUTDECODERSByArunAvudainayagamMay2006Chair:JohnM.SheaMajorDepartment:ElectricalandComputerEngineeringSpatialdiversitytechniquesusemultipletransmitandreceiveantennasantennaar-raystoimproveperformanceinwirelessenvironmentswithoutrequiringadditionalband-widthorlossinthroughput.However,thespacingbetweenantennaelementsdependsonthecarrierwavelength,andthismightoftenexceedthesizeofmodernmobileradios.Thus,alternativeapproachesarerequiredtoharnessspatialdiversityinsmallterminals.Recently,cooperationamongusershasbeenproposedasanalternativemeanstoachievediversityinwirelessnetworkswithsmallradios.Inthisproposal,wedevelopcollaborationschemesforscenariosinwhichradiosinanetworkcooperatetoimproveperformance.Sincera-diosinawirelessnetworkaretypicallyseparatedinspace,thedifferentnodescanpooltheirresourcestogethertoformavirtualantennaarray.Theelementsoftheantennaarraycanthencollaboratebyexchanginginformationwitheachotherinordertoachievedi-versitygains.Theinformationexchangedbythecollaboratingnodesiscalledcooperationoverhead.Ourschemesaretargetedtowardsbandwidth-limitedsystemsinwhichthecoop-erationoverheadshouldbesmall.Weprovideaframeworkcalledcollaborativedecodingtohelpdesignschemesthathavelowcooperationoverheadandstillachieveperformanceclosetothatofoptimalcombiningschemes.Wepresentacollaborationtechniquecalled x

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improvedleast-reliablebitsI-LRBcollaborativedecodingthatprovidesahigherlevelofadaptationthanpreviouslyproposedcooperativeschemes.TheI-LRBschemeutilizesreliabilityinformationandinformationaboutcompetingpathsinsoft-inputsoft-outputde-coderstoadaptivelyselecttheamountofinformationthatisneededtocorrectaparticularpartofamessage,aswellaswhichbitsshouldbeexchanged.Simulationresultsshowthattheproposedapproachoffersasignicantperformanceadvantageoverexistingcoopera-tiontechniques.Forexample,I-LRBcanprovidea30%-60%improvementinthroughputwithrespecttotraditionalcooperationschemesinbandwidth-constrainedsystems. xi

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CHAPTER1INTRODUCTIONMultipathfadingisoneofthemostcommonproblemsassociatedwithwirelesscom-munications.Reectionsfromandrefractionthroughvariousobjectsinthechannelcausemultipleattenuatedanddelayedcopiesofthetransmittedsignaltoconstructivelyorde-structivelycombineatthedestination.Fadingcancausesevereuctuationsinthesignal-to-noiseratioSNR,whichinturnaffectssystemperformance.Varioustechniqueslikeequalization,error-controlcodinganddiversitycombiningareusedindependentlyorinconjunctiontocombatfading.Diversitytechniquestypicallyimproveperformancebymakingmultipleindependentcopiesofthetransmittedsignalavailabletothedestina-tion.Thesemultiplecopiescanthenbeoptimallycombinedusingvarioustechniqueslikemaximal-ratiocombiningMRCorequal-gaincombiningEGC[ 1 ].Temporaldiversityistypicallyachievedthrougherror-controlcoding.Frequencydiversityisachievedbyus-ingvariousphysicallayertechniqueslikefrequencyhoppingormulti-carriermodulation.Recentadvancesinspace-timecodinghaveproventhatprocessinginthespatialdo-mainisanefcientapproachtoachievediversityindelay-limitedandbandwidth-limitedapplications.Space-timecodesexploitthemultipathnatureofthewirelessmediumtocombatthedetrimentaleffectsoffading.Spatialdiversitycanbeachievedbyusingmulti-pleantennasatthetransmitterand/orreceiver.However,forsignicantgainsthespacingbetweentheantennaelementsshouldbeatleasthalfthewavelengthoftheRFcarrier.Thisprohibitstheuseofantennaarraysinmostsmall,portableradios.Thisisthereasonwhyantennaarraysarenotusedinthecellulardownlinkorinadhocnetworks.Inrecentyears,anumberofnetwork-assisteddiversitytechniqueshavebeenstud-ied[ 2 9 ].Intheseapproaches,theusersdependonthenetworktoprovidediversityatthephysicallayer.Thebroadcastnatureofthewirelesschannel,whereinanynodewithin 1

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2 rangeofthetransmittercanlistentothetransmission,isexploitedinthisnetwork-basedapproachtospatialprocessing.ThispropertyofthewirelessmediumisreferredtoasthewirelessbroadcastadvantageWBA[ 10 ].Sincenodesinanetworkarespatiallysepa-rated,thedifferentnodesthatreceivethetransmissionfromasourcecanbeconsideredtobetheelementsofavirtualantennaarray.Sincetheelementsarenotphysicallycon-nected,thisisreferredtoasadistributedarray.Thedifferentuserscanthencollaboratewitheachothertoachievediversitygains.Diversityachievedwhenusersinanetworkcollaboratetoimproveeachother'sperformancehasbeentermedcooperativediversityormultiuserdiversity.Wewilltousethetermusercooperation[ 2 ]torefertotheprocessofcollaborationbetweenthevarioususersofthenetwork.Alotofworkonusercooperationincludingmostinformation-theoreticandafewpracticalschemesisbasedonsimplerepetitioncoding[ 2 6 ].Thebasicideaoftheseschemesisthatanyuserthatlistenstothetransmissionfromthesourceforwardstheinfor-mationeitherthecodedbitsafterdecodingorquantizedversionsofthereceivedsymbolvaluestothedestination.Theamountofinformationexchangedbythecollaboratingnodesisreferredtoasthecooperationoverhead.Theuseofrepetitioncodesmakesthesetechniquesinefcientintermsoftheoverhead.Cooperationthroughtheuseofmorepow-erfulerrorcorrectioncodeshasalsobeenproposedin[ 7 8 ].Thedisadvantageoftheseschemesisthattheydonoteasilyscaletolargenetworkswithmorethantwocooperatingnodes. 1.1 ObjectivesandMainContributionsSincethewirelessmediumisbandwidth-limited,thecooperationoverheadisaveryimportantissueandonethathasnotbeenaddressedsofarintheliterature.Theobjectiveofthisworkistoprovideaframeworktohelpdevelopcooperationstrategiesthatareefcientintermsofthecooperationoverhead.Itisalsoimportantthatthecollaborationtechniquesextendnaturallytomultiplecooperatingnodes.Usingthisframework,calledcollabora-tivedecoding,wedevelopstrategiesthatprovideclose-to-optimalperformancewithonly

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3 afractionoftheoverheadrequiredbyconventionalcooperationschemes.Unlikepreviouscooperationstrategies,collaborativedecodingprovidesaconvenientapproachtotradeper-formanceforoverhead,andcollaborativedecodingscaleseasilytomultiplecooperatingnodes.Conventionalcodedcooperationstrategiesarebasedondistributedencodingofames-sageamongthecollaboratingnodes.Collaborativedecodingisbasedonadistributedde-codingofanencodedmessageamongcollaboratingnodes.Allschemesinthisdissertationusesoft-inputsoft-outputSISOdecoders.ThemagnitudeoftheoutputoftheSISOde-coderiscalledthereliabilityandisanindicationofthecorrectnessofthedecodedbit.Inallourschemes,thenodesexchangeinformationforonlyafractionofthemessagebitsbasedonthereliabilityinformation.Thedesignofacollaborativedecodingschemethenconsistsofthechoiceofthebitstobeexchangedandwhatinformationistobeexchangedamongthenodes.Inexistingcooperationstrategies,themessagesexchangedbycollaboratingnodesispredeterminedandxed.Collaborativedecodingadaptsthecontentinvolvedincooperationtoeachchannelinstantiation.Thus,bytailoringthemessagesexchangedbycollaboratingnodestothepotentialbiterrors,collaborativedecodingaimstolowerthecooperationoverhead.Thecontributionsofthisworkaretwo-fold.Firstthedissertationfurthersunder-standingofthefundamentaloperationofthemaximumaposterioriMAPconvolutionaldecoderwiththemax-log-MAPimplementation.Inthiscontextthemaincontributionsinthisdissertationarethefollowing: 1. Weprovideaclosed-formapproximationtothedensityanddistributionfunctionofreliabilitiesattheoutputofamax-log-MAPSISOdecoder.Thisclosed-formesti-mateisparameterizedbyasinglenumericalquantitythatcanbedeterminedanalyt-ically.Theestimatescanbecanbeusedtoanalyzereliability-basedsystems. 2. Usingtheseclosedformapproximationsweprovideanapproximationtothebiterrorrateofamax-log-MAPdecoderintermsofasingleQ-function.Thisistherstsuchresultintheliterature.

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4 3. Weinvestigatethecorrelatednatureofthesoft-outputofamax-log-MAPdecoder.Themax-log-MAPdecodercomputesthesoft-outputforatrellissectionbycon-sideringthemaximum-likelihoodMLpathandacompetingpaththatdiffersfromtheMLpathintheinputforthattrellissection.Weshowthatthetime-correlatedreliabilitiesoccurinamax-log-MAPdecoderbecausethesamecompetingpathisconsideredincomputingthesoft-outputforadjacentbits. 4. WeprovideanefcientapproachtoexplicitlycomputetheMLandcompetingpathsbyusingcomputationsthatarealreadyperformedinthedecoder.Thesecondareaofcontributioninthisdissertationisinapplyingtheknowledgegainedabouttheoperationofthedecodertothedesignofcollaborativedecoding.Themaincontributionsinthisareaarethefollowing: 1. Wedesignacooperationstrategycalledimprovedleast-reliable-bitsI-LRBcollab-orativedecodingthathasthefollowingfeatures: Achievesfulldiversityinthenumberofcooperatingnodes. Requiresafractionoftheoverheadinvolvedinfullmaximal-ratio-combining. Easilyscalestomultiplerelays. Offerstheabilitytoeasilytradeperformanceforoverhead. 2. I-LRBisadaptiveintwolevels.Thetrellissectionsforwhichinformationiscom-binedareadaptedtoeachchannelinstantiation.Foreachtrellissection,theamountofinformationcombinedisadaptedtothereliabilityofthattrellissection. 3. I-LRBexploitscorrelatedbitreliabilitiesbycomputingcompetingpathsinthede-coder,andutilizesknowledgeofthecompetingpathstoreducecooperationover-head. 1.2 OutlineoftheDissertationThisdissertationisorganizedasfollows.InChapter 2 ,wesummarizevariousim-portantresultsintheliteraturethatpertaintotheideaofusercooperation.Theseresultsprovideabasisforcomparisonwithourschemes,andmakesiteasiertoemphasizethedistinctionbetweenourapproachandtheexistingschemes.Weprovideanintroductiontosoft-inputsoft-outputSISOdecodersinChapter 3 .AgoodunderstandofSISOdecoding

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5 isimportanttounderstandtheoperationofourtechniques.Wealsoprovideamathematicalcharacterization/approximationofthestatisticsoftheSISOdecoderoutput.InChapter 4 ,weintroducetheconceptofcollaborativedecodinginwhichvarioususerscooperateinthedecodingprocessandachievespatialdiversity.ThoughtheseschemesaresuitableforAWGNchannels,weshowthattheseschemesarenotsuitableforfadingchannels.Wedevelopguidelinestohelpdesigncooperativediversityprotocolsforfadingchannels.Wealsodevelopvariousguidelinesforthedesignofcooperativeprotocolsinthischapter.InChapter 5 ,westudythecorrelatednatureoftheoutputoftheSISOdecoder.WeshowthaterroreventsencounteredintheSISOdecodercanbeusedtocapturethiscorrelation.Wealsopresentatechniquetoefcientlycomputetheseerroreventswithminimalmod-icationstothedecoder.BasedonthedesignguidelinespresentedinChapter 4 andthetechniquepresentedinChapter 5 ,inChapter 6 wedesignacollaborationschemethatuti-lizesthecorrelatedoutputoftheSISOdecodertoreducethecooperationoverhead.ThedissertationisconcludedinChapter 7

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CHAPTER2BACKGROUNDANDRELATEDRESEARCHInthischaptertheideaofcooperativecommunicationsisintroducedandimportantreferencesrelatingtothisbroadareaaresummarized.Theobjectiveofthechapteristofamiliarizethereaderwiththedifferentapproachestousercooperationandthevariousis-suesinvolvedinthedesignofsuchschemes.Westartbyintroducingtheveryrstideasofcooperationandsomerecenttechniquesthatwereproposedintheinformationtheorycom-munity.Wethenhighlightsomeofthemorepracticalapproachesthathavebeenstudiedinrecentyears. 2.1 Information-TheoreticStrategiesStudiesontherelaychannelinthelate1960scanbeconsideredtocontaintherstinstancesofcooperation.TherelaychannelshowninFigure 2-1 wasrstintroducedandstudiedbyvanderMeulenin1968[ 11 ].Inthissetting,anintermediatenode,calledtherelay,listenstothetransmissionfromthesourcetothedestination,processesthisinforma-tion,andtransmitsadditionalinformationabouttheinitialtransmissiontothedestination.Thedestinationusestherelaytransmissiontoresolveanyambiguityabouttheoriginaltransmission.Thetransmissionfromtherelayisdonejointlywiththesource;i.e.,relaytransmissionforblockiissuper-imposedonblocki+1sentbythesource.Thus,therelaycooperateswiththesourcetoimprovereceptionatthedestination.In1979,CoverandElGamal[ 12 ]studiedthecapacityofrelaychannelsunderdif-ferentscenarios.CoverandElGamalputforwardthreedifferentapproachestoachieveusercooperationinarelaychannel.Infacilitation,therelaypassivelyaidsthecommuni-cationbetweenasourceanddestinationbynottransmitting,therebyreducinginterferencetotheoriginalcommunication.Incooperation,therelaydecodesthetransmissionfrom 6

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7 Figure2-1:Therelaychannel. thesourceandprovidesadditionalinformationabouttheinitialtransmissiontothedesti-nationtoaidinrecoveringtheoriginalmessage.Inobservation,therelayjustforwardstheobservedsymbolvaluestothedestination.Theobservationschemewasintroducedtoovercomeaproblemwiththecooperationscheme.Incooperation,therelaypartitionsthesetofvalidcodewordsintobinsusingtheSlepian-Wolfpartitioningtechnique[ 13 ]andtransmitsthebinindexofthepartitioncontainingthesourcemessage.Thedestinationthenusesthesetofcodewordsinthecor-respondingpartitiontoresolveanyambiguityaboutthetransmittedmessage.However,thecomputationofthebinindexrequirescorrectdecodingattherelay,andthusthisschemeislimitedbytheratebetweenthesourceandtherelay.Theobservationschemecanovercomethisproblembecauseitdoesnotrequirecorrectdecodingattherelay.Recentstudiesoftherelaychannelcanbefoundin[ 14 18 ].TheCoverandElGamalschemeisextendedtomultiplenodesinGuptaandKumar[ 14 ].InCoverandElGamal[ 12 ]andGuptaandKumar[ 14 ],thenodescooperateusingblock-Markoven-codingandSlepian-Wolfpartitioning[ 13 ].Thedestinationdecodesthemessageusingtwotransmittedblocks;i.e.,uponreceivingblocki,thedecoderestimatesthemessageinthepreviousblocki)]TJ/F15 11.955 Tf 12.775 0 Td[(1.Thenodesusecodebooksofdifferentsizesattherelayandthesource,makingitdifculttoextendthethisstrategytomultiplenodes.AtechniquethatusescodebooksofthesamesizeisproposedinWillems[ 19 ]andisextendedtomultiplerelaysinKrameretal.[ 15 ].Theuseofequalsizecodebooksmakesiteasiertoextend

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8 Figure2-2:Themultipleaccesschannelwithcooperatingencoders. thisschemetomultiplerelays.AnotheradvantageofthisschemeisthattheuseofthecomplicatedSlepian-Wolfbinningtechniqueisavoided.Thedisadvantagemanifestsitselfasalargedecodingdelayincurredduetothebackwardsdecodingtechniqueusedatthedestination.ThecodingisdoneoverBblocksandthedecodingprocesscanstartonlyafterreceivingalltheBblocks.Thedecodingisdoneinthebackwarddirection,startingwiththelastblockandendingattherstblock.AstrategythatavoidsSlepian-Wolfparti-tioningandinwhichallrelaysusecodebooksofthesamesizebutinwhichthedecoderlagisreducedtojustoneblockasintheCoverandElGamalschemeisproposedinXieandKumar[ 16 ].Anextensionofthisschemetomultiplerelayshasalsobeenstudied[ 17 18 ].Theinformationtheorycommunityalsostudiedcooperationfromtheviewofamul-tipleaccesschannel.Intheearly1980susercooperationinthesettingofamultipleaccesschannelwasstudied[ 20 21 ].Unliketherelaychannel,boththeencodershavedatatosendtothedestination.InWillems[ 20 ],theuserscooperateoveraseparatechannelbeforesend-ingtheirmessages.TheneedforaseparatesetofchannelsforcooperationiseliminatedinWillemsandvanderMeulen[ 21 ].ThescenarioconsideredisshowninFigure 2-2 .Thecooperativetransmissionworksasfollows.Eachencodercooperateswiththeotherandlearnsthecodewordthattheotherencoderisgoingtosendinthecurrenttransmission.Thus,intransmissioni,eachencoderhasknowledgeofallthepreviousi)]TJ/F15 11.955 Tf 11.055 0 Td[(1codewordsoftheotherencoder.Theencodingprocessworksasfollows.Withoutlossofgenerality,we

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9 assumethatencoder1startstheencodingprocessfortransmissioni.Encoder1formsitscodewordasafunctionofitsowndataandthecodewordssentbyencoder2inallprevioustransmissions.Encoder2learnsthecurrentcodewordofencoder1andformsitsowncode-wordasafunctionofitsowndataandthecurrentandpreviouscodewordsofencoder1.Thisstrategyrequiresperfectcooperationbetweentheencoders;i.e.,eachencodershouldlearnthecodewordsoftheotherencoderwithouterrors.NotethatthemodelofWillemsetal.reducestotherelaychannelifoneoftheencodershasnodataofitsowntosend.Thenthatparticularencoderwillactasarelayfortheotherencoder.Theauthorsprovethatencodercooperationincreasesthecapacityofthemultiple-accesschannelwhencomparedtonon-cooperativetransmission.Thetechniquesdescribedinthissectiondependoninformationtheoreticconceptslikerandombinning,typical-setandbackwardsdecoding.Thesearenotviableforpracticalimplementation.Inthenextsection,wereviewafewpracticalcooperationstrategiesbasedonsimplerepetitioncodingideas. 2.2 RepetitionBasedCooperationAftertheworkofWillems,theideaofusercooperationwaslargelyignoreduntilthelate90s.Theadvancesinspace-timecoding[ 22 23 ]provedthatexploitingspatialdiversitywiththeuseofmultipletransmitandreceiveantennascanleadtosignicantimprovementsindatarate.However,smallportableradiosdonotpermittheuseofmultipleantennas.Usercooperationisanaturalwaytoachievespatialdiversitybypoolingtheresourcesofmanyradios,eachequippedwithasingleantenna.UsercooperationinawirelessscenariowasrstinvestigatedbySendonaris,ErkipandAazhang[ 2 5 6 24 ].Sendonarisetal.studytheideaofusercooperationinthesettingofacellularCDMAsystem.ThesystemmodeltheyuseisidenticaltothemodelshowninFigure 2-2 .However,duetothewirelesssettingthelinksbetweenthetwoencodersthecooperationchannelsareimperfectchannelsthatexperiencefading.

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10 ThecooperationmodelisbasedontheideaofWillemsetal.,whereinacodewordsentbyoneuserdependsonthecodewordsentbytheotheruser.InSendonarisetal.[ 2 ],theauthorsconsidermorepracticalaspectsofusercooperationinthisscenario.Inparticular,theauthorsbeginbyusinginformationtheorytoevaluatetheeffectsofcooperationonoutageprobability,diversity,andcellularcoverage.Thentheauthorsproposeandanalyzethefollowingpracticalcooperationscheme.LetXitandcitdenotethesignalstransmittedbyuseriandthespreadingcodeusedbyuseriattimet.ThenthesignalstransmittedbythetwousersareX1t=a11b1tc1t;a12b1tc1t;a13b1tc1t+a14^b2tc2tX2t=a21b2tc2t;a22b2tc2t;a23^b1tc1t+a24b2tc2t;wherebjiisuseri'sjthbitand^bjiisthecorrespondingestimateattheothernode.Theparametersfajigrepresenthowmuchpowerisallocatedtoeachbit.Thus,inthersttwoperiodseachusertransmitsitsownbits.Inthethirdperiod,eachencodersendsalinearcombinationofitsownbitsofthesecondperiodanditsestimateofitspartner'sbitsofthesecondperiod.Sincethebasisofcooperationisanestimateoftheotherencoder'scodeword,theauthorsallocaterateandpowertoguaranteeerror-freecommunicationonthecooperationchannels.Thus,eachencoder'sestimateofthecodewordoftheotherencoderisperfect.Withthissystemandcooperationmodeltheauthorsprovethatitispossibletoincreasethemaximumsum-capacityofthenetworkifthetransmitterhasknowledgeofthechannelphase.Lanemanetal.[ 3 4 ],introducetwobroadclassesofcooperationtechniquescalleddecode-and-forwardandamplify-and-forward.Theirapproachissimilartotherelay-channel-basedcooperationtechniquesofCoverandElGamal.Inthedecode-and-forwardscheme,therelaysrstdecodethesourcemessageandthenforwardthere-encodedinfor-mationbitstothedestination.ThisisakintoCoveretal.'scooperationschemewiththe

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11 Figure2-3:Thedecode-and-forwardcooperationscheme. relaytransmissionconsistingofre-encodedinformationbitsinsteadofSlepian-Wolfbinindices.Theoperationofthedecode-and-forwardschemeisshowninFigure 2-3 .Theschemeworksintwophasesasintheconventionalrelaychannel.Intherstphase,thesourceencodestheinformationbitsrepresentedbyanemptyrectangle,andtransmitsthecodedbitsrepresentedbyasolidrectangle.Thedestinationandtherelayreceivenoisyversionsofthecodedbits.Inthesecondphase,therelaydecodestheinformationbits,re-encodesthemusingthesamecodeusedatthesource.There-encodedcodewordisthensenttothedestination.Attheendofthesecondphase,thedestinationhastwoindependentnoisycopiesofthecodewordsentbythesourceassumingthattherelaydecodedcorrectly.Thesetwoindependentcopiescanbecombinedusingvariouscombiningschemeslikemaximal-ratiocombiningMRCorequal-gaincombiningEGC[ 1 ].Thisschemecanbeconsideredasaninstanceofrate-1=2repetitioncodingsincethedestinationreceivestwoindependentcopiesofthesamemessage.However,therepetitionisdonebytherelayinsteadofthesourceitself.Theeffectivenessofcooperativecommunicationschemesisoftenassessedintermsoftheeffectsoncapacityandonthediversityachieved.Theabilitytoachievediversityisquantiedintermsofthediversityorder,whichisdenedastheasymptoticslopeofthebitorblockerrorratecurveonlog-logscale.IfthereareMelementsinanantennaarray,then

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12 Figure2-4:Theamplify-and-forwardcooperationscheme. atmostMindependentcopiesofthemessagearereceived.Hence,themaximumdiversityorderthatcanbeachievedisM.Anyschemethatachievesdiversityorderthatisequaltothenumberofcooperatingnodesissaidtoachievefulldiversity.AsinthecooperationschemeproposedbyCoveretal.,thedecode-and-forwardschemedependsoncorrectdecodingattherelay.Itisprovedthatthedecode-and-forwardschemescanprovideallofthecapacitybenetsofferedbycooperativetransmission,butcannotachievefulldiversityinthenumberofcollaboratingnodes[ 3 4 ].Thereasonisthatadiversitychanneliscreatedonthelinkbetweentherelayandthedestinationonlywhentherelaydecodessuccessfully,andhencethisschemeislimitedbythechannelbetweenthesourceandtherelay.NotethattheschemeofSendonarisetal.alsofallsunderthedecode-and-forwardclassofcooperationschemessinceaperfectestimateofthepartner'sbitsisrequiredforcooperation.Theoperationoftheamplify-and-forwardschemeisillustratedinFigure 2-4 .Therstphaseisidenticaltorstphaseinthedecode-and-forwardscheme.Inthesecondphase,therelaydoesnotperformdecoding.Instead,therelayamplies/scalesitsobservationsthereceivedsymbolvaluessubjecttoapowerconstraintandforwardsittothedestination.Ifyisthemessagereceivedfromthesource,thetransmissionoftherelaycanbeexpressedasx=y;.1

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13 wheretheamplicationfactorisconstrainedbys P jj2P+N0:.2HerePisthemaximumtransmitpowerofthesourceandrelay,representsthefadingamplitudebetweenthesourceandrelay,andN0=2isthenoisevariance.Onaverage,thescalingfactorconstrainsthetransmissionpoweroftherelaytoitsmaximumallowedvalueP.Unlikethedecode-and-forwardscheme,therelayalsoampliesitsownreceivernoise.ThisisidenticaltoCoveretal.'sobservationschemeiftheamplicationfactorissettounityi.e.,iftherelayshavenopowerconstraint.Asinthedecode-and-forwardscheme,thedestinationhastwoindependentnoisyversionsoftheoriginalcodewordthatcanbeoptimallycombined.Thus,fromtheperspectiveofthedestination,itstillappearsasthougharate-1=2repetitioncodeisusedatthesource.Itisshownthatamplify-and-forwardschemescanachievefulldiversityinthenumberofcooperatingnodes[ 3 4 ].Inthedecode-and-forwardschemetherelayjusttransmitsthebinarycodeword.Intheamplify-and-forwardscheme,therelaymustamplifythereceivedsymbols,andretransmittheseampliedsoftvalues.Thissoft-amplicationprocesswillnotbepracticalinmanyrealsystems.Instead,therelaywouldhavetoquantizethereceivedsymbolvaluesandthentransmitthequantizedbitstothereceiver.Theinformationexchangedbytherelaysinordertoimproveperformanceisreferredtoasthecooperationoverhead.ThenifBbitsareusedforquantization,thenthecooperationoverheadoftheamplify-and-forwardschemesisBtimesthedecode-and-forwardcooperationoverhead.However,theamplify-and-forwardschemedoesnotdependoncorrectdecodingatanyoftherelays.Thus,theamplify-and-forwardschemeachievesfulldiversityatthecostofoverhead.Theschemesintroducedinthissectionarebasedonsimplerepetitioncodingideas.Therelaysjustrepeattheirestimateoftheoriginalcodewordortheirreceivedsymbolvalues.Inthenextsectionwereviewafewcooperationstrategiesthatarebasedonbettererror-correctioncodes.

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14 Figure2-5:Codedcooperationusingrate-compatiblepuncturedconvolutionalcodes. 2.3 CodedCooperationCooperativediversitythroughtheuseofbettererror-correctioncodesiscalledcodedcooperation[ 7 ].Otherschemesforcodedcooperationhavealsobeenproposed[ 8 9 ].Codedcooperationschemescanbedividedintotwomainclasses.Intherstclassoftechniques,distributedencodingisperformedamongthecooperatingnodes,anddecodingtakesplaceonlyatthedestination.Inthesecondclass,encodingisperformedonlyatthesource,anddecodingtakesplaceinacollaborativemanneramongthecooperatingnodes.Inthissection,wereviewtwocodedcooperationschemesthatbelongtotheformerclass.Techniquesbelongingtothelattercategorywillbeintroducedinthefollowingchapters.HunterandNosratiniastudytheideaofusercooperationusingrate-compatiblepunc-turedconvolutionalRCPCcodes[ 7 25 26 ].RCPCcodeswereintroducedbyHage-nauer[ 27 ]asameanstoachieveincrementalredundancyinARQschemes.TheoperationoftheRCPC-basedcodedcooperationschemeisillustratedinFigure 2-5 .Forthisexam-ple,wehaveshowntheuseofarate-1=3convolutionalmothercode.Thedataisencodedwiththemothercodeinthesource.Thenapartofthecodewordthecenterpartintheexampleispuncturedoutandtheremainingcodebitsaretransmitted.Thus,therelayandthedestinationreceivenoisyversionsofarate-1=2codeword.Therelaydecodesthishigh-ratetransmissionandthenre-encodestheinformationbitsusingthemothercode.Thentherelaypuncturesthosesectionsofthecodewordthatweretransmittedbythesourcein

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15 Figure2-6:Codedcooperationusingturbocodes. therstphasetherstandlastpartsintheexample.Therelaythentransmitstheremain-ingpartofthecodewordthepartthatwaspuncturedintherstphasetothedestination.Thus,thedestinationeffectivelyhasallpartsoftheoriginalmothercode.Thus,therelaytransmissionhelpstransformtheinitialhighratecoderate-1=2intoadecodablelowerraterate-1=3transmissiontherebyimprovingperformanceatthedestination.ZhaoandValenti[ 8 ]investigatecooperationusingturbocodes[ 28 29 ].TheirschemeisshowninFigure 2-6 .AturboencoderalsoshowninFigure 2-6 consistsoftworecursivesystematicconvolutionalRSCencoders.Theinformationbitsarefeddirectlyintooneoftheencodersandapermutedversionoftheinformationbitsisfedintotheother.Thetwosetsofparitybitsalongwiththesystematicinformationbitsformthecodeword.Thecooperationschemeworksasfollows.ThesourceencodestheinformationbitswithanRSCencoderandtransmitstheparitybitsalongwiththesystematicbits.Therelaydecodesthistransmissionusingaconvolutionaldecoder.Itpermutesitsestimateoftheinformationbitsandre-encodesitwiththesameRSCcodeusedatthetransmitter.Thisproducesthesecondsetofparitybitsthatmakeuptheturbocode.Thissecondsetofparitybitsissenttothedestinationbytherelay.Thus,thedestinationhasineffectreceived

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16 acodewordencodedbyaturbocode.Itcanmakeuseofthepowerful,iterativeturbodecodingalgorithm[ 28 29 ]toimproveperformance.Notethatboththeschemesreliesoncorrectdecodingattherelayandthusfallinthedecode-and-forwardcategory.Thus,likeanyofthedecode-and-forwardschemes,thesetechniquesarenotguaranteedtoachievefulldiversity.ItisprovedthattheRCPC-basedcooperationschemeiscapableofachievingfulldiversityonlywhentherelayisabletodecodecorrectly[ 26 ].Anotherbigdrawbackofthecodedcooperationschemesthatutilizedistributeden-codingisscalability.Theseschemesdonotscaleeasilytomultiplerelays.Whenthereismorethanonerelay,itisnotimmediatelyobviousonhowthedistributedencodingshouldbedone.Inthesequel,wepresentcollaborativedecoding,whichisacodedcooperationschemethatisbasedondistributeddecodingthatscalesnaturallytoanynumberofcoop-eratingnodes.

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CHAPTER3SOFT-INPUTSOFT-OUTPUTDECODINGThischapterpresentsabriefoverviewofsoft-inputsoft-outputSISOdecoding.AgoodgraspofSISOdecodingconceptsisrequiredtounderstandourcollaborativedecodingschemethatispresentedinthefollowingchapters.AmathematicalcharacterizationoftheoutputofaparticularimplementationofthemaximumaposterioriMAPdecoderisalsopresented.Thischaracterizationaidsintheanalysisofoneofourapproachestousercooperation. 3.1 TheLog-MAPandMax-log-MAPAlgorithmsDecodersthatoperateonoatingpointsoftinputsandproduceoatingpointoutputsarecalledSISOdecoders.Thesignofthesoft-outputisthehard-decisionandthemagni-tudeofthesoft-outputiscalledthereliabilityofthehard-decision[ 30 ].Thereliabilityisanindicationofthecorrectnessofthehard-decision;i.e.,ahighvalueofthereliabilityimpliesahighprobabilityofthedecisionbeingcorrectandvice-versa.Inthisproposal,werestrictourattentiontoSISOMAPdecoders.Itiswellknownthatbit-by-bitMAPdecodingproducestheminimumprobabilityofbiterroramongalldecodingalgorithms.TheinputstoatypicalMAPdecoderareaprioriprobabilitiesoftheinformationbitsandchannelsymbols.Theaprioriprobabilitiesareusuallyinitializedtoequallylikelyvalues.Thesoft-outputofaMAPdecodercorrespondstotheaposterioriprobabilityAPPofaninformationbituibeing0or1,Pui=0jrorPui=1jr.Duetoreasonsofspeedandnumericalstability,MAPdecodersaretypicallyimplementedinthelog-domainLog-MAPdecoders.Theoutputofalog-MAPdecodercorrespondstolog-likelihoodratiosLLRsoftheAPPs.TheLLRforeachinformationbituiiscomputedasfollowsLuijr=lnPui=0jr Pui=1jr=lnPc2C+Pcjr Pc2C)]TJ/F77 11.955 Tf 8.745 4.56 Td[(Pcjr;.1 17

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18 whereristhereceivedcodeword,C+isthesetofallcodewordswithui=0andC)]TJ/F77 11.955 Tf -424.298 -22.114 Td[(isthesetofallcodewordswithui=1.Notethatck2f+1;)]TJ/F15 11.955 Tf 9.298 0 Td[(1g.TheoutputLLRisalsoreferredtoasthesoftinformation.AssumingthatallthecodewordsareequallylikelyandusingBaye'srule,thesoftinformationforcodewordstransmittedonaadditivewhiteGaussianchannelAWGNwithnoisevariance2=N0=2canbewrittenasLuijr=lnXc2C+Prjc)]TJ/F15 11.955 Tf 11.955 0 Td[(lnXc2C)]TJ/F77 11.955 Tf 8.247 15.088 Td[(Prjc;=lnXc2C+exp)]TJ 13.151 8.088 Td[(kr)]TJ/F54 11.955 Tf 11.955 0 Td[(ck2 22)]TJ/F15 11.955 Tf 11.955 0 Td[(lnXc2C)]TJ/F15 11.955 Tf 8.247 15.088 Td[(exp)]TJ 13.15 8.088 Td[(kr)]TJ/F54 11.955 Tf 11.955 0 Td[(ck2 22: .2 AsuboptimalimplementationoftheLog-MAPdecoder,calledtheMax-Log-MAPdecoder,isobtainedbyusingtheapproximationlnPxi=maxlnxitoevaluatetheLLRin 3.2 .Thus,foraMax-Log-MAPdecoderthesoft-outputisgivenby,Luijr=minc2C+kr)]TJ/F54 11.955 Tf 11.955 0 Td[(ck2 22)]TJ/F15 11.955 Tf 13.338 0 Td[(minc2C)]TJ/F78 11.955 Tf 8.247 25.369 Td[(kr)]TJ/F54 11.955 Tf 11.955 0 Td[(ck2 22;.3SincetheunionofC+andC)]TJ/F77 11.955 Tf 10.387 1.793 Td[(spansthespaceofallvalidcodewords,oneofthetermsin 3.3 correspondstotheEuclideandistancebetweenrandthemaximum-likelihoodMLdecodingsolution.Thus,thereliabilityforbitiicanbeexpressedasi,jLuijrj=1 22minjkr)]TJ/F54 11.955 Tf 11.955 0 Td[(cjik2)-222(kr)]TJ/F54 11.955 Tf 11.955 0 Td[(cMLk2;.4wherecjiisacodewordcorrespondingtoaninputsequencethatdiffersfromtheMLin-putsequenceintheithbit.SincethedistancebetweenrandtheMLcodewordissmallerthanthedistancebetweenrandanyothercodeword,thedifferencein 3.4 isalwaysposi-tive.Thus,theMax-Log-MAPdecoderassociateswiththeithbit,theminimumdifferencebetweenthemetricassociatedwiththeMLpathandthebestpaththatdiffersfromtheMLpathintheinputoftheithtrellissection[ 31 ].Ahighvalueofreliabilityimpliesthat

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19 theMLpathandthenextbestpatharefarapart,andhencethereisalowerprobabilityofchoosingtheotherpathandmakingabiterror.Ithasalsobeenshownviasimulationin[ 32 33 ]thatreliabilityisameasureofthecorrectnessofthebitdecision.Thus,abitwithhighreliabilityismorelikelytohavedecodedcorrectlythanabitwithlowreliability.Notethatthescalingofthereliabilitybythenoisevariancein 3.4 doesnotaffecttheperformanceoftheMax-Log-MAPdecoderandisjustanimplementationconsideration.Ifchannelestimatesareavailabletothedecoder,thescalingcanbeperformed. 3.2 TheDensityFunctionofReliabilitiesAssociatedwithaMax-Log-MAPDecoderReggianiandTartara[ 34 ]providedtherstcharacterizationofthesoftinformationintermsofitsprobabilitydensityfunctionPDF.ReggianiandTartara[ 34 ]examinethepro-jectionofnoiseinthedirectioncorrespondingtoanerroreventandinterpretthisrandomvariableasadistanceinEuclideanspacetoderivethePDF.Hereanerroreventdenotesasequencethattranslatesonecodewordintoanother,wherethepaththroughthecodetrellisthatisinducedbytheerrorsequenceisonlyinthesamestateastheoriginalcodewordattheendpointsofthesequence.Fortherestofthepaper,therandomvariableresultingfromtheprojectionofnoiseontoadirectionspeciedbyanerroreventwillbereferredtoastheprojectionrandomvariablePRV.ReggianiandTartara[ 34 ]presenttwoapproachestoobtainthePDF.Intherstapproach,thePDFisderivedbasedontheassumptionthatdifferentPRVsprojectionofnoiseontodirectionsspeciedbydifferenterroreventsareindependent.ThePDFobtainedusingtheindependenceassumptionresultsinconserva-tivereliabilityestimatesthatarelowerthantheactualvalues.Theauthorssuggestincor-poratingthecorrelationbetweenthePRVsintothePDFtoavoidconservativeestimates.Inthesecondapproach,theauthorsobtainacovariancematrixinvolvingthecorrelationbetweendifferentPRVsanduseitinajointmultivariatedistributiontoobtainthePDF.ThoughthePDFobtainedusingthesecondapproachproducesgoodreliabilityestimates,theexpressionforthedensityfunctionisverycomplicated.Evenwiththeindependenceassumption,thePDFobtainedusingthistechniquecannotbeexpressedinclosed-formand

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20 involvesproductsandsummationsthatdependontheenumerationofallpossibleerrorevents.Thus,thePDFgiveninReggianiandTartara[ 34 ]isnotattractiveforuseintheanalysisofreliability-basedtechniques.Wenowpresentastreamlinedderivationofthedensitiesofreliabilitiesattheoutputofamax-log-MAPdecoderbyworkingintheconventionalHammingspaceReggianiandTartara[ 34 ]workinEuclideanspaceanduseahighsignal-to-noiseratioSNRapprox-imationtoobtainthePDF.AsimpletechniquetoaccountforthecorrelationbetweenthePRVsisalsopresented.Usingthistechnique,wecanavoidtheuseofcomplicatedjointmultivariatedistributions.Thoughthisapproachproducesgoodestimatesofthedensityfunctionandotherstatisticsofthereliability,theexpressionisstillcomplicatedtobeoffurtheruseinanalysis.Tothisend,wealsopresentanadhocestimateofthePDFthatismathematicallytractable.Thisclosed-formestimateofthePDFisparameterizedbyasinglequantitythatcanbenumericallyevaluated.WeshowthatourtechniqueproducesanaccurateapproximationofthetruePDF. 3.2.1 AHighSNRApproximationtotheDensityFunctionofReliabilitiesThereliabilityoftheoutputofamax-log-MAPdecoderisgivenin 3.4 .SincecjiisacodewordcorrespondingtoaninputsequencethatdiffersfromtheMLinputsequenceintheithbit,cjicanbeexpressedas,cji=cML+eji;.5whereejiisanerroreventgeneratedbyaninputsequencewithbitiequalto1.SincethesymbolsofcjiandcMLtakeonvaluesinf+1;)]TJ/F15 11.955 Tf 9.298 0 Td[(1gandtheerroreventtransformsonecodewordintoanother,thecomponentsofejitakeonvaluesinf+2;0;)]TJ/F15 11.955 Tf 9.299 0 Td[(2g.Using 3.5 in 3.4 ,wegeti=minjkejik2)]TJ/F15 11.955 Tf 11.955 0 Td[(2r)]TJ/F54 11.955 Tf 11.955 0 Td[(cMLTeji:.6Notethatwehavedroppedthescalingbythenoisevariance1=22in 3.6 .Afterderivingthedensityanddistributionfunctionsusing 3.6 ,asimpletransformationcanbeused

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21 toaccountforthescalingin 3.4 .AthighSNRs,theMLdecoderwillndthecorrectcodewordinputsequence.ThusforhighSNRswecanexpressthereceivedsequenceasr=cML+e;.7whereeN)]TJ/F15 11.955 Tf 5.48 -9.683 Td[(0;N0 2I.Thisassumptionissimilartotheapproachin[ 34 ],inwhichtheauthorsobtaintheconditionaldensityfunctiongivencorrectdecodingofabit.Using 3.7 in 3.6 wegeti=minjkejik2+2eTeji:.8Notethataccordingtoourterminology,ejiisanerrorevent,whereaseTejiisthePRVi.e.,theprojectionofthenoiseontothedirectionoftheerroreventeji.LetZj,kejik2+2eTeji:.9SinceZjisjustalinearcombinationofGaussiannoisesamples,ZjisalsoaGaussianrandomvariable.ItiseasytoseethatZjN4dj;16djNo 2;.10wheredjistheHammingweightnumberofnon-zeroelementsofeji.Thus,thereliabilitycanbeexpressedastheminimumoverasequenceofGaussianrandomvariableswithdistributionsgivenby 3.10 .AssumingthatalltheZjsareinde-pendentlydistributed,thecumulativedensityfunctionCDFofcanbewrittenasF=1)]TJ/F78 11.955 Tf 11.955 11.358 Td[(YjProbZj>=1)]TJ/F25 7.97 Tf 16.956 14.944 Td[(dmaxYd=dminQ)]TJ/F15 11.955 Tf 11.955 0 Td[(4d p 16d2ad; .11 whereadisthemultiplicityoferroreventsofweightdandQxrepresentstheGaussiancomplementarydistributionfunction.ThePDFcanbeobtainedbydifferentiatingtheCDF.

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22 Usingtheproductruleofdifferentiation,thePDFisobtainedasf=dmaxXdj=dminadj 4p dj2exp)]TJ/F15 11.955 Tf 13.151 8.088 Td[()]TJ/F15 11.955 Tf 11.955 0 Td[(4dj2 32dj2Q)]TJ/F15 11.955 Tf 11.955 0 Td[(4dj p 16dj2adj)]TJ/F24 7.97 Tf 6.587 0 Td[(1dmaxYdi=dmindi6=djQ)]TJ/F15 11.955 Tf 11.955 0 Td[(4di p 16di2adi: .12 Thus,evenunderthesimplifyingassumptionofindependentPRVs,thedensityfunctionobtainedfromrstprinciplesisverycomplicatedandnotsuitedforuseintheanalysisoftechniquesinvolvingreliabilities.FortheMax-Log-MAPdecoderwiththenoisescalingimplementedasin 3.4 ,theCDFandPDFofthereliabilitycanbeobtainedbyasimpletransformationasF;=F)]TJ/F15 11.955 Tf 5.479 -9.684 Td[(22;f;=22f)]TJ/F15 11.955 Tf 5.48 -9.684 Td[(22:.13Thesubscriptisusedintheaboveexpressionstoindicatethatthesoft-informationisscaledbythenoisevarianceintheMax-Log-MAPdecoder.Sinceisnon-negativeandcontinuous,themeanofthereliabilitycanthenbeevaluatednumericallyas2,E[]=Z1)]TJ/F50 11.955 Tf 11.955 0 Td[(F;d=Z10dmaxYd=dminQ22)]TJ/F15 11.955 Tf 11.956 0 Td[(4d p 16d2add: .14 3.2.2 OntheCorrelationBetweenOutputErrorEventsInSection 3.2 ,wemodelthereliabilityastheminimumofanumberofGaussianrandomvariablesthatareassumedtobeindependent.ThisassumptionisvalidonlyifallthePRVsareindependent.However,thisisnotavalidassumption.ItispossiblethatthedifferenterroreventsejiassociatedwiththePRVseTejisharethesamepaththrough

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23 thetrellisatcertaintimeinstants.Ateachofthesetrellissections,thePRVssharecom-monnoisesamplesfromthevectoreandthus,thePRVsarecorrelated.Becauseofthiscorrelation,theexpressionsforthePDFandmeanofthereliabilitiesgivenby 3.13 and 3.14 cansignicantlydifferfromthesimulationresults,aswillbeshowninSection 3.6 .Thus,thecorrelationsamongtheoutputerroreventsshouldbeconsideredinorderfortheanalyticalexpressionstoagreewiththesimulationresults.In[ 34 ],theauthorsaccountforthiscorrelationbyobtainingthejointmultivariatedistributionofZjandusingthisdistribu-tiontocomputethedensityfunction.However,thisapproachwouldinvolvecomputingacovariancematrixinvolvingdifferentpairsoferroreventsandusingthiscovariancematrixinthedensityfunction.Thisapproachresultsinaverycomplicatedexpression.Evenwiththeindependenceassumption,thedensityfunctionin 3.12 iscomplicated.Further,theapproachusingthemultivariatedistributionoffersnofurtherinsightintothebehaviorofthereliabilities.NotethatthecorrelationbetweendifferentPRVsarisebecausetheysharecommonnoisesamples,whichisaconsequenceoftheassociatederroreventsdifferingfromthecorrectcodewordinacommonsetofsymbols.Weintroduceasimpleapproachtoac-countforthecorrelationbetweenPRVsbycomputingthecorrelationamongdifferenterrorevents.Werstdenethecorrelationbetweentwoerroreventse1ande2oflengthsl1andl2respectivelyasCe1;e2=Pminl1;l2i=1e1;ie2;i maxl1;l2;.15whereej;ireferstotheithbitoferroreventejandthe`'operatordenotestheXNORoperation.Forexample,`11101011'and`111010000111'aretwoerroreventsoflength8and12respectively,andthecorrelationbetweenthetwoeventscanbecomputedusing 3.15 tobe0:5.WeaccountforthecorrelationbetweenoutputerroreventsbyeliminatingsomeoftheerroreventsthatarehighlycorrelatedandusingthereducedsetoferroreventstocomputethePDF/CDFofreliabilities.WedeneacorrelationthresholdTCorr,and

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24 whenevertwoerroreventshaveacorrelationvaluegreaterthanTCorr,thelongerofthetwoeventsiseliminatedfromtheeventset.Thelongerofthetwoeventsisremovedfromtheeventsetbecauseperformanceisusuallydominatedbythelowweighterrorevents.ThisprocessiscontinueduntilallremainingpairsoferroreventshavecorrelationlessthanTCorr.Wenormalizethecorrelationbythelongerofthetwoerroreventlengthstoensurethateventswithverydissimilarlengthshavealowvalueofthecorrelation.Thiseliminatesthepossibilityofdiscardingalongeventwhichmayshareacommoninitialpaththroughthetrelliswithasmallerrorevent.Thus,acondensedeventsetisobtainedwithinwhichtheeventshavelowcorrelationvalue.WeexpectthesmallcorrelationbetweentheeventsinthecondensedsettohaveanegligibleeffectontheindependenceassumptionusedinderivingthePDF.ItwillbeshowninSection 3.6 thatifthesummationin 3.12 isperformedoverthecondensedeventset,theresultingvaluesarestrikinglyclosetothesimulationresultsforproperlychosenvaluesofTCorr.Thus,theneedforjointmultivariatedistributionsinvolvingthecovariancematrixofoutputerroreventsisavoidedusingthistechnique. 3.3 AMathematicallyTractableDensityFunctionTheexpressionsforthedensityfunctionofthereliabilitygivenby 3.12 and 3.13 arecomplicatedandnotconvenientforuseinmathematicalanalysisofreliability-basedtech-niques.WeaddressthisissuewithanadhocestimateofthePDFbasedonthefollowingobservations: Themeanofthereliabilitiesobtainedfrom 3.14 isveryclosetothesimulationresults.ThisfactwillbesubstantiatedinSection 3.6 Giventhecorrectdecoderoutput,theconditionaldistributionofthesoftoutputforabitisapproximatelyGaussianwithvarianceapproximatelyequaltotwicethemeancf.[ 35 36 ].

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25 Thus,wesuggestmodelingthereliabilityastheabsolutevalueofaGaussianrandomvariablethatsatisesthesymmetrycondition,i.e.,=jXj;XN;2;.16whereisthemeanobtainedbynumericallyevaluating 3.14 .ThecumulativedistributionfunctionCDFcaneasilyfoundtobeF=8>><>>:Q)]TJ/F25 7.97 Tf 6.675 -4.428 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[( p 2)]TJ/F50 11.955 Tf 11.955 0 Td[(Q)]TJ/F25 7.97 Tf 6.675 -4.428 Td[(+ p 2;>00;otherwise: .17 Differentiatingwithrespectto,thePDFofthereliabilityis,f=8>><>>:exp)]TJ/F26 7.97 Tf 5.479 -9.683 Td[()]TJ/F18 5.978 Tf 7.782 4.025 Td[()]TJ/F20 5.978 Tf 5.757 0 Td[(2 4+exp)]TJ/F26 7.97 Tf 5.48 -9.683 Td[()]TJ/F18 5.978 Tf 7.782 4.025 Td[(+2 4 2p ;>00;otherwise: .18 Unliketheexpressionin 3.13 ,thedensityfunctionin 3.18 doesnotinvolvesummationsandproductsandisexpressedinclosed-form.Indeed,wehavetoresorttonumericalcomputationtoobtainthemean,,butforproblemsinvolvingexplicitprobabilitiesofreliabilities, 3.18 ismathematicallymoretractablethan 3.13 .InSection 3.6 weprovideresultsthatshowthisGaussianapproximationisextremelyaccurate. 3.4 AClosed-FormExpressionfortheBit-Error-RateofSISODecodersInthissection,wedemonstrateoneapplicationoftheapproximatedensityfunctionofreliabilitiespresentedinthepreviouschaptersee 3.18 .Wewillusethedensityfunctiontoderiveaclosedformexpressionforthebit-error-rateofmax-log-MAPdecoding.TheprobabilityofabitdecodingincorrectlyconditionedonitsreliabilityisgivenbyPbj=1 1+e:.19

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26 TheprobabilityofbiterrorcanbeobtainedbyintegratingPbjoverthedensityofgivenin 3.18 .Pb=Z101 1+ef .20 =1 p 4Z10exp)]TJ/F51 11.955 Tf 8.136 -9.684 Td[()]TJ/F24 7.97 Tf 13.151 5.699 Td[()]TJ/F25 7.97 Tf 6.586 0 Td[(2 4+exp)]TJ/F51 11.955 Tf 8.136 -9.684 Td[()]TJ/F24 7.97 Tf 13.151 5.699 Td[(+2 4 1+e; .21 =1 p 4Z10exp)]TJ/F51 11.955 Tf 8.136 -9.684 Td[()]TJ/F24 7.97 Tf 13.151 5.699 Td[(+2 41+e 1+e; .22 =Q p 2=Qr 2: .23 Thus,thebiterrorratecanbeexpressedasaQ-function 1 withtheargumentdependingsolelyonthemeanofreliability.Themeanofthereliabilitycanbecalculatedusing 3.14 .Theprobabilityofbiterrordependsontheweightdistributionoftheerroreventsofacode.Themeanofthereliabilityencapsulatesallpropertiesofthecodeintoasinglequantitytherebyleadingtoaconvenientexpressionforthebit-error-rate.Theexpressionofthebiterrorrategivenin 3.23 canbeusedinvariousreceiver-drivenstrategiesthatrelyonthereceiverhavinganestimateofthebit-error-rate.Forexample,considerareal-timestreamingaudio/videostreamingapplication.Theseapplicationsareloss-tolerantbutdelay-intolerant.Thus,anARQschemeforsuchascenariocanbedesignedasexplainedbelow.Afterdecoding,thereceiverestimatesthenumberofbitsthathavedecodedinerrorbyusing 3.23 .Sincetheseapplicationscantoleratesomeloss,thereceiverrequestsforare-transmissiononlywhenthenumberofbitsinerrorexceedsacertainthreshold. 3.5 ExtensiontoBlock-FadingChannelsWenowextendtheresultspresentedtoblock-fadingquasi-staticfadingchannels.Inablock-fadingenvironment,allbitsinapacketexperiencethesamechannelgain.For 1weconsiderthistobeclosedformsincetheQ-functioniswidelyusedincommunica-tiontheoryandcanbecomputedaccuratelyandefciently

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27 suchascenario,wecharacterizethereliabilityattheoutputofamax-log-MAPdecoderconditionedonaparticularrealizationofthechannel.Assumingcoherentdetectionthechannelgainsareknowntothedecoder,theentirecodespaceisrotatedandscaledbythechannelgain.ThenMLorMAPdecodingisperformedonthereceivedvectorinthenewcodespace.Thus,thereliabilityconditionedontheblock-fadingchannelgaincanbeexpressedusing 3.4 asij=1 22minjkr)]TJ/F50 11.955 Tf 11.955 0 Td[(cjik2)-222(kr)]TJ/F50 11.955 Tf 11.955 0 Td[(cMLk2:.24NotethatcjiandcMLarecodewordsintherotatedcodespace.Usingthesameap-proachasbefore,wecanassumethattheMLcodewordisthetruetransmittedcodeword.Thuswehave,r=cML+e;eN;2I=cML+e0;e0N;2 2I .25 Using 3.25 and 3.5 in 3.24 ,thereliabilityconditionedonthefadingcoefcientisij=2 22minjkejik2+2e0Teji:.26Proceedingasin 3.8 3.13 ,theconditionalCDFandPDFofreliabilitiescanbeobtainedasFj;=F22 2;fj;=22 2f22 2;.27whereFandfaredenedin 3.11 and 3.12 respectively.TheconditionalmeanofsoftinformationcanthenbeobtainedasE[j]=2 2;.28wherexisdenedin 3.14

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28 Figure3-1:Themeanofreliabilitiesasafunctionofthesignal-to-noiseratiowhenthecorrelationbetweentheoutputerroreventsareignored. 3.6 NumericalResultsInthissection,theanalyticalexpressionsderivedintheprevioussectionsarecom-paredwithsimulationresults.Forallresults,non-recursive,non-systematicconvolutionalcodesCCwithablocksizeof1000bitsareused.TheMax-Log-MAPimplementationoftheBCJR[ 37 ]algorithmisusedfordecoding.InourimplementationtheMax-Log-MAPmetricisscaledbythenoisevarianceasin 3.4 .InFigure 3-1 ,themeansofthereliabilitiesobtainedusing 3.14 arecomparedwiththeactualmeansobtainedfromsimulation.Thecomparisonisshownforarate1=2,constraintlength3convolutionalcodeandarate1=3,constraintlength7convolutionalcode.Theconstraintlength3CChasgeneratorpolyno-mials1+D2and1+D+D2or;78inoctalnotation.Theconstraintlength7CChasgeneratorpolynomials54;624;7648.Itisobservedthattheanalyticalexpressionpro-ducesestimatesthataresmallerthantheactualvalues.AsexplainedinSection 3.2.2 andinReggianiandTartara[ 34 ],theassumptionthatalltheoutputerroreventsareindependentleadstoover-countingwhichcausestheanalyticalresultstoproduceconservativeresults.

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29 Table3-1:Erroreventmultiplicityofthe5;7convolutionalcode d ad ad AllEvents TCorr=0.7 dmin=5 1 1 6 2 2 7 4 4 8 8 8 9 16 9 10 32 5 11 64 11 12 128 5 13 256 14 14 512 13 dmax=15 1024 20 AtlowSNRs,thereisalargergapbetweenanalyticallyobtainedvaluesandthesimulationresultswhencomparedtohighSNRs.ThisisbecausetheperformanceatlowSNRsisdominatedbyblocksthatdecodeincorrectlyandhencetheassumption 3.7 isviolated.Totightenthegapbetweentheanalyticalandsimulationresults,itisrequiredtocon-siderthecorrelationbetweenerrorevents.ThenumberoferroreventswithweightdeventmultiplicityisshowninTable 3-1 forthe;78CCItisseenthateliminatingeventsthathaveacorrelationvaluehigherthanthecorrelationthresholdTCorr=0:7inthiscaseresultsinacondensedsetoferrorevents.Weexpectthatusingthiscondensedsetofeventswithlowcorrelationwillreducetheover-countingproblemcausedbytheindependenceassumption.ThemeanofthereliabilitiesafteraccountingforthecorrelationbetweenoutputerroreventsasexplainedinSection 3.2.2 isshowninFigure 3-2 .Ifthesummationin 3.12 isperformedoveracondensedsetoferroreventsasshowninTable 3-1 ,andthemeanthencomputedusing 3.14 ,itcanbeseenfromFigure 3-2 thattheanalyticalresultsareveryclosetothetruevaluesevenatlowSNRs.ThePDFofreliabilitieseqn. 3.13 forthe;78codeiscomparedwiththetruePDFinFigure 3-3 .ThetruePDFwasobtainedexperimentallybysimulatingthedecoding

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30 Figure3-2:Themeanofreliabilitiesasafunctionofsignal-to-noiseratioaftertakingintoaccountthecorrelationbetweenoutputerrorevents. ofanumberofblocksthatweretransmittedoveranAWGNchannel.ThereliabilityofeachbitwasrecordedandthetruePDFwasestimatedfromthehistogramoftherecordedreliabilities.ItcanbeseenfromFigure 3-3 thatresultsclosetothetruePDFcanbeobtainedwhenthecorrelationbetweenoutputerroreventsisconsideredinthecomputationofthedensityfunction.ThecorrelationisconsideredbyevaluatingthePDFin 3.12 overthecondensedsetoferroreventsshowninTable 3-1 .NotethattheanalyticalPDFismuchclosertothetruePDFathigherSNRs.ThePDFobtainedusingthesimple,adhocestimatein 3.18 isshowninFigure 3-4 .Themean,,thatspeciesthePDFisobtainednumericallyfrom 3.14 .ItisobservedthatthisadhocexpressionproducesresultsthatareclosertothetruePDFwhencomparedtotheexpressionin 3.12 .UnlikethePDFgivenin 3.12 ,theadhocestimateproducesresultsthatareveryclosetothetruePDFevenatlowSNRs.ThecorrelationbetweenerroreventscanbeaccountedforintheadhocPDFestimatebyevaluatingoveracondensedsetoflow-correlationerror-eventsasshowninTable 3-1 .Asbefore,accountingforthe

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31 Figure3-3:ThePDFofreliabilitiesofthe;78CCfortwodifferentsignal-to-noiseratios. correlationproducesbetterresultswhencomparedtotreatingthePRVsasindependentrandomvariables.Theclosed-formapproximationfortheprobabilityofbiterrorPbgivenin 3.23 iscomparedtosimulationresultsinFigure 3-5 .Resultsareshownforboththememory2andmemory6codes.SincePbdependssolelyon,resultsareshownfortwoapproachestocompute.Intherstapproach,iscalculatedanalyticallyusingtheexpressiongivenin 3.14 .Inthesecondapproach,isobtainedthroughsimulation.ThisapproachshowstheutilityoftheapproximationtoPbinapplicationswhichrequirethereceivertohaveanestimateofitsbit-error-rate.Forbothscenariositisseenthattheclosed-formapproxi-mationisveryclosetothesimulationresults.FortheresultsinFigure 3-5 ,alltheerroreventswereconsideredcorrelationbetweenerroreventswereignoredwhenthemeanofthereliabilitieswascalculatedanalyticallyusing 3.14 .Ifthemeaniscomputedafterac-countingforthecorrelationbetweenerrorevents,theclosed-formapproximationproduces

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32 Figure3-4:ThePDFofreliabilitiesofthe;78CCobtainedusingthesimpler,mathe-maticallytractableexpressiongivenin 3.18 betterresultswhencomparedtoignoringthecorrelation.ThisisnotshowninFigure 3-5 forthesakeofclarity.ThemeanofreliabilitiesobtainedwhenthecodedbitsaretransmittedoveraRayleighblock-fadingchannelisshowninFigure 3-6 forthe;7convolutionalcodes.Theana-lyticalvaluesareobtainedbyintegratingtheconditionalmeanin 3.28 overthedensityofthefadingamplitudesgivenbyf=2e)]TJ/F25 7.97 Tf 6.587 0 Td[(2uwhereuistheunitstepfunction.

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33 Figure3-5:Theprobabilityofbiterrorformax-log-MAPdecodingofconvolutionalcodesevaluatedusingtheclosed-formapproximationgivenin 3.23 Figure3-6:Themeanofreliabilitiesofthe;7convolutionalcodesasafunctionofsignal-to-noiseratioofablock-fadingchannel.

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CHAPTER4CODEDCOOPERATIONTHROUGHCOLLABORATIVEDECODINGThecodedcooperationschemessummarizedinSection 2.3 exploittheencoderstruc-tureoftheerrorcontrolcodethatisused;i.e.,thenodescooperativelyencodethemessagetoformamorepowerfulcodewordatthereceiver.Inthischapterweintroduceacoopera-tivestrategythatinvolvescollaborationinthedecodingprocess;i.e,thedestinationandtherelayscooperativelydecodethemessagefromthesource.ThesystemmodelisshowninFigure 4-1 .Adistanttransmitterbroadcastsamessagetoaclusterofreceivingnodes,oneormanyofwhichcouldbetheintendeddestination.Ifanyoftheothernodesdecodescorrectly,itcanuseofoneofthetraditionaldecode-and-forwardschemestoforwardthemessagetothedestination.Themoreinterestingscenarioiswhennoneofthenodesde-codescorrectly.Inthiscase,wecannotuseanyschemethatdependsoncorrectdecodingattherelays.Theamplify-and-forwardschemecouldbeused,butthecooperationoverheadisveryhigh.Thus,alternativetechniquesarerequiredtominimizetheoverhead.Thefundamentaldrawbackofthedecode-and-forwardbasedapproachisthateachrelayformsitstransmissionbasedonitsowndecodingdecisionsandnotonthedecodingdecisionsattheothernodes.Forexample,iftheintendeddestinationoroneoftheothernodesinFigure 4-1 hasdecodedallbuttwoofthebitscorrectly,thenitisnotnecessaryfortheotherrelaystoforwardinformationasinthepreviouslydescribeddecode-and-forwardschemes.Iftherelayshavesomeinformationaboutthedecisionsmadeattheothernodes,theycanaccordinglyscheduletheirtransmissionstominimizethecollaborationoverhead.Thus,inthemodelshowninFigure 4-1 ,thereisnodistinctionmadebetweentherelayandthedestination.Thenodescooperatewitheachothertoobtainsomeinformationaboutthedecisionsmadeattheothernodes.Thisinformationhelpsminimizethecooperationoverheadrequiredforcorrectdecodingatoneofthenodes. 34

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35 Figure4-1:Systemmodelforcollaborativedecoding. Thesystemmodelalsorepresentsabroadcastscenariowhereinallnodesareinterestedinthemessagefromthetransmitter.Aftercollaboration,thenodethatdecodescorrectlycanpassthisinformationontoothernodes.AniterativetechniquetoachievecooperationinthisbroadcastscenariowithtwocooperatingnodeswasproposedbyWongetal.[ 9 38 ].ThebasicprincipleofthistechniquewithtwocooperatingnodesisshowninFigure 4-2 .TheschemeproceedsintwostagesasinthepreviouscooperationschemesseeChapter 2 .Intherststage,thedistanttransmitterbroadcastsitsmessagetothecooperatingcluster.Stage2proceedsinmultipleiterationswiththeiterationscontinuinguntiloneofthenodesdecodescorrectlyoraxednumberofiterationselapse.Ineachiteration,thenodesrstexchangesomecoordinationinformation.ThecoordinationinformationisshownasthesoliddiamondinFigure 4-2 .Thecoordinationinformationgiveseachnodesomeinfor-mationaboutdecodingattheothernode.Basedonthisinformation,eachnodetransmitssomeinformationabouttheinitialmessagesentbythesourcetotheothernode.Onre-ceivingthismessage,eachnodeperformsdecodingandifeithernodedecodescorrectly,cooperationisterminated.Ifnot,thecooperationprocedureisrepeatedasshowninFig-ure 4-2 .Theprocessofiteratingbetweendecodingandinformationexchangeisreferredtoascollaborativedecoding.Aninformation-theoreticstudyofthesystemmodelshowninFigure 4-1 withtwocooperatingnodeswasstudiedbyDraperetal.[ 39 ].Thenodesstartcollaboratingafter

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36 Figure4-2:Principleofinteractive/collaborativedecodingwithtwonodes. theyreceivetheentireblocksentbythetransmitter.Aroundofconversationisdenedasamessageexchangebetweenthenodeswithonemessagetransmittedfromeachnodetotheother.Thesumofthetransmissionratessum-ratebetweenthenodesrequiredforcor-rectdecodingisusedastheperformancecriteria.Alowsum-rateimplieslowcooperationoverhead.Theauthorsprovethatcollborationwithmultipleroundsofconversationbe-tweenthetwonodescanguaranteecorrectdecodingintheShannonsense:arbitrarilylowerrorprobabilityastheblocklengthgoestoinnitywithalowersum-rateoverheadthancollaborationconsistingofoneroundofconversation.Thisisbecauseinformationsentinanearlieriterationservesasside-informationatthereceivingnodeandthetransmittingnodecanusemoreefcientcodingtechniquesthatuseside-informationatthetransmitterandreceivertoencodefutureconversations.WithreferencetoFigure 4-2 ,notethatthereisnospeciccoordinationinformationinthisscheme.Theinformationtransmittedbyanodeinoneiterationservesascoordinationinformationforthenextiteration.Thus,eachnodetailorsitstransmissionbasedoninformationaboutdecodingattheothernodethatitreceivedinthepreviousiteration. 4.1 CollaborativeDecodingthroughReliabilityExchangeWongetal.[ 9 38 ]presentaniterativeapproachtocooperationforthescenariode-pictedinFigure 4-1 .Thebasicprincipleoftheirapproachisasfollows.Onreceiving

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37 themessagefromthetransmitter,thenodesperformMAPdecodingontheirreceivedsym-bols.Eachnodeusesthebitreliabilitiestodeterminewhichbitsareunreliableandrequestsadditionalinformationaboutthesebitsfromothernodes.TheothernodestransmittheirestimatesoftheaposterioriLLRforthesebits.Theoriginalrequesterusesthisinforma-tionasthecorrespondingaprioriinformationinitsMAPdecoderandperformsdecodingagain.Thisprocessofinformationrequestanddecodingisrepeatedforafewiterations.Forsimplicity,theprocessofexchangingsoft-informationwillbehenceforthreferredtoasreliabilityexchange.Theinformationthatpassesbetweenthedifferentnodeswillbereferredtoastheoverheadincollaborativedecoding.WebuildontheschemeofWongetal.[ 9 38 ]byinvestigatingtheperformanceandoverheadfortwodifferentclassesofreliabilityexchangeschemesformultiplenodesgreaterthantwo. 4.1.1 CollaborativeDecodingthroughtheReliabilityExchangeoftheLeastReliableBitsInthissection,weprovideresultsforanextensionoftheschemeproposedin[ 9 ].Foralltheresultsinthischapter,arate1/2nonrecursiveconvolutionalcodewithgeneratorpolynomials1+D2and1+D+D2isusedtoencodetheinformationsequence.Forconve-nience,werefertothiscodeasthe;7code,where5and7aretheoctalrepresentationsofthegeneratorpolynomials.TheencodedmessagesaretransmittedoveradditivewhiteGaussiannoisechannelsusingbinaryantipodalsignalingandarecoherentlydemodulated.EachreceiverdecodesthereceivedmessageusingtheBCJR[ 37 ]algorithm.Eachnodethenrequestsreliabilityinformationforacertainpercentageoftheleastreliableinforma-tionbitsbybroadcastingthebitindicesofthosebits.Eachnodethatreceivesthebitindicesreplieswithitsestimateofthesoftinformationforthosebits.ThenodethatrequestedtheinformationthenusesthesereliabilitiesasaprioriinformationandrunstheBCJRalgo-rithmagain.InWongetal.[ 9 ]itisshownthatforapacketsizeofapproximately1000bitsencodedwitha;7convolutionalcode,collaborativedecodingwithtworeceiversprovidesperformanceveryclosetoMRCatvaluesofEb=N0greaterthan5dB.Threeiter-ationsofcollaborativedecodingwasperformedbyrequestingsoftinformationfor7:5%of

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38 theleastreliableinformationbitsineachiteration.Thereasonforrequestingtheleastreli-ablebitsLRBsisthatmostofthebitsthatdecodeincorrectlyhavelowreliabilityvalues.UsingMRCwouldrequireexchangingallofthereceivedcodedsymbols.Theoverheadinbits,denotedbyMRCcanbecalculatedasMRC=N Rcq;.1whereNisthesizeoftheinformationmessageinbits,Rcisthecoderateandqisthenumberofbitsrequiredtorepresentaoatingpointchannelsymbol.Notethat 4.1 representstheoverheadcontributionofasinglenode.Usingthecollaborativedecodingschemementionedabove,theoverheadcontributionofasinglereceivercanbesplitintotwoparts.TherstpartconsistsofthebitindicesthatareceiverbroadcaststorequestthesoftinformationoftheLRBs,andthesecondpartconsistsofthesoftinformationthatanodetransmitseachtimeitreceivesanLRBrequestfromanothernode.Thus,theoverheadforthisschemecanbeexpressedasLRB=NIaNdlog2Ne+NR)]TJ/F15 11.955 Tf 11.955 0 Td[(1q;.2whereNIisthenumberofiterationsofcollaborativedecoding,aisthefractionofin-formationaboutwhichreliabilityinformationisrequested,NRisthenumberofreceiversinvolvedincollaborativedecodinganddxeisthesmallestintegergreaterthanorequalx.TherstterminthesummationontheR.H.Sof 4.2 accountsforthebitindicesthatneedtobetransmittedtorequestsoftinformation,andthesecondterminthesummationaccountsforthebitsrequiredtosendoutthesoftinformationeachnodereceivesNR)]TJ/F15 11.955 Tf 12.051 0 Td[(1LRBrequests.Notethatthesizeoftherequestscanbefurtherreducedthroughsourcecod-ingorbyexploitingthetime-correlationbetweenthereliabilitiesofthebitsinerror[ 33 ].Both 4.1 and 4.2 refertotheoverheadperreceiver.Alltheschemesinthispaperwillbecomparedusingtheoverheadcontributionperreceiver.

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39 Table4-1:OverheadofLRB-1fordifferentnumberofnodes. Numberofnodes Overheadbits %reduction relativetoMRC 2 2025 77:5% 5 4050 55:0% 10 7425 17:5% Generallyvebitsareenoughtorepresentaoatingpointchannelsymbolaccu-rately[ 40 ],[ 41 ].Forapacketsizeof900bits,theoverheadforMRCcanbecalculatedusing 4.1 as9000bits.Tenbitsarerequiredtorepresenteachbitindexinpacketof900bits,andifweperformthreeiterationsofcollaborativedecodingwithsoftinformationof5%oftheLRBsbeingrequested,theoverheadis2025bitsfortwonodesusing 4.2 .Thusweseethatperformingcollaborativedecodingreducestheoverheadby77:5%whencomparedtotheMRCoverhead.Performingcollaborativedecodingwiththreeiterationsof5%LRBexchangewillbehenceforthreferredtoasschemeLRB-1.LRB-1hascollaborativedecodingoverheadof22:5%ofMRCoverheadforapacketsizeof900bitsandaclustersizeoftwonodes.Fortherestofthechapter,theoverheadforcollaborativedecodingwillbereportedasapercentagewithreferencetotheMRCdecodingoverhead.NotethattheoverheadperreceiverforLRB-1increaseswiththenumberofreceivers.TheoverheadforLRB-1fordifferentnumberofnodesisshowninTable 4-1 .ThereductioninoverheadrelativetoMRCdecreaseswithanincreaseinthenumberofnodes.InFigure 4-3 ,theperformanceofLRB-1isshownfordifferentnumberofcollaborat-ingreceivers.Wenotethattheperformancesaturatesformorethanfourreceivers.Thisindicatesthatbiasingtheleastreliablebitswithalotofaprioriinformationfromtoomanyreceiverswillnotimprovetheperformancesignicantly.Thisisbecausetherearesomeincorrectlydecodedbitsthatmayhaverelativelyhighreliabilities.Thisisagainsubstan-tiatedinthenextsection.Whenleast-reliablebitsareexchanged,theincorrectbitswith

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40 Figure4-3:Performanceoftwocollaborativedecodingschemesinwhichreceiversrequestinformationforasetofleast-reliablebits. highreliabilitiesmayneverbecorrectedregardlessofhowmuchinformationisprovidedfortheLRBs.AnobviousmethodtoimprovetheperformanceofLRB-1istoincreasethepercentageofLRBsrequested.FromFigure 4-3 ,weseethatrequesting10%ofLRBreliabilitiesinsteadof5%givesanimprovementinperformanceofapproximately1dBforaclusterofsixcollaboratingnodes.However,thisincreasesthecollaborativedecodingoverhead,andoursimulationsshowthattheperformancesaturatesformorethanfourreceiverseveninthiscase.Anotherdisadvantageofrequestingmoreinformationisthatasthenumberofreceiversincreases,thetimeforinformationexchangealsoincreases.Eachreceiverhastosendoutasetofbitindicesrequestingreliabilityinformation,andthenalltheotherreceivershavetorespond.Tocoordinatethisinformationexchange,agoodMACprotocolwillhavetobedesigned.Thislatencywouldnotbeacceptableincertainapplications.AsimpleextensiontoLRB-1istotransmitallthesoftinformationviaabroadcastchannelandtohaveeachnodeuseallthereceivedsoftinformation,evenifthatnodewas

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41 nottheoriginalrequester.Sincethenodesotherthantheonethatrequestedinformationalsoreceivethesoftinformation,theycanmakeuseofitasaprioriinformationintheirnextroundofSISOdecoding.Thus,forthenodesthatdidnotrequesttheinformation,reliabilityinformationforasetofbitswithrandomreliabilitiesisobtained.Thisschemewith5%requestandthreeiterationswillbereferredtoasLRB-2.LRB-2hasthesameoverheadasLRB-1.TheresultsinFigure 4-3 showthatLRB-2evenoutperformsLRB-1with10%LRBexchange.Further,LRB-2doesnotsufferfromthesaturationproblemlikeLRB-1.Hence,LRB-2wouldbeabetterchoiceifexchangingLRBswastheschemechosentoperformcollaborativedecoding.Thebiggestdisadvantageofthisschemeisthattheper-receiveroverheadgrowslin-earlywiththenumberofreceiverscf. 4.2 .Thus,ifthenumberofnodesislarge,evenrequestingaverysmallpercentageofLRBsoftinformationmightcausetheoverheadtobecomelargerthantheMRCoverhead.Inthenextsection,anexchangeschemeispre-sentedthathasanoverheadwhichisindependentofthenumberofreceivers. 4.1.2 CollaborativeDecodingthroughtheReliabilityExchangeoftheMostReliableBitsOnewaytosignicantlyreducetheoverheadistopreventanodefromtransmittingsoftinformationmorethanonce.From 4.2 ,weseethatforblocksizesofapproximately1000andmorethantenreceivers,multipletransmissionsofsoftinformationcontributestowardsmorethan82%oftheoverheadperreceiver.Supposethat,afterSISOdecoding,eachreceiverselectsacertainsetofbitsandbroadcaststhereliabilitiesofthesebitstotheothernodes.Itisimportanttoensurethatthenodesbroadcastgoodreliabilityin-formation,i.e.,reliabilityinformationaboutbitsthataredecodedcorrectly.Thecriticalstepinthisschemeistodeterminethesetofbitsforwhichanodewillbroadcastthesoftinformation.Sinceeachnodeonlysendsoutsoft-informationonlyonce,thecollaborativedecodingoverheadperreceiverisgivenbyMRB=NIaNdlog2Ne+q:.3

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42 Figure4-4:Reliabilitydensityfunctionsassociatedwithcorrectlyandincorrectlydecodedbits. Notethatforthisscheme,unliketheLRB-basedschemes,theoverheadperreceiverisindependentofthenumberofreceivers.Ifanodebroadcaststhesoftinformationforabitthatwasdecodedincorrectly,usingthisvalueasaprioriinformationwoulddegradetheperformanceoftheothernodes.Themotivationbehindourapproachforselectingbitscomesfromobservingthedensityfunc-tionsofthereliabilitiesassociatedwithcorrectlyanderroneouslydecodedbits.Figure 4-4 showsthedensityfunctionofthereliabilitiesfora;7convolutionalcodewithablocksizeof900bits.Wenotethatthetrendfollowedbythedensityfunctionsisasexpected;i.e.,mostoftheincorrectlydecodedbitshavelowreliabilitiesandthecorrectlydecodedbitshavearelativelyhighreliability.WeobservethatatanEb=N0of0dB,themaximumvalueofthereliabilityofabitthatdecodesincorrectlyisabouthalfofthemaximumvalueofthereliabilityofabitthatdecodescorrectly.ForvaluesofEb=N0greaterthan3dB,morethan50%ofthebitsthatdecodecorrectlyhavereliabilitiesgreaterthanthemaximumreliability

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43 Figure4-5:Performanceoftwocollaborativedecodingschemesinwhichreceiversbroad-castinformationaboutasetofmost-reliablebits. oftheincorrectlydecodedbits.Hence,ifanodebroadcastsasmallpercentageofitsmostreliablebitsMRBs,itisverylikelytosendoutgoodsoftinformation.Thesebitswillcorrespondtoasetofbitswithrandomreliabilitiesattheothernodes.Performingthreeiterationsof10%MRBreliabilityexchangewillbereferredtoasschemeMRB-1.ThecollaborativedecodingoverheadperreceiverofMRB-1iscalculated,using 4.3 ,as45%thatofMRC.ThoughreliabilityinformationisexchangedformorebitsthaninLRB-1andLRB-2,theoverheadisstillsmallerthaninLRB-1andLRB-2formorethanvenodes.Oursimulationsshowedthattheperformanceimprovementislessthan2dBevenwithtennodeswhencomparedtotheperformanceofasinglereceiver.ThisisshowninFigure 4-5 .Thesmallerperformanceimprovementcanbeattributedtothesetofbitsthatarebroadcastineachiteration.AttheendoftherstSISOdecoding,reliabilitiesof10%ofthemostreliablebitsarebroadcast.Sincewearebiasingcertainbitswithgoodaprioriinformation,attheendofthenextSISOdecoding,thereliabilitiesforthesebitswill

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44 Figure4-6:Bitindicesofreliabilitiesexchangedasafunctionofiteration. becomelarge,anditisverylikelythatthesebitswilllieinthe10%MRBset.Soreliabilitiesofthesebitswillbebroadcastinthenextiteration.Butsincetheothernodeshavealreadyreceivedtheinformationaboutthesebits,biasingthemwithmoreaprioriinformationwillnotimprovetheperformancesignicantly.ThiscanbeobservedinFigure 4-6 inwhichweshowthebitindicesbroadcastineachiterationforonepacketof900bits.AnasteriskonabitpositionimpliesthatreliabilityinformationaboutthatbitwaseitherrequestedLRB-1ortransmittedMRBschemes.WeseethatforMRB-1,averylargeportionofbitsarebroadcastagainineveryiteration.Inthreeiterations,thereliabilitiesof102bitsarebroadcastagainamongthetotalof270bitstransmitted.Thisconstitutesaround37%ofthetotalbitssent.AsthevalueofEb=N0increases,therearefewerbitsinerrorandinordertoimprovetheperformance,theseerroneouslydecodedbitsneedtobebiasedwithreliableaprioriinformation.Ifagoodpercentageofthebitsarerepeated,therewillbealowprobabilitythataprioriinformationwillbereceivedforallofthebitsthatareinerror.Asimplemethodtoeliminatethisproblemistogivethenodesmemorytorememberthesetofbitsforwhichsoftinformationistransmittedorreceived.Thisensuresthatbits

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45 thatarealreadylikelytohavegoodreliabilitiesafteroneiterationdonotgetbiasedwithmoreaprioriinformationinthenextiteration.Otherbitsarenowgivenanopportunitytoreceivereliabilityinformation.Thisscheme,whichisjustMRB-1withmemory,willbereferredtoasMRB-2.InMRB-2,eachnodesortsitsbitsinascendingorderofreliabilityaftertherstSISOdecoding.Theneachreceiverbroadcasts10%oftheMRBsforwhichsoftinformationwasnottransmittedbyanynodeinthepreviousiterations.Thus,ineachiterationanewsetofbitsgetreliabilityinformation.ThisisillustratedinFigure 4-6 .InMRB-2,therearenobitsforwhichsoftinformationistransmittedinmorethanoneiteration.TheperformanceofMRB-2isshowninFigure 4-5 .Ifinanyoftheiterations,anodeisnotabletondabitaboutwhichaprioriinformationhasnotbeentransmittedearlier,itdoesnotsendoutanyreliabilities.Thus,theoverheadinMRB-2islessthanorequaltotheoverheadinMRB-1,buttheperformanceofMRB-2ismuchbetterthanthatofMRB-1.NotethataddingmemorytoLRB-1willnotimprovetheperformancesignicantly.Thisisbecauseineachiteration,aprioriinformationbiasestheleastreliablebitsandtheirreliabilityincreasesafterSISOdecoding.Thus,inthenextiterationanewsetofbitswillconstitutethesetofLRBs.Hence,thereisonlyanegligibleoverlapinthesetofLRBsineachiteration.ThiscanbeobservedinFigure 4-6 ,inwhichLRB-1hasjustonebitthatisrepeatedinthreeiterations.AgoodsuboptimalvariantofMRB-2sendshard-decisionsoftheMRBsinsteadofthesoftdecisions.Thisreducestheoverheadfortransmittingsoftinformationfromqcf. 4.3 bitsperinformationbittoonebitperinformationbit.Thus,forMRB-2withthreeiterationsof10%reliabilityexchange,thecollaborativedecodingoverheadisonly33%forapacketof900bits.Forareasonablylargenumberofreceivers,theharddecisionsfromdifferentreceiversformaprioriinformationthatissufcienttobiastheinformationbitstoproducecorrectdecisionsattheoutputoftheSISOdecoder.TheperformanceofthisschemeforsixreceiversisillustratedinFigure 4-7

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46 Figure4-7:Performanceofsuboptimalvariantstwocollaborativedecodingschemesinwhichharddecisionsareexchangedinsteadofsoftinformation. Notethatthistechniquecanbeextendedtoanyoftheschemesdiscussedearlier.TheperformanceofLRB-2withhard-decisionsisalsoshowninFigure 4-7 .Weseethatalossofapproximately0:5dBcanbeexpectedforthesuboptimalschemewhencomparedtotheoriginalscheme.BycomparingtheperformanceoftheLRBandMRBschemesinFigure 4-3 andFigure 4-5 ,itisseenthatLRB-2outperformsMRB-2forasmallnumberofreceiversandMRB-2performsbetterwhenthenumberofcooperatingnodesislarge.Thisisbecauseanimprovementinperformanceisobtainedwhenbitsthataredecodedincorrectlygetgoodaprioriinformation.IntheLRBschemes,thebitsthatarelikelytobedecodedincorrectlyarespecicallytargetedleadingtoaimprovementinperformance.TheMRBschemesaremoreoptimisticinnature.Eachnodebroadcastsreliableinformationthatmayormaynotbeusefultotheothernodes.Thus,atoneoftheothernodes,goodreliabilitiesarereceivedforasetofbitswithrandomreliabilities.Theremayormaynotbeanincorrectlydecodedbitinthesebitpositions.Whenafewreceiverscollaborateitisnotlikelythatall

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47 theunreliablebitsreceiveaprioriinformationfromothernodes.Thus,theperformanceofMRB-2isrelativelyworseforasmallnumberofcooperatingusers.However,foralargenumberofreceivers,itislikelythatmanyofthebitsinerrorarecovered.Forexample,with10%ofMRBexchangeandmorethaneightreceivers,itislikelythatinformationwillbeexchangedforalmostallthebitsinablockof900bits,andhencetheperformanceofMRB-2isbetterthanLRB-1foralargenumberofreceivers.TheschemesthatworkwiththeMRBsalsorequireless-complexchannelaccesstech-niques.Ifthenumberofnodesarexed,asimpleroundrobinofallthenodescanbeusedtoallowthemtobroadcastreliabilitiesofacertainpercentageoftheirMRBs.Fordynam-icallyformedadhocnetworks,aclusterheadcouldbechosenthatassignstheorderinwhichthenodesbroadcastthereliabilities.WhencoupledwiththefactthattheoverheadofLRBschemescanbecomeprohibitiveforlargecooperatinggroups,thesimplicityofMRB-2anditsperformanceinreasonablybigcooperatinggroupsmakesitmoresuitedforpracticalimplementation. 4.2 GuidelinesfortheDesignofCollaborativeDecodingSchemesNotethatthereliabilityexchangeschemesdescribedearliercanbeconsideredtolieintherealmofthedecode-and-forwardschemeswiththerelaystransmissionsconsistingofthesoftinformation.However,unlikethedecode-and-forwardschemesmentionedearlier,collaborativedecodingdoesnotdependoncorrectdecodingatthecooperatingnodes.TheSISOdecodersinourschemesusebit-by-bitMAPdecodingliketheBCJR[ 37 ]algorithm,andhencecorrectdecodingisnotneededtoextractusefulinformationforasmallsubsetofbits.Thus,ourreliabilityexchangeschemesareanimprovementoverthedecode-and-forwardschemessincetheyarenotlimitedbythecapacitybetweenthesourceandthecooperatingnodes.AlltheresultsshowninthischaptercorrespondtotransmissionoverAWGNchan-nels.ThereisnodiversityinitstruesenseonAWGNchannelssinceallthechannelsareequivalent.Thus,inordertostudythediversitybenetsofcollaborativedecodingusing

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48 Figure4-8:PerformanceoftheMRB-2schemewitheightnodesonablock-fadingchannel. reliabilityexchange,weneedtostudytheperformanceofthereliabilityexchangeschemesoverfadingchannels.TheprobabilityofpacketerrorforcollaborationusingMRB-2witheightreceiversexperiencingblock-fadingisshowninFigure 4-8 .Itisseenfromthesimu-lationresultsthattheMRB-2schemedoesnotachievefulldiversitythecurvesforMRCandMRB-2arenotparallel.ItisobservedthattheperformanceofMRB-2schemewitheightreceiversisover5dBworsewhencomparedtoMRC.Forthesakeofcomparison,theperformanceofaschemethatexchangessoft-informationforalltheinformationbitsisalsoshown.ThisisthebestperformancethatanyoftheMRBschemescanachieve.Theperformanceofthisschemeisaround3dBworsethanMRC.Thus,thebestMRBschemerequiresmorethantwicetheSNRtoachievethesameperformanceasMRC.ThereasonforthepoorperformanceoftheMRBschemeisasfollows.Ifallnodesexperienceseverefading,thenitisdifculttoextractusefulsoftinformationforuseintheMRBscheme.Thereasonfornotachievingfulldiversitycanbeattributedtothefactthatreliabilityex-changefallsundertherealmofthedecode-and-forwardschemes.Laneman[ 3 ]provedthat

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49 thedecode-and-forwardschemesarenotcapableofprovidingallthediversityadvantagesassociatedwithcooperativeschemes.However,amplify-and-forwardschemesareguaranteedtoachievefulldiversityad-vantages[ 3 ].Thus,itseemsnecessarytoexchangeinformationforthecodedbitsasintheamplify-and-forwardschemesinformationisexchangedfortheinformationbitsinre-liabilityexchangeinordertoachievefulldiversity.MRC,whichisaninstanceofthiscategory,alsoexchangesreceivedsymbolvalues.However,MRCcombinesinformationinaninefcientmannerwithrespecttooverhead.Becauseoftheuseoferrorcorrectioncodes,therearecertainbitstrellissectionsaboutwhichreliabledecisionscanbemadewithouttheexchangeofinformation.TheLRBschemesmakeuseofthisobservationtorequestforinformationforonlythosetrellissectionsthatarelikelytobeinerror.ButtheLRBschemesrequestforthesameamountofinformationforallthetrellissections.Whentheindexofatrellissectionistransmittedbythenoderequestinginformation,alltheothernodesinthecooperatingclusterwillrespond.Thus,alltrellissectionsinthesetofLRBsreceivethesameamountofinformation.However,itisnotclearifatrellissectionthatdecodesincorrectlywithahighreliabilityrequiresthesameamountofinformationinor-dertocorrectthedecisionasatrellissectionthatdecodesincorrectlywithalowreliability.Ideally,theamountofinformationrequestedshouldbeadaptedtothereliability.Infadingchannels,somenodeshavebetterchannelstotheoriginaltransmitterandhencehavemadeagreaternumberofcorrectbitdecisions.Suchnodesshouldsharemoreinformationwithothernodeswhencomparedtotherelayswithbadchannels.AllthenodestransmitanequalamountofinformationifMRCoroneoftheLRB/MRBschemesisused.Theseobservationsleadtothreeprinciplesthatshouldbekeptinmindwhiledesign-ingcooperativeprotocols: P1. Inordertoobtainfulldiversityadvantages,itisnecessarytoexchangeinformationclosesttotheRFfrontend,i.e.,thereceivedsymbolvaluessoftdemodulatorout-puts.

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50 P2. Theinformationexchangedinthecooperatingclustershouldbeadaptedtotheerrorsineachpacket.Ifreliabilitiesareusedtoadaptthecollaborationcontent,theamountofinformationrequestedforeachtrellissectionshouldbebasedonthereliability. P3. Nodeswithgoodchannelsshouldsharemoreinformationthannodeswithbadchan-nels.NotethatMRCandthereliabilityexchangeschemesdescribedinthischapterviolatealloftheseprinciples.InChapter 6 ,wepresentanimproved-LRBI-LRBschemethatisbasedontheseprinciples.ItwillbeshownthatI-LRBachievesfulldiversitysameasMRCinthenumberofcooperatingnodes.TheperformanceofI-LRBisbetterthancomparableamplify-and-forwardbasedcollaborativeapproacheswhilestillachievedper-formanceclosetothatofMRCwithafractionofthecollaborationoverhead.TheI-LRBschemeexploitsthecorrelatedreliabilitiesattheoutputofaSISOdecoderinordertore-ducetheoverhead.Inthenextchapter,weinvestigatetheunderlyingdecodermechanismthatleadstocorrelatedbitreliabilities.ThisunderstandingwillproveusefulinthedesignoftheI-LRBscheme.

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CHAPTER5ONCORRELATEDBITERRORSATTHEOUTPUTOFAMAX-LOG-MAPDECODERWebeginthischapterbyrstmotivatingtheneedtounderstandthecorrelatedna-tureofreliabilities.WewilluseLRBasanexampletoshowhowcorrelatedreliabilitiescanhelpindecreasingtheoverheadassociatedwithcollaborativedecoding.IntheLRBschemes,theoutputoftheSISOdecoderisusedtoidentifythebitswithlowreliabilities,andinformationisrequestedforsuchbitsbecausethesebitsaremorelikelytobeinerror.However,asnotedin[ 33 ],errorsattheoutputofadecoderaretypicallytime-correlated.SincetheLRBsaremorelikelytobeinerror,theLRBsarealsocorrelatedi.e.,ifabithasdecodedwithalowreliabilityitislikelythattheadjacentbitsalsohavealowreliability.ThiscanalsobeobservedinFigure 4-6 whereitisseenthatineachiteration,informationisrequestedforsetsofconsecutivebits.Thus,inLRBanodewillrequestforinformationaboutasetofconsecutivetrellissectionswithlowreliabilities.Thisisaconservativeap-proachbecausecorrectingoneLRBwillhaveaneffectontheneighboringLRBsduetothecorrelatednatureoftheoutputofthedecoder.Inotherwords,iftwobitsarestronglycorrelated,itislikelythatcombininginformationforonebitwillinuencethedecisionattheotherbit.Thus,itisnotnecessarytorequestforadditionalinformationfortheentiresetofconsecutiveLRBs.Inordertodecreasethecooperationoverheadassociatedwithcollaborativedecoding,itisnecessarytounderstandtheinteractionbetweendecodedbitreliabilities.Inthischapter,weshowthattheerroreventthatseparatestheMLcodewordandacompetingcodewordinthemax-log-MAPdecodercansuccinctlycapturethecorre-latednatureofbiterrors.WealsoshowhowthiserroreventcanbeefcientlycomputedusingcomputationsthatarealreadyperformedintheBCJRmax-log-MAPdecoder.Inthe 51

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52 nextchapter,weusethiserroreventthatseparatestheMLandcompetingpathtodesignacollaborativedecodingschemethatimprovesontheperformanceofLRB. 5.1 TerminologyandNotationTheterminologyandnotationintroducedherearespecictorate1=2convolutionalcodes.Itisstraight-forwardtogeneralizethesetoratek=ncodes. inputandoutputlabels:Aninputlabelisusedtoindicatetheinputthatcausesaparticularstatetransitioninthecode-trellis,andanoutputlabelisusedtoindicatethecorrespondingoutputcausedbythatstatetransition. pathandevent:Asequenceofvalidstatetransitionsinthetrellisiscalledapaththroughthetrellis.Notethateverycodewordrepresentsapaththroughthetrel-lis.Becausethecodeislinear,thedifferencebetweenanytwocodewordsisapaththroughthetrellis.Suchapathisalsocalledanevent. validstate:Avalidstateliesonanypaththroughthetrellis.Becausethetrellisstartsandstopsintheall-zerosstate,noteverystateisavalidstateneartheendsofthetrellis. metric:TheEuclideandistancebetweenthereceivedvectorrandanycodewordc,kr)]TJ/F54 11.955 Tf 11.173 0 Td[(ck2,isreferredtoasthemetric 2 .Notethatthemetricisamaximum-likelihoodMLdecisionstatisticforadditivewhiteGaussiannoiseAWGNchannels.ThenotationusedinthisChapterisgiveninTable 5-1 5.2 RevisitingMax-log-MAPDecodingofConvolutionalCodesThesoft-outputofamax-log-MAPdecoderforcodewordstransmittedonanAWGNchannelwithnoisevariance2canbewrittenasseeChapter 3 ,eq. 3.3 Luijr=minc2Ci+kr)]TJ/F54 11.955 Tf 11.955 0 Td[(ck2 22)]TJ/F15 11.955 Tf 13.338 0 Td[(minc2Ci)]TJ/F78 11.955 Tf 8.247 28.063 Td[(kr)]TJ/F54 11.955 Tf 11.955 0 Td[(ck2 22:.1 2Notethatametricisassociatedwithaparticularcodeword.Inotherwords,eachcodewordhasadifferentmetric.

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53 Table5-1:Notationusedinthischapter N Block-sizethenumberofsectionsinthecode-trellis. ui Theinputtotheencoderattimei;i.e.,theinputlabelfortrellissectioni.Forbinarycodesconsideredinthispaperui2f0;1g.Wewillrefertouiastheinformationbit. c i=[c0i;c1i] Theoutputoftheencoderattimei.Thisisatwo-dimensionalvectorcon-sistingofthetwoparitybitsoutputbytheencoderateachtime.IfBPSKisusedformodulationthen c ji2f)]TJ/F15 11.955 Tf 27.438 0 Td[(1;1g;8j2f0;1g.Sinceevery c icorre-spondstoaparticularbranchinthetrellis, c iwillusedastheoutputlabelsforthebranchesinthetrellisattimei.Wewilluseparitybitsorcodedbitstorefertotheoutputlabelsatanyparticulartimeinthetrellis. c=[ c 1;:::; c N] Avalidcodewordoutputlabelsonapaththroughthetrellis.Appropriatesubscriptswillbeusedtoindicatethecodewordbeingconsidered. r i=[r0i;r1i] Thereceivedvectorcorrespondingto c i. r Thereceivedvectorcorrespondingtoc. cba [ c a; c a+1;:::; c b)]TJ/F24 7.97 Tf 6.587 0 Td[(1; c b].rbaissimilarlydened. ulc Inputlabelattrellissectionlincodewordc. cl Componentlincodewordc.Notethatthisreferstoaparticularbitinthecorrespondingoutputlabel. Ci+ fc:uic=0gi.e.,thesetofallcodewordswithinputlabel0attrellissectioni. Ci)]TJET1 0 0 1 52.44 -18.78 cmq[]0 d0 J0.398 w0.199 0 m0.199 28.892 lSQ1 0 0 1 6.177 18.78 cmBT/F51 11.955 Tf 0 0 Td[(fc:uic=1gi.e.,thesetofallcodewordswithinputlabel1attrellissectioni. C Thesetofallvalidcodewords.C=Ci)]TJ/F78 11.955 Tf 9.221 11.922 Td[(SCi+. S Setofstatesinthetrellis.Forthememory-twocodeconsideredinthispaper,therearefourstates. Therefore,S=f0;1;2;3g. sk Stateoftheencoderattimek.Notesk2S. S!s Thesetofvalidstatesattimek)]TJ/F15 11.955 Tf 12.219 0 Td[(1thathavebranchesleadingintostatesattimek. Ss! Thesetofvalidstatesattimek+1thathavebranchesemergingfromstatesattimek. skc Thestatethatcodewordcpasses 1 throughattimek. is logPsi=s;ri1 is0;s logPsi=s; r ijsi)]TJ/F24 7.97 Tf 6.587 0 Td[(1=s0 is logPrNi+1jsi=s N;2 representsaGaussiandistributionwithmeanandvariance2.

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54 Notethatthemaximum-likelihoodMLcodeword/pathcMLisacodewordthatisclosesttothereceivedvectorr,cML=argminc2Ckr)]TJ/F54 11.955 Tf 11.955 0 Td[(ck2:ItispossiblethatthereismorethanoneMLcodewordalthoughthisoccurswithproba-bilityzerofortheunquantizedAWGNchannel,inwhichcasewearbitrarilychooseoneofthepathsastheMLcodeword.Denition1.Competingcodeword/pathcicomp:ThecompetingpathattrellissectioniisthepaththatisclosesttothereceivedvectoramongallpathsthatdifferfromtheMLpathintheinputlabelfortrellissectioni,cicomp=argminfc2C:uic6=uicMLgkr)]TJ/F54 11.955 Tf 11.955 0 Td[(ck2:.2AsinthecaseoftheMLcodeword,theremaybemorethanonecodewordthatsatis-es 5.2 ,inwhichcasethetieisbrokenbychoosingoneofthecodewordsarbitrarily.NotethatalthoughthereisonlyonecML,theremaybemanydifferentcicompfordifferentvaluesofi.Thenthereliabilityforbiti,whichisthemagnitudeofthesoftinformationin 5.1 ,canbeexpressedasi,jLuijrj=1 22kr)]TJ/F54 11.955 Tf 11.955 0 Td[(cicompk2)-222(kr)]TJ/F54 11.955 Tf 11.955 0 Td[(cMLk2:.3NotethatcicompisreferredtoascjiinChapter 3 .Wehavereplacedthenotationtosimplifyexposition,andtostressthefactthatcicompiscompetingwithcMLforthehard-decisionontrellissectioni.SincethedistancebetweenrandtheMLcodewordissmallerthanthedistancebetweenrandanyothercodeword,thedifferencein 5.3 isalwayspositive.AhighvalueofreliabilityimpliesthattheMLpathandthenextbestpathwiththeoppositeinputlabelforbitiarefarapart,andhencethereisalowerprobabilitythatthedecoderchosethewrongpathandmadeabiterror.Thus,reliabilityisameasureofthecorrectness

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55 ofthebitdecision.Thishasalsobeenshownviasimulationresultsin[ 32 33 ].Abitwithhighreliabilityismorelikelytohavedecodedcorrectlythanabitwithlowreliability.TheI-LRBschemethatisdescribedinSection 6.4 utilizesboththebitreliabilitiesandknowledgeofcMLandcicompindeterminingwhichinformationshouldbeexchangedinthecollaborativedecodingprocess.Inthenextsection,wedetailhowcMLandcicompcanbedeterminedforaparticulartrellissection. 5.2.1 ObtainingtheMLandCompetingPathusingtheBCJRAlgorithmFollowingthedevelopmentin[ 42 ],thesoftinformationin 5.1 canbeexpressedasLuijr=maxCi+i)]TJ/F24 7.97 Tf 6.587 0 Td[(1s0+is0;s+is)]TJ/F15 11.955 Tf 10.352 0 Td[(maxCi)]TJ/F78 11.955 Tf 12.994 28.064 Td[(i)]TJ/F24 7.97 Tf 6.586 0 Td[(1s0+is0;s+is;.4whereks,ks0;s,andksaredenedinTable 5-1 .Itcanalsobeshownthatsee[ 42 ]is=maxs02S!si)]TJ/F24 7.97 Tf 6.587 0 Td[(1s0+is0;s .5 i)]TJ/F24 7.97 Tf 6.586 0 Td[(1s=maxs02Ss!is0+is;s0 .6 is0;s/k r i)]TJET1 0 0 1 17.993 2.89 cm0 g 0 GBT/F77 11.955 Tf 0 0 Td[(c ik2; .7 wheres02S!sands02Ss!aredenedinTable 5-1 ,0=0andN=0.Thus,itisseenfrom 5.7 thatis0;sisproportionaltothebranchmetriccf.[ 43 ],P r ij c i,usedintheViterbialgorithmwheretheconstantofproportionalitydependsononlythechannelcoefcientandsignal-to-noiseratio.Lettheorderedpairofstatessi)]TJ/F24 7.97 Tf 6.587 0 Td[(1;sithatmaximizesthersttermin 5.4 bes+i)]TJ/F24 7.97 Tf 6.587 0 Td[(1;s+i.Lets)]TJ/F25 7.97 Tf 0 -8.012 Td[(i)]TJ/F24 7.97 Tf 6.587 0 Td[(1;s)]TJ/F25 7.97 Tf 0 -8.012 Td[(ibetheorderedpairofstatesthatmaximizesthesecondterm.Bycompar-ing 5.1 and 5.4 ,itisseenthatoneoftheorderedpairsofstatess+i)]TJ/F24 7.97 Tf 6.587 0 Td[(1;s+iors)]TJ/F25 7.97 Tf 0 -8.012 Td[(i)]TJ/F24 7.97 Tf 6.586 0 Td[(1;s)]TJ/F25 7.97 Tf 0 -8.012 Td[(icorrespondstocML,whiletheotherorderedpaircorrespondstocicomp.Forexample,ifmaxCi+i)]TJ/F24 7.97 Tf 6.587 0 Td[(1s0+is0;s+is>maxCi)]TJ/F78 11.955 Tf 12.995 28.064 Td[(i)]TJ/F24 7.97 Tf 6.586 0 Td[(1s0+is0;s+is;

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56 thensi)]TJ/F24 7.97 Tf 6.586 0 Td[(1cML=s+i)]TJ/F24 7.97 Tf 6.587 0 Td[(1,sicML=s+i,andsi)]TJ/F24 7.97 Tf 6.586 0 Td[(1cicomp=s)]TJ/F25 7.97 Tf 0 -8.012 Td[(i)]TJ/F24 7.97 Tf 6.587 0 Td[(1,sicicomp=s)]TJ/F25 7.97 Tf 0 -8.012 Td[(i.Thus,whencomputingsoft-outputfortrellissectioni,itispossibletoidentifythebranchesthroughthetrellisattimeithatcorrespondtotheMLpathandthecompetingpath.WenowintroducetwotheoremsthatwillenableustoobtaincMLandcicompinastraight-forwardmannerusingthecomputationsperformedbythedecoder.Theorem1:ThebranchselectiontheoremGiventhestateinthecodetrellisattimek,sk=s0andthevectorofreceivedsymbolsr,thefollowingstatementsaretrue:aTrace-back:Thestate-transitions!s0,wheresk)]TJ/F24 7.97 Tf 6.587 0 Td[(1=s=argmaxs2S!s0fk)]TJ/F24 7.97 Tf 6.586 0 Td[(1s+ks;s0g,isabranchonthecodewordcgivenby,c=argminfc2C:skc=s0gkrk1)]TJ/F54 11.955 Tf 11.955 0 Td[(ck1k2.bTrace-forward:Thestate-transitions0!s,wheresk+1=s=argmaxs2Ss0!fk+1s0;s+k+1sg,isabranchonthecodewordcgivenbyc=argminfc2C:skc=s0gkrNk+1)]TJ/F54 11.955 Tf 11.955 0 Td[(cNk+1k2.Proof:ToprovetheTrace-backprocedureinTheorem1,werstprovethefollowingLemma.Lemma:ks/minc2C:skc=skrk1)]TJ/F54 11.955 Tf 12.523 0 Td[(ck1k2foranystatesattimekthatisonthepathofavalidcodewordProof:Bymathematicalinduction.Notethat0=0.Then1iscomputedusing 5.5 as1=0+1;0.8becausethereisonlyonevalidstateleadingintostate0attime1.Similarly,1=0+1;2.Thelemmadoesnotapplytotheotherstatesattime1becausetheyarenotvalidstatesforarate1/2convolutionalcodeinitializedtostate0attime0.Sousing 5.7 ,thelemmaholdsfork=1.

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57 Assumethatthelemmaholdsfortimek)]TJ/F15 11.955 Tf 11.955 0 Td[(1.Thenks=maxs2S!sk)]TJ/F24 7.97 Tf 6.587 0 Td[(1s+ks;s .9 /maxs2S!smaxc2C:sk)]TJ/F18 5.978 Tf 5.757 0 Td[(1c=s)-222(krn1)]TJ/F54 11.955 Tf 11.955 0 Td[(ck)]TJ/F24 7.97 Tf 6.586 0 Td[(11k2+ks;s .10 /minc2C:skc=skrk1)]TJ/F54 11.955 Tf 11.955 0 Td[(ck1k2; .11 where 5.10 followsfromtheassumptionabouttheclaim,andthelastequationfollowsfrom 5.7 .Thus,theclaimistruefortimek.Theprincipleofinductioncompletestheproof.Remark:Fromthelemma,ksisproportionaltothepartial-pathmetriclogPrk1jck1[ 43 ]ofthesurvivingpathatstatesattimekintheViterbialgorithmwhenthebranchmet-ricistheEuclideandistance.Proofofthetrace-backtheorem:Comparethetrace-backtheoremand 5.5 .Thetrace-backtheoremchoosesthepreviousstatesi)]TJ/F24 7.97 Tf 6.587 0 Td[(1thatcorrespondstothebranchinvolvedincomputingthealphaforthecurrentstatesi.Sinceksisproportionaltothepartialpathmetricofthesurvivingpathleadingtosk=s,thebranchinvolvedincomputingksispartofthecorrespondingsurvivingpath.Thus,conditionedonthecurrentstate,thetrace-backtheoremchoosesthepreviousstateasthestateattimek)]TJ/F15 11.955 Tf 12.003 0 Td[(1onthesurvivingpathattimek.Theproofofthetrace-backprocedurefollowsbecausethesurvivingpathhasthebestpartial-pathmetricminkrk1)]TJ/F54 11.955 Tf -422.083 -23.908 Td[(ck1k2amongallpathscthatpassthroughsk=s.Thetrace-forwardtheoremcanbeprovedinasimilarmannerbycomparingthetrace-forwardtheoremwith 5.6 .Theorem2:Theconditionalpathselectiontheorem.Givenastatetransitionattimei,i.e.,si)]TJ/F24 7.97 Tf 6.587 0 Td[(1=s0andsi=s,letCrepresentthesetofallpathsthroughthetrelliscodewordspassingthroughthistransitionattimei.Thatis,C=fc2C:si)]TJ/F24 7.97 Tf 6.586 0 Td[(1c=s0;sic=sg:Thenthesequenceofstatetransitions

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58 fs00;s01;:::;s0i)]TJ/F24 7.97 Tf 6.586 0 Td[(2;s0;s;si+1;:::;sNggivenbys0k)]TJ/F25 7.97 Tf 6.586 0 Td[(i=argmaxs2S!s0k)]TJ/F20 5.978 Tf 5.756 0 Td[(i+1fk)]TJ/F25 7.97 Tf 6.587 0 Td[(is+k)]TJ/F25 7.97 Tf 6.587 0 Td[(i+1s;s0k)]TJ/F25 7.97 Tf 6.586 0 Td[(i+1g;i=2;3;:::;k .12 sk+i=argmaxs2Ssk+i)]TJ/F18 5.978 Tf 5.756 0 Td[(1fk+is+k+isk+i)]TJ/F24 7.97 Tf 6.587 0 Td[(1;sg;i=1;2;:::;N)]TJ/F50 11.955 Tf 11.955 0 Td[(k .13 correspondstothecodewordcthatisclosesttothereceivedvectorramongallthecode-wordsinC,c=argminc2Ckr)]TJ/F54 11.955 Tf 11.955 0 Td[(ck2:Proof:Theprooffollowsbyrepeatedapplicationofthetrace-backandtrace-forwardtheorems.Asmentionedearlier,thestatetransitionsfromtimei)]TJ/F15 11.955 Tf 12.227 0 Td[(1toithatcorrespondtotheMLpathandthecompetingpathcanbeobtainedduringthecomputationofthesoft-outputforbiti.Giventhestatessi)]TJ/F24 7.97 Tf 6.586 0 Td[(1cML,andsicML,theMLcodewordcMLcanbeobtainedusingtheconditionalpathselectiontheorem.ThecodewordoutputbytheconditionalpathselectiontheoremisclosestinEuclideandistancetothereceivedvectoramongallpathsthatpassthroughsi)]TJ/F24 7.97 Tf 6.587 0 Td[(1cML,andsicML,andisthustheMLpath.Similarly,thecompetingpathcanbeobtainedusingtheconditionalpathselectiontheoremgivensi)]TJ/F24 7.97 Tf 6.587 0 Td[(1cicomp,andsicicomp.AsnotedintheLemma,thetrace-backtheoremalwayschoosesthepreviousstatesi)]TJ/F24 7.97 Tf 6.586 0 Td[(1thatcorrespondstothebranchinvolvedincomputingthealphaforthecurrentstatesi.Similarly,thetrace-forwardtheoremalwayschoosesthenextstatesi+1thatcor-respondstothebranchinvolvedincomputingthebetaforthecurrentstatesi.Thisob-servationenablesanefcientmodicationoftheBCJRalgorithmthatenablescomputingcMLandcicompforanytrellissectioni.Duringthecomputationoftheis;s2S;i2f1;:::;Ng,recordthestatesi)]TJ/F24 7.97 Tf 6.586 0 Td[(1=s0thatmaximizesi)]TJ/F24 7.97 Tf 6.586 0 Td[(1s0+is0;sasthepreviousstatefors.Similarly,duringthecomputationoftheis;s2S;i2f1;:::;Ng,recordthestatesi+1=s0thatmaximizesi+1s0+is;s0asthenextstatefors.Thengiventhebranchinthecode-trelliscorrespondingtocMLorcicompforattimei,theentirecodeword

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59 canbeobtainedbyrecursivelyfollowingthestatesrecordedinthisway.ThusnoadditionalcomputationsarerequiredtocomputecMLandcicomp.Byrecordinginformationaboutthestatesthatleadtothemaximumvaluesin 5.5 and 5.6 duringtheBCJRalgorithm,cMLandcicompcaneasilybeobtainedthroughaseriesoftable-lookups.Duringthetrace-backortrace-forwardprocedure,ifsi)]TJ/F25 7.97 Tf 6.587 0 Td[(kcML=si)]TJ/F25 7.97 Tf 6.586 0 Td[(kcicompforsomek,thenthesequenceofstate-transitionsobtainedforanytimebeforekwillbethesameforcMLandcicomp.Similarly,iforsi+kcML=si+kcicomp,thenthesequenceofstate-transitionswillbethesameforcMLandcicompforanytimeafterk.ItwillbeshowninChapter 6 thatI-LRBonlyrequiresknowledgeoftrellissectionswherecicompandcMLdiffer.Thus,itissufcienttoexecutethetrace-backandtrace-forwardproceduresuntilsikcML=sikcicomp.Itiswellknownthatthesoft-output/reliabilitiesofadjacentbitsinaconvolutionalcodearecorrelated[ 34 ].Formax-log-MAPdecoders,itwasfoundthroughsimulationsthatgroupsofneighboringbitshavethesamereliability.SincerandcMLarethesameforalltrellissections, 5.3 impliesthatbitsdecodingwiththesamereliabilityshouldhavethesamecompetingpath.Byusingthetechniquedescribedabove,andobservingthecompetingpathsforadjacentbitsthatdecodedwiththesamereliabilityitisveriedthatthecompetingpathsareindeedthesameforthosebits.Thus,formax-log-MAPdecoders,thestrongcorrelationbetweenthereliabilitiesofadjacentbitsisreectedinthechoiceofthesamecompetingpathinthecode-trellisforthosebits. 5.2.2 OntheUtilityofCompetingPathsintheDesignofCollaborativeDecodingInthissection,weprovideabriefoutlineofhowexplicitknowledgeofthesecompet-ingpathscanhelpreducecooperationoverhead.Itiswellknownthaterrorsattheoutputofaconvolutionalcodearebursty,andsimilarlythesoft-output/reliabilitiesaretempo-rallycorrelated[ 34 33 ].Itwasshownthatthereasonforthiscorrelationisthatbitsthatareclosetoeachotherinthetrellismayoftensharethesamecompetingcodeword/path.Formax-log-MAPdecoding,suchbitshaveexactlythesamereliability,ascanbeseenfrom 5.3

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60 AparticularbitdecodesincorrectlyiftheMLpathisnotthetransmittedcodeword.Thus,alltheneighboringbitsthatalsochoosethesameMLpathwillalsodecodeinerrorleadingtotime-correlatederrors.Thefactthaterrorsinconvolutionalcodesoccurinburstswasalsonotedin[ 33 ].ConsidertheuseoftheLRBscheme.ConsideraLRBsaybitithatdecodedinerrorwithcMLandcicompasthecompetingpaths.ThismeansthatbitichosethewrongpaththroughthetrellisastheMLpath.Inthiscase,itislikelythatthenextclosestpathwithrespecttothereceivedvectorwiththeoppositebitdecisioncorrespondstothetruetrans-mittedcodewordi.e.,itislikelythatcicompisthetruecodeword.Nowsupposethatreceivingadditionalinformationaboutbitifromothernodesisabletocorrectthedecision.Thisim-pliesthattheadditionalinformationchangedtheMLpathtobecicomp.Assumethattherewereadjacenttrellissectionsik,thathadoriginallydecodedwiththesamereliabilitysamechoiceofcMLandcicompbeforerequestingadditionalinformationforbiti.SinceadditionalinformationchangedthechoiceoftheMLpathtocicompforbiti,thenitislikelythatalltheadjacenttrellissectionsthatoriginallydecodedwiththesamereliabilityasbitiwillalsochoosecicompastheMLpath.Thiswillalsocorrecttheerrorsattheseadjacenttrellissections.Thus,bycombininginformationforonetrellissectionitispossibletocor-rectbitdecisionsatothertrellissectionsalso.TheI-LRBtechniqueintroducedinthenextchapterusesthisideatoreduceoverheadbynotrequestinginformationforallthetrellissectionsthatdecodewiththesamecompetingpath.I-LRBrequestsminimaladditionalinformationthatwillipthedecisionfromcMLtocicomp,therebycorrectingallbiterrorsassociatedwiththeincorrectchoiceofcicomp.

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CHAPTER6IMPROVEDLEAST-RELIABLE-BITSCOLLABORATIVEDECODINGFORBANDWIDTH-CONSTRAINEDSYSTEMSInthissectionwedescribetheImprovedLRBI-LRBcollaborativedecodingschemeforconvolutionallyencodedcommunications.WemodifytheLRBschemedescribedinChapter 4 tosatisfythedesignguidelinesmentionedinSection 4.2 .WeuseLRBasourbaselineschemebecausetheinformationexchangedintheLRBschemestargetsthetrellissectionsthataremostlikelytohavedecodedinerror.InadditiontoconformingwiththedesignguidelinesofSection 4.2 ,theI-LRBschemealsoexploitscorrelatedreliabilitiestoreducethecooperationoverhead.ThesystemmodelforcollaborativedecodingisshowninFigure 4-1 .Adistanttrans-mitterbroadcastsapackettoaclusterofreceivingnodes.ARQisnotpossiblebecauseofthepowerlimitationsofthemobilesandthedistancetothetransmitter.Cooperationoverheadiscriticalinbandwidth-constrainedsystems.Inthesesystems,thecooperationoverheadisupperboundedbyamaximumvalue.Thisconstraintmaybenecessaryinor-dertoprovideaminimumthroughputguaranteetothedistanttransmitter.Sincethereisnofeedbackchanneltothedistanttransmitter,itwillcontinuetotransmitmessagesatacertainrate.Insuchascenario,cooperationexchangeofmessagescannotcontinueindenitely.Ifthecollaborationproceedsforalongtime,thentheprocessofcollaborativedecodingwillinterferewithadditionaltransmissionsfromthesource.Thereforeitisnecessarytoconstrainthecooperationoverheadinordertoensurethatcollaborationdoesnotconictwithtransmissionsfromthesource.Werstbeginwithabroadperspectiveofcollaborativedecodinginbandwidth-constrainedenvironments.Afterintroducinganbaselineamplify-and-forwardschemeMRCvariantforsuchsystems,wedeveloptheI-LRBscheme. 61

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62 6.1 CollaborativeDecodingwithConstrainedOverheadsIncollaborativedecoding,themessageatthesourceispacketizedandencodedwithacodethatpermitsSISOdecoding.Thecodewordisthenbroadcasttoaclusterofreceivingnodesthatwillattempttodecodethemessage.Thereceivedmessageforsymboliatnodejcanbemodeledasri;j=ajxi+ni;j;.1wherexiisthetransmittedsymbolattimei;ajisthechannelcoefcientatreceivingnodej,whichweassumeisxedovereachpacket;andni;jiswhiteGaussiannoise.Inallthatfollows,weconsiderrateR=1=2codes,butitisstraight-forwardtogeneralizetheworktoothercoderates.Ifanynodeintheclusterdecodesthemessagecorrectly,thenweconsiderthemes-sagetobesuccessfullyreceived.Ifnoneofthenodesdecodesthepacketcorrectly,thenthenodesbegintheprocessofcooperatingtoreceivethemessage.InthecollaborativedecodingschemespresentedinChapter 4 ,thenodesusetheoutputsoftheSISOdecoderstoselectwhichinformationshouldbeexchangedandwhichnodesshouldtransmitthatinformation.TheaposterioriprobabilityAPPloglikelihoodratioLLRattheoutputofaSISOdecoderisarealnumberandiscommonlyreferredasthesoftoutput.Thesignandmagnitudeofthesoftoutputforaninformationbitrepresenttheharddecisionandthereliabilityofthatdecision,respectively[ 30 ].Thesamplemeanofthereliabilitiesatnodej,j,isameasurementoftheoverallreliabilityofthedecoder'sdecision.Weassumethatthenodesexchangethejsaftertherstdecoderiterationandthatcombiningoccursatthenodewiththelargestj,whichwerefertoasthebestnode.ThenodesthenbroadcastinformationaboutaselectedsetofthereceivedsymbolsasinA-Ftothebestnode.Thecooperativeprocesscangothroughseveraliterations,eachofwhichconsistsofthreeparts.Intherstpartoftheiteration,thenodesidentifyinformationtobeexchanged.Inthesecondpart,aselectedgroupofnodeswilltransmitthatinformationtothebestnode.Inthenalpartofeachiteration,thenodesdecodethe

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63 messageandcheckwhetherithasdecodedcorrectly.Theprocessstopsifanyofthenodeshasdecodedthemessagecorrectlyorifthelimitonthenumberofiterationsisreached.Ineachiteration,weconstrainthemaximumnumberofbitsthatcanbetransmittedinthecooperativeprocess.Thismaybenecessaryinmanysystemstoensurethatthecooper-ativeprocessdoesnotconictwiththetransmissionofadditionalpacketsfromthesource.WespecifytheconstraintasaportionofthetotalinformationexchangedinmaximalratiocombiningMRC.LetNbetheinformationblocksize,Rbethecoderate,Nrxbethenumberofreceivers,andqbethenumberofbitsusedtoquantizethechannelobservations.ThenthecooperationoverheadforMRCisMRC=NqNrx=Rbits.ThelargeMRCwillbenotacceptableformanyapplications.Hence,weconstraintheamountofinformationthatcanexchangedinthecooperatingclustertobeafractionpofmrc.Notethatthisplacesalimitonthemaximumamountofinformationexchangeinthecooperativeprocessforaparticularpacket;however,theactualamountofinformationexchangeforanyparticularpacketmaybemuchlessbecauseweallowthecooperativeprocesstoterminatewheneverthepacketisdecodedcorrectly.Ineachiteration,weconstraintheoverheadtopmrc=Niter,whereNiteristhetotalnumberofiterationsallowed.NotethattherearethreemaindifferencesbetweenthecollaborativedecodingschemedescribedinChapter 4 andthecollaborativedecodingschemedescribedaboveforbandwidth-constrainedsystems. InthecollaborativedecodingschemeofChapter 4 ,combiningisperformedatallnodes.Forexample,allnodesintheLRBschemerequestforadditionalinforma-tionabouttheirLRBs.Incollaborativedecodingforbandwidth-constrainedsystems,combiningisperformedonlyatthebestnode.ThissolvestheproblemofoverheadbeingproportionaltothesizeofthecooperatingclusterintheLRBschemes. IncollaborativedecodingschemeofChapter 4 ,informationfromallothernodesiscombinedineachiteration.Forexample,allnodesbroadcasttheAPPsfortheLRBs

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64 requestedbyanodeintheLRBscheme.Incollaborativedecodingforbandwidth-constrainedsystems,informationiscarefullychosenfromaselectsubsetofnodes.ThisiskeepinginaccordancewithdesignprincipleP3giveninSection 4.2 UnlikecollaborativedecodingofChapter 4 ,theamountofinformationcombinedineachiterationisconstrainedincollaborativedecodingforbandwidth-limitedsys-tems.Wenextdescribethetwomaincooperativeschemesthatwillbecomparedinthischapter.Therst,whichwecallconstrained-overheadincrementalMRCCOI-MRC,isaniterativeformofmaximal-ratiocombininginwhichtheoverheadisconstrainedasex-plainedabove.Thesecondschemeisacollaborativedecodingschemecalledtheimprovedleast-reliablebitsI-LRBscheme.Becauseofthecomplexityofthisscheme,werstprovideanoverviewofitinSection 6.1.2 ,andcontrastitwiththeLRBschemedescribedinSection 4.1.1 .WethendevelopthetoolsrequiredforI-LRB,andprovideadetaileddescriptionofI-LRBinSection 6.4 6.1.1 Constrained-overheadIncrementalMRCConsiderrstanimplementationoffullMRCinagroupofcollaboratingradios.Eachnodeotherthanthebestnodescalesitsreceivedsymbolsbythefadinggain,quantizesthem,andtransmitsthemtothebestnode.Asmentionedabove,thiswouldresultinalargeoverhead.AvariantofthisschemethatcanofferevenbetterperformancethanMRCwithloweroverheadisincrementalMRCI-MRC.InincrementalMRC,thecooperationisdoneoverseveraliterations. 1 Initerationi,thenodewiththei+1thlargestitrans-mitsinformationaboutallofitsreceivedsymbolstothebestnode 2 .Thenthebestnode 1Wethankananonymousreviewerofapreviouspaperforproposingthiscooperativescheme.2Notethatforquasi-staticfadingchannelsthevalueofiisgenerallydominatedbythefadingcoefcient.Iftwonodeshavesimilarfadingcoefcients,thisapproachallowsustochoosetheonewhosereceivedinformationprovidesmorecondenceindecoding.

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65 combinesthatinformationwithitsownreceivedsymbolsandanypreviouslyreceivedin-formation,decodesthemessage,andcheckswhetherthemessagehasdecodedcorrectly.Ifthemessagedecodescorrectly,thecooperativeprocedureterminates,andthustheaver-ageoverheadofI-MRCistypicallymuchlessthanMRC.Inaddition,becausedecodingisperformedaftereachinformationexchange,I-MRCcanachieveaslightlylowererrorprobabilitythanMRC.AlthoughI-MRChasaloweraverageoverheadthanMRC,theoverheadineachiter-ationconsistsofallofthereceivedsymbolsfromonenode,andthemaximumoverheadisthesameasMRC.Asexplainedabove,itmaybenecessarytoconstrainthemaximumover-head.Thus,weintroduceaconstrained-overheadI-MRCCOI-MRCscheme.InCOI-MRC,theoverheadisconstrainedtopNqNrx=Rbits.WeallowatotalofNiter=Nrx)]TJ/F15 11.955 Tf 11.716 0 Td[(1iterations,soineachiteration,pNqNrx=RNiterbitsareexchanged.TheinformationineachiterationrepresentsasetofpNNrx=RNiterreceivedsymbolsfromthebestnodethathasnotpreviouslytransmittedallofitsreceivedsymbols.Thesetofsymbolsisuniformlyselectedfromtheremainingsetofsymbolsatthatnode.Onceallofthesymbolsatanodehavebeentransmitted,thenthenextbestnodeintermsofiwilltransmitinformationforitsreceivedsymbols.Aftereachroundofinformationexchange,thebestnodeusesMRCtocombinethenewinformationwithitspreviouslyreceivedinformation.Thebestnodethendecodesthemessage.Ifthemessagedecodescorrectlyorifthemaximumnumberofiterationshasbeenreached,collaborationends.Otherwise,anotheriterationofinformationexchangeisperformed. 6.1.2 OverviewofImprovedLeast-ReliableBitsCollaborativeDecodingTheMRC-basedschemesareeffectiveapproachesforcooperation.However,theseschemesaredumbschemesinthesensethattheydonotutilizeinformationthatisavail-ablethatcouldimprovetheperformanceforthesameconstraintonthecollaborativeover-head.SISOdecodersoffertheabilitytoassesswhichbitdecisionsarereliableandwhich

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66 areunreliable.Byrstexchanginginformationthatcanimprovetheunreliablebitdeci-sions,wemaybeabletoachieveabettertradeoffbetweenoverheadandperformance.Theschemethatweproposeisbasedontheleast-reliablebitsLRBschemesthatweredescribedinSection 4.1.1 .IntheseLRBschemes,eachnodeidentiesthesetofbitswiththeleastreliabilitiesi.e.,smallestmagnitudeoftheAPPLLRandrequestsinformationforthesebitsfromeveryothernode.OurtechniqueimprovesonthepriorLRBschemesinseveralways: 1. Werequestinformationatonlythebestnode,sothattheoverheadfromtheinforma-tionrequestsisreduced. 2. WeutilizethefactthatthesetofLRBsisoftencorrelated,andwedeveloptechniquestoavoidrequestingtoomuchinformationbecauseofthiscorrelation. 3. ThesetofnodesthatrespondtoarequestsentbythebestnodestransmitquantizedvaluesoftheirreceivedsymbolsandnottheAPPsasintheLRBscheme.ThissatisesP1ofthedesignguidelinesinSection 4.2 4. Theamountofinformationrequiredtocorrectabitdependsonitsreliability,sowepresentatechniquetoadapttheamountofinformationbasedonabit'sreliability.ThissatisesP2ofthedesignguidelinesinSection 4.2 5. Allnodesdonotrespondtoarequestsentbythebestnode.Whenthebestnoderequestforadditionalinformationaboutatrellissection,onlythenextbestnodethathasnotalreadytransmittedinformationaboutthattrellissectiontransmitsitsreceivedsymbols.Sincecodedsymbolsarecombinedstartingwiththesecondbestnode,itislikelythatthesecondbestnodewilltransmitmorecodedsymbolsthananothernode.ThissatisesP3ofthedesignguidelinesinSection 4.2 6. Notallbitsthatsurroundanunreliablebitwillnecessarilyhelptocorrectthatbit,sowepresentatechniquetoselectthesetofbitswhicharemostlikelytocorrectanunreliablebit.

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67 7. LRBattemptstoreducecooperationoverheadbytargetingindividualtrellissectionsthatdecodedincorrectly.I-LRBreducescooperationoverheadbytargetingcompet-ingpathsthroughthetrellisthatpotentiallydecodingincorrectly.Bytargetingthecompetingpaths,I-LRBhasthepotentialtocorrectallthebiterrorsassociatedwiththispath.WerefertothenewapproachastheimprovedLRBI-LRBscheme.Inthispaper,wedemonstratehowthegoalsoftheI-LRBschemecanbeachievedforconvolutionallyen-codedcommunicationsbyutilizinginformationgeneratedinthemax-log-MAPimplemen-tationoftheBCJRdecodingalgorithm.ThedetailsofI-LRBwithconvolutionalcodesaregiveninSection 6.4 .RecallthatinI-LRB,thebestreceiversortsthetrellissectionsaccordingtothereliabil-ities,andrequestsinformationfromtheothercollaboratingnodestoimprovethedecodingofsomesetofleastreliablebits.TheLRBswilloftenoccuringroupsbecausetheyarecausedbythesameerrorevent,andthusitisonlynecessarytoprovideenoughinforma-tiontocorrecttheerroreventtocorrectallofthebiterrorscausedbythatevent.Moreover,weshowthatsomeofthereceivedsymbolscorrespondingtoaLRBmaynotbeusefulinresolvingthemostlikelyerrorevent.Intherestofthischapter,werstproposeasim-pleanalyticaltechniquethatcanbeusedtodeterminehowmuchinformationneedstobetransmittedforeachleastreliablebit.Wethendescribehowthedecodercanuseinforma-tionabouttheMLandcompetingpathstodecidewhichinformationcanmostefcientlycorrectanybiterrorsintheLRBs.Finally,weprovideadetaileddescriptionoftheI-LRBschemeforconvolutionallyencodedcommunications. 6.2 EstimationofRequestSizeDuringthecollaborativedecodingprocess,thedecodermustactundertheassumptionthatanyLRBisinerror,wheninfacttheerrorprobabilityforeventheleastreliablebitisgenerallylessthan0.5otherwise,wewouldjustinvertthatbitdecision.GiventhereliabilityofaLRB,thedecoderneedstoestimatetheamountofinformationthatshould

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68 berequestedtocorrectthebit.ThemostlikelyerroreventforbitiistheeventthatseparatescMLandcicomp,whichisgivenbyei=cMLcicomp;whererepresentstheXORadditionorsubtractioninabinaryeldoperator.Forlinearconvolutionalcodes,asconsideredinthispaper,eiisacodeword.Thereliabilityin 5.3 canbefurthersimpliedasi=1 2rTcML)]TJ/F54 11.955 Tf 11.955 0 Td[(cicomp:.2IfthechannelfromthedistanttransmittertothecollaboratingclusterinFigure 4-1 doesnothaveunitchannelgains,thenthereliabilityatthejthreceivercanbeexpressedasi;j=1 2ajrTcML)]TJ/F54 11.955 Tf 11.955 0 Td[(cicomp;.3wherewehavesuppressedthedependenceofcMLandcicompontheparticularreceivernum-ber,j.Thedecodertriestoestimatetheamountofinformationrequiredtochangethedeci-sionfromtheMLpathtothecompetingpathassumingthatthiswillcorrecttheerror.LetcMLkandcicompkdenotethekthparitybitontheMLpathandcompetingpathforinfor-mationbiti,respectively.IfcMLk=cicompk,thenthatparitybitdoesnotprovideanydistinctionbetweenthetwopathsinthetrellis.Thus,requestinginformationaboutsuchparitybitsfromtheothercollaboratingnodeswillnotbehelpfulinresolvingbetweenthesetwopaths.Inthemostlikelycase,inwhicheithercMLorcicompisthecorrectcodeword,thedecoderwillonlyimproveitsdecisionifadditionalinformationisreceivedforthoseparitybitsforwhichthedecisionsofMLandcompetingcodewordaredifferent.Denition2.CandidatesetofparitybitsSifortrellissectioni:ThesetofparitybitsforwhichthedecisionsoftheMLcodewordcMLandcompetingcodewordcicomparedifferent.

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69 Si=fk:cMLk6=cicompkg=fk:eik=1g.4Oncethecandidatesetofparitybitsisobtained,thedecodertriestoestimatethenumberofparitybitsfromthecandidatesetSithathavetoberequestedfromothernodesinorderforthedecodertodecideinfavorofcicompinsteadofcML.Letrbethereceivedvectorafterrequestingcodedsymbolsfromanotherreceiver 3 ,sayreceiver2.ThedecoderestimatestheminimumnumberofadditionalcodedsymbolsthatwillchangethedecisionfromcMLtocicompwithprobabilitygreaterthansomethreshold.Thatis,afterreceivingtheadditionalinformation,wedesireahighprobabilitythatkr)]TJ/F54 11.955 Tf 11.955 0 Td[(cicompk2
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70 all-zerosCWhasbeentransmitted,inwhichcasecicompl=1andcMLl=)]TJ/F15 11.955 Tf 9.299 0 Td[(1,8l2Si.Sincetheall-zerosCWisthetruetransmittedcodeword,r0lNa2;2.Thus,Xi,Xl2Si;jj=2a2r0lcMLl)]TJ/F54 11.955 Tf 11.955 0 Td[(cicomplN)]TJ/F15 11.955 Tf 9.299 0 Td[(4a22;16a222:Thusthedecoderestimatesthataftertherst-retransmission,correctdecodingismadeifXi<)]TJ/F15 11.955 Tf 9.299 0 Td[(22i.Thedecoderestimatesthenumberofcodedbitsforwhichinformationisrequiredfromanotherreceiverasfollows,minPXi<)]TJ/F15 11.955 Tf 9.298 0 Td[(22i .8 minQ2i)]TJ/F15 11.955 Tf 11.955 0 Td[(2a22 2p a222; .9 whereisthenumberofparitybitsretransmittedandisapredenedthreshold.Thus,thedecoderestimatesthenumberofbitstoberetransmittedastheminimumnumberthatwouldcausethedecodertodecideinfavorofcicompinsteadofcMLwithaprobabilitythatisatleast.Thisprovidestheminimumnumberofbitsthatismostlikelytocorrectbitiifitisinerror.PXi<)]TJ/F15 11.955 Tf 9.298 0 Td[(22iwillbereferredtoasthecorrectionprobabilityaftercombiningPc.Thus,thereceiverrequeststheminimumnumberofcodedbitssuchthatPcexceeds.Therefore,byrequestcodedsymbolsforparitybits,I-LRBhasthepotentialtocorrectallthebitsthatchosethesamecompetingpathi.e,decodedwiththesamereliability.Forexample,assume=1,andthatthereare3bitsthatdecodedwiththesamecompetingpath.Supposetheadditionalcopiesofthis1codedsymbolisenoughtoipthedecisionfromcMLtocicomp,thenthebitdecisionatthe3trellissectionsthatoriginallychosecicompasthecompetingpathwillalsochange.Thus,I-LRBhasthepotentialtocorrect3biterrorsbyrequestingonly1codedsymbol.NotethatLRBwouldhaverequestedfor3reliabilityvalues.

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71 Table6-1:InstantaneousSNRestimationfortrellissectionsbasedontheaverageoftheinstantaneousSNRsoftheparitybitsinthecandidateset Outputlabel Outputlabel Estimate ontrellissectioni lontrellissectioni oftheinstantaneousSNR forcML forcicomp fortrellissectioni 11 )]TJ/F15 11.955 Tf 9.298 0 Td[(1)]TJ/F15 11.955 Tf 11.955 0 Td[(1 jr0ij+jr1ij=2 )]TJ/F15 11.955 Tf 9.299 0 Td[(11 1)]TJ/F15 11.955 Tf 11.955 0 Td[(1 00 )]TJ/F15 11.955 Tf 9.299 0 Td[(11 11 jr0ij )]TJ/F15 11.955 Tf 9.299 0 Td[(11 )]TJ/F15 11.955 Tf 9.298 0 Td[(1)]TJ/F15 11.955 Tf 11.955 0 Td[(1 jr1ij 6.3 EstimationoftheRequestSetAfterthedecoderestimatesfromthecandidateset,itneedstoselectthesubsetofparitybitsinSiforwhichinformationwillberequestedfromanotherreceiver.WeestimateaninstantaneousSNRforeachtrellissectioninvolvedintheerroreventeithatseparatescMLandcicomptodecidethecandidatesetforcollaborativeexchange.Thereceiversortsthetrellissectionsintheerror-eventaccordingtotheinstantaneousSNRs,andrequestsforparitybitsfromthetrellissectionswithlowSNRs.TheconceptofinstantaneousSNRwasproposedin[ 44 ]foruseinselectingwhichsymbolsshouldberetransmittedinanARQscenario.Severaldifferentschemeswereconsideredin[ 44 ],andtheonedescribedherewasfoundtoofferthebestperformance.Ifforaparticulartrellissectioni, c MLand c icompdifferinonlyoneparitybit,thentheinstantaneousSNRofthatsectionisequaltotheabsolutevalueofthereceivedsymbolcorrespondingtothatparitybit.Ifforaparticulartrellissectioni, c MLand c icompdifferinbothparitybits,thentheinstantaneousSNRofthetrellissectionistheaverageoftheinstantaneousSNRsofthetwoparitybits.ThereceiverselectsparitybitscorrespondingtotrellissectionswiththelowestSNRsfromthecandidateset.TheinstantaneousSNRofaparticulartrellissectionfordifferentoutputlabelsoncMLandcicompisgiveninTable 6-1 .NotethatallpossibleoutputlabelscanbeobtainedbyinterchangingtheoutputlabelsontheMLandcompetingpathsineachrowofTable 6-1

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72 6.4 DetailedDescriptionofI-LRBCollaborativeDecodingWiththeaboveapproachestoestimatetherequestsizeandtherequestset,wecandescribeI-LRBcollaborativedecodingindetail.Uponinitiationofcollaboration,thenodesbroadcasttheirstodeterminethebestreceiver.Startingwiththebestreceiver,letthereceiversbenumberedRX1toRXNrx.ThesecondbestreceiverRX2,transmitsitsfadingcoefcienta2toRX1.RX1needsthefadingcoefcienttoestimatethenumberofcodedsymbolsthathavetoberequested.LetthenumberofiterationsincollaborativedecodingbedenotedbyNiter.Fortheresultspresentedinthispaper,wesetNiter=Nrx)]TJ/F15 11.955 Tf 12.729 0 Td[(1.Giventheoverheadconstraint,RX1limitsthenumberofbitsthatcanbeexchangedineachiterationtopMRC=Niter.Ineachiteration,RX1sortstheinformationbitsaccordingtothereliabilities,andobtainsthecompetingpathforeachLRBusingthetechniquedescribedinChapter 5.2.1 .ThenforeachLRB, 1. RX1estimatesusing 6.9 2. RX1obtainsthecandidatesetandthesetofparitiestoberequestedbasedontheinstantaneousSNRs. 3. RX1broadcasts,andtheindicesoftheparitybitsthatneedcodedsymbolsfromanothernode. 4. Foreachbitindex,thebestnodethathasnotpreviouslytransmittedinformationforthatbitwilltransmitinformationforthatbit.Eachnodescalesitreceivedsymbolsbythechannelcoefcientandbroadcaststhatinformationforabit.If>jSijthenumberofcodedsymbolsrequiredismorethanthesizeofthecandidateset,thencodedsymbolsareobtainedfromthenextbestreceiveruntilatotalofsymbolsaretransmitted.Consideranexampletoillustratetoillustratestep4above.AssumethatthecodewordshowninboldinFigure 6-1 isthecompetingpathforbitiandthattheMLpathistheall-zerospath.Assumethatthisistherstiterationinwhichbitsinthiscandidatesethavebeen

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73 Figure6-1:Thecode-trellisforthe;7convolutionalcodewithexamplesofthenotationusedinthischapter. selectedtoreceiveinformationfromthecollaboratingnodes.Forthesakeofexposition,assumethattrellissectionsi)]TJ/F15 11.955 Tf 12.234 0 Td[(1,i,andi+1haveincreasinginstantaneousSNRsinthatorder.If=2,informationabout c i)]TJ/F24 7.97 Tf 6.586 0 Td[(1willbeobtainedfromRX2.If=3,informationabout c i)]TJ/F24 7.97 Tf 6.587 0 Td[(1andc1iwillbeobtainedfromRX2.If=7,codedsymbolsforalltheparitybitsinthecandidatesetareobtainedfromRX2,andcodedsymbolsfor c i)]TJ/F24 7.97 Tf 6.586 0 Td[(1areobtainedfromRX3.OncetheappropriatenumberofcodedsymbolsarecombinedfortheLRB,RX1requestsforcodedsymbolsforthenextLRBthathasadifferentcompetingpath.Theremaybeotheradjacenttrellissectionwiththesamereliability.Butbyrequestingforparitybits,allthebitsthathadoriginallydecodedwiththesameMLandcompetingpathsarecorrectedwithaprobabilitythatisgreaterthan.Hence,ifmultipletrellissectionshavethesamecompetingpathandhencethesamereliability,itisenoughtoconsideronlyoneofthemtocomputetherequestsetandrequestsize.IftheinformationreceivedisabletochangethedecisionfromcMLtocicomp,thenallthetrellissectionthatchosethesamecompetingpathwillbecorrected.Thus,I-LRBexploitstime-correlatedreliabilitiesbynotrequestinginformationforalladjacentbitsthatdecodedwiththesamereliability.

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74 Aspreviouslydescribed,codedsymbolsforaparticulartrellissectionfromaparticu-larcollaboratingnodesareonlytransmittedonce.Usingthepreviousexample,assumethatthebranchfromstate2tostate1hasalreadyreceivedcodedsymbolsfromRX2becausethisbranchwaspartofadifferentcompetingpathforsomeotherbitthathadareliabilitylessthanthatofbiti.Thenwhen=3,informationforfor c i)]TJ/F24 7.97 Tf 6.586 0 Td[(1andc1i+2willbeobtainedfromRX2assumingthatcodedsymbolsforthesebitshavenotbeenobtainedfromRX2earlier.Also,ifcodedsymbolsforc1iisrequiredinthenextiteration,itshouldbeob-tainedfromRX3,andnotRX2.ThisprocedureisrepeateduntilatotalofpMRC=Niterbitsareexchangedwithinthecluster.NotethatthisincludesthebitsrequiredtoindextheparitybitsrequestedbyRX1.Inpractice,alloftheinformationrequestscanbeperformedatthebeginningofaniteration,followedbyeachreceiver'sresponsestartingfromRX2toRXNrx.RX1combinesallofthereceivedinformationwithitspreviouslyreceivedin-formationusingMRConabit-by-bitbasis.IfRX1isabletodecodecorrectlyorthemaximumnumberofiterationshasbeenreached,thenthecollaborativedecodingprocessterminates.Otherwise,anotheriterationofcollaboratingdecodingisperformed. 6.5 ResultsInthissection,wepresenttheperformanceofourcollaborativedecodingscheme.Foralltheresultsinthischaper,arate1=2,memory-three,non-recursive,non-systematicconvolutionalcodewithgeneratorpolynomials1+D2and1+D+D25;7inoctalnotationisusedforencodingatthedistanttransmitter.ThemessageconsistsofN=900-bitpackets.Foralltheresults,thechannelbetweenthedistanttransmitterandtheclusterofcooperatingnodesisassumedtobeaquasi-staticRayleighfadingchannel,wherethefadingisconstantovereachpacket.Forallresults,thenumberofcollaboratingiterationsNiter=Nrx)]TJ/F15 11.955 Tf 11.955 0 Td[(1.TheblockerrorrateforI-LRBandCOI-MRCisshowninFigure 6-2 fordifferentnumberofcollaboratingnodes.Fortheseresults,a5%overheadconstraintwithrespectto

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75 Figure6-2:Probabilityofblockerrorfordifferentnumberofcollaboratingnodeswhentheoverheadconstraintisxedat5%oftheoverheadforMRC. theoverheadrequiredforMRCwasimposed.ItisobservedthatI-LRBoutperformsCOI-MRCforallsizesofthecooperatingclustershown.ItisalsoseenthatthegainofferedbyI-LRBincreasesasthenumberofcollaboratingnodesincrease.Forexample,withatargetblockerrorrateof10)]TJ/F24 7.97 Tf 6.587 0 Td[(2,I-LRBoutperformsCOI-MRCbyapproximately2dBwhenthereare8collaboratingnodes.Theperformanceofonlyonenodenocooperationisalsoshownforthesakeofcomparison.Asinglereceiverachievesablockerrorrateof10)]TJ/F24 7.97 Tf 6.586 0 Td[(2ataround23dBEb=N0.Hence,cooperationusingI-LRBprovidesagainofaround21dB.ThecorrespondingthroughputforthisscenarioisshowninFigure 6-3 .ItisseenthatthroughputforI-LRBislargerthanthethroughputforCOI-MRCforallthecases.Atasignal-to-noiseratioSNRof2dB,andwitheightcollaboratingreceivers,I-LRBincreasesthethroughputbyalmost30%withrespecttoCOI-MRC,andby350%withrespecttoasinglenode.

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76 Figure6-3:Throughputfordifferentnumberofcollaboratingnodeswhentheoverheadconstraintisxedat5%oftheoverheadforMRC. TheblockerrorrateofCOI-MRCandI-LRBiscomparedinFigure 6-4 fordifferentoverheadconstraintswhenthereareeightcollaboratingnodes.ThecorrespondingaveragecooperationoverheadisshowninFigure 6-5 .ItisseenthatI-LRBperformsbetterthanCOI-MRCbothintermsofblockerrorrateandcooperationoverhead.Inotherwords,I-LRBachievesalowerblockerrorratewithalowercooperationoverhead.ThethroughputofeightcollaboratingnodesisshowninFigure 6-6 fordifferentoverheadconstraints.ItisseenthatI-LRBoffersconsistentlyhigherthroughputthanCOI-MRC.ThethroughputofI-MRCCOI-MRCwithnooverheadconstraintandthatofasinglereceiverarealsoshown.ThoughI-MRChasthebestblockerrorrateamongalltheschemesseeFigure 6-4 ,ithasalowerthroughputwhencomparedtoI-LRBorCOI-MRC.Thus,itisclearthatI-MRCachievesgoodblockerrorrateperformanceatthecostofhigheroverhead.ItisalsoobservedthatthethroughputofI-LRBdecreaseswhentheoverheadconstraintisrelaxed.Thisimpliesthatthegaininblockerrorrateisnotsignicantasmorecombiningisal-lowedinthecooperatingcluster.Theincreaseinoverheadcausedbyrelaxingtheoverhead

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77 Figure6-4:ProbabilityofblockerroroverheadforCOI-MRCandI-LRBwitheightcoop-eratingnodes,anddifferentconstraintsontheoverhead. Figure6-5:AveragecooperationoverheadforCOI-MRCandI-LRBwitheightcooperatingnodes,anddifferentconstraintsontheoverhead. constraintover-shadowsthedecreaseinblockerrorrate,leadingtoalowerthroughput.Thus,theI-LRBschemeiscapableofprovidingalargeincreaseinthroughputwithavery

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78 Figure6-6:ThroughputforCOI-MRCandI-LRBwitheightcooperatingnodes,anddif-ferentconstraintsontheoverhead. smalloverhead.ThisisbecauseI-LRBtargetsthetrellis-sectionswhicharelikelytobeinerror,andadaptstheamountofinformationcombinedforthesesectionsbasedontheirreliabilities.TheaveragenumberofiterationsrequiredbytheCOI-MRCandI-LRBschemesisshowninFigure 6-7 .ItisseenthatcollaborativedecodingisterminatedfasterinI-LRBthaninCOI-MRC.SincetheamountofinformationcombinedineachiterationisthesameinI-LRBandCOI-MRC,andsinceI-LRBrequiresfeweriterations,theoverheadofI-LRBissmallerthanthatofCOI-MRCasshowninFigure 6-5 .Forexample,atanSNRof0dBanda5%overheadconstraint,I-LRBrequiresfewerthanhalfthenumberofiterationsrequiredbyCOI-MRC.ItcanbeveriedfromFigure 6-5 thattheoverheadofI-LRBisindeedaround50%ofCOI-MRCat0dBforthe5%constraint.

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79 Figure6-7:Averagenumberofiterationspercollaborativedecodingattemptrequiredbyeightreceivers.

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CHAPTER7CONCLUSIONANDDIRECTIONSFORFUTURERESEARCH 7.1 ConclusionInthiswork,wehavestudiedtheideaofusercooperationfromadecodingperspec-tive.Ourobjectiveistoachievegoodperformancewithlowcollaborationoverhead.Ourschemesarebasedonaverysimpleidea.Ifthecooperatingnodeshavesomeinformationaboutdecodingatothernodes,thecooperationoverheadcanbesignicantlyreduced.Weintroducedaframeworkcalledcollaborativedecodingtohelpdevelopcooperationstrate-giesthatareefcientintermsofthecooperationoverhead.Inanycollaborativedecodingscheme,thecooperatingnodesiteratebetweenaprocessofinformationexchangeandde-coding.Theinformationexchangeportionofcollaborationprovidesinformationaboutdecodingataparticularnodetoothernodes.Theothernodesusethisinformationtode-cidewhatinformationtotransmit.Collaborativedecodingreliesontheuseofsoft-inputsoft-outputSISOdecoders.ThemagnitudeoftheoutputoftheSISOdecoderiscalledthereliabilityandisanindicationofthecorrectnessofthedecodedbit.Incollaborativede-coding,thenodesusereliabilityinformationfromtheSISOdecodertoadaptthemessagesthatareexchangedamongthecooperatingnodes.Thisisthebiggestdifferencebetweencollaborativedecodingandconventionalcooperationstrategieswhereintheinformationexchangedduringcollaborationispredeterminedandxed.Unlikepreviouscooperationstrategies,collaborativedecodingprovidesaconvenientapproachtotradeperformanceforoverhead,andcollaborativedecodingscaleseasilytomultiplecooperatingnodes.Wealsodevelopguidelinesforthedesignofcollaborativedecodingstrategies.Weusetheseguidelinesanddesignanovelapproachcalledimprovedleast-reliable-bitI-LRBcollaborativedecodingforuser-cooperationinbandwidth-limitedscenarios.TheI-LRBschemehastheadvantageoverpreviouslyproposedcooperationstrategiesinthatitadapts 80

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81 theinformationexchangedincollaborativeprocessbasedontheaposterioriprobabilitiesatthedecodingnode.TherearetwolevelsofadaptationinI-LRB.First,I-LRBadaptsthesetofbitsforwhichinformationisrequestedbasedonthereliabilities.Second,foreachchosentrellissection,I-LRBadaptsthenumberofcoded-symbolsexchangedbasedonthereliability.I-LRBreducestheoverheadbynotcombiningcodedsymbolsforallofthetrellissectionsthatcorrespondtoasingleerrorevent.TheadvantagesoftheI-LRBschemecomefromexploitinginformationgeneratedintheBCJRdecoder.Weshowthattemporalcorrelationinreliabilitiesariseduetothesamechoiceofcompetingpathsfordifferenttrellissections.Weshowthatthecompetingpathscanbeexplicitlycalculatedusingcomputationsthatarealreadyperformedinthedecoder.Byobservingcompetingpathsthatoccurinthedecoder,I-LRBcanrequestfortheminimumnumberofcodedsymbolsthatcancorrectallthetrellissectionsthatchoosethatcompetingpathintheirreliabilitycomputation.SimulationresultsshowthatI-LRBachievesalowerprobabilityofblockerrorwithaloweraveragecollaborativeinformationexchangethantheCOI-MRCscheme.TheresultsshowthatI-LRBcanprovidea30%-60%improvementinthroughputwithrespecttotraditionalcooperationschemes.Theoverheadrequiredforthisimprovementislessthan5%oftheoverheadoftraditionalcombiningschemeslikeMRC.Thus,I-LRBoffersanefcientapproachforcollaborationwhenthemaximumcollaboratingoverheadisconstrained. 7.2 DirectionsforFutureResearchWenowpresentareasofpotentialresearchthatcanbepursuedusingtheideaspresentinthisdissertation. Theperformanceofcollaborativedecodingcanbestudiedinmultipleaccesswirelessnetworksbyabstractingtheresultsinthisdissertationintoanetworksimulator.Sincecollaborationamongagroupofnodesintroducesinterferenceinthenetwork,itisnotclearifcollaborativedecodingcanactuallyimprovethethroughputoftheentirenetwork.

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82 Sofarwehaveusedxedconvolutionalcodesthatareguaranteedtoachievetheminimumbiterrorrateamongalltheconvolutionalcodeswiththesameconstraintlength.Wehavenotconsideredtheissueofcodedesign.Itisnotclearifaconvo-lutionalcodethatprovidesthebesterror-performanceinasingle-userscenariowillalsoprovidethebesterrorperformanceinacooperativesetting.Forexample,wefoundthatarecursivesystematicconvolutionalRSCcodeprovidesalowerprob-abilityofblockerrorthananequivalentnon-recursiveconvolutionalcodeforthesamecooperationoverhead.Itwillbeinterestingtostudythereasonbehindthisobservation. Wehavestudieduser-cooperationinabandwidth-constrainedsetting.User-cooperationcanalsobestudiedinanerror-costrainedsystem.Inthesesystems,thenodesshouldachieveacertainbit/blockerrorratethroughcooperation.Wecancomparecollabo-rativedecodingtoconventionalcooperationschemestoseewhichtechniqueachievestherequirederrorratewiththelowestoverhead.Onewaytodesignacollaborativedecodingschemeforthissystemiscomputethenumberoftrellissectionsforwhichinformationistoberequestedinordertoachievethetargeterrorrate.Theclosedformexpressionforthebiterrorratecanbeusedtocomputethis.Usingtheclosedformapproximationitisseenthatatargetbiterrorratetranslatestoatargetmeanofthereliabilities.Thus,afteraroundofdecoding,eachnodecancomputeitsmeanofthereliabilitiesandthenestimatehowmuchinformationtorequestinordertoachievethetargetmean.

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REFERENCES [1] J.G.Proakis,DigitalCommunications,4thed.,McGraw-Hill,NewYork,2000. [2] A.Sendonaris,E.Erkip,andB.Aazhang,Increasinguplinkcapacityviausercoop-erationdiversity,inProc.1998IEEEInt.Symp.Inform.Theory,Boston,Aug.1998,p.156. [3] J.N.Laneman,CooperativeDiversityinWirelessNetworks:AlgorithmsandAr-chitechtures,Ph.D.dissertation,M.I.T,September2002. [4] N.Laneman,D.Tse,andG.Wornell,Cooperativediversityinwirelessnetworks:Efcientprotocolsandoutagebehavior,IEEETrans.Inform.Theory,vol.50,no.12,pp.3062,Dec.2004. [5] A.Sendonaris,E.Erkip,andB.Aazhang,UsercooperationdiversitypartI:Systemdescription,IEEETrans.Commun,vol.51,pp.1927,Nov.2003. [6] A.Sendonaris,E.Erkip,andB.Aazhang,UsercooperationdiversitypartII:Im-plementationaspectsandperformanceanalysis,IEEETrans.Commun,vol.51,pp.1939,Nov.2003. [7] T.E.HunterandA.Nosratinia,Cooperativediversitythroughcoding,inProc.2002IEEEISIT,Laussane,Switzerland,July2002,p.220. [8] B.ZhaoandM.Valenti,Distributedturbocodeddiversityfortherelaychannel,IEEElectronicsLetters,vol.39,pp.786,May2003. [9] T.F.Wong,X.Li,andJ.M.Shea,Iterativedecodinginatwo-nodedistributedarray,inProc.2002IEEEMilitaryCommunicationsConferenceMILCOM,Anaheim,CA,Oct.2002,vol.2,pp.704.2.1. [10] J.Wieselthier,G.Nguyen,andA.Ephremides,Algorithmsforenergy-efcientmul-ticastinginadhocwirelessnetworks,inProc.IEEEMilitaryCommunicationsCon-ference,1999,pp.1414. [11] E.C.vanderMeulen,TransmissionofInformationinaT-TerminalDiscreteMemo-rylesschannel,Ph.D.dissertation,UniversityofCalifornia,Berkeley,CA,September1968. [12] T.M.CoverandA.A.ElGamal,Capacitytheoremsfortherelaychannel,IEEETrans.Info.Theory,vol.IT-25,pp.572,Sept.1979. 83

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86 [41] G.MontorsiandS.Benedetto,Designofxed-pointiterativedecodersforconcate-natedcodeswithinterleavers,IEEEJ.Select.AreasCommun.,vol.19,pp.871,May2001. [42] W.E.Ryan,ConcatenatedcodesanditerativedecodinginWileyEncyclopediaofTelecommunicationsJ.G.Proakised.,WileyandSons,NewYork,2003. [43] S.LinandD.J.Costello,ErrorControlCoding:FundamentalsandApplications,PrenticeHall,EnglewoodCliffs,NJ,1983. [44] A.Avudainayagam,A.Roongta,andJ.M.Shea,Improvingtheefciencyofreliability-basedhybrid-ARQwithconvolutionalcodes,inProc.2005IEEEMil-itaryCommunicationsConference,AtlanticCity,NJ,Oct.2005,pp.1.

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BIOGRAPHICALSKETCHArunAvudainayagamreceivedtheB.Edegreeinelectronicsandcommunicationengi-neeringin2000fromAnnaUniversity,India,andtheM.Sdegreeinelectricalandcomputerengineeringin2001fromtheUniversityofFlorida.SinceJanuary2002,hehasbeenwork-ingtowardshisPh.D.degreeattheUniversityofFlorida.HeinternedwiththeWCDMAsystemstestgroupatQualcomm,Inc.fromMay2005-Dec.2005,wherehewasinvolvedinaccessstratumperformanceevaluationofQualcomm'sWCDMAmodemchips.Hisresearchinterestsincludewirelesscommunications,codesongraphsandbeliefpropaga-tion,iterativedecodingtechniques,cooperativecommunication,andappliederrorcontrolcoding. 87


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COLLABORATIVE DECODING: ACHIEVING COOPERATIVE DIVERSITY IN
WIRELESS NETWORKS USING SOFT-INPUT SOFT-OUTPUT DECODERS















By
ARUN AVTUDAINAYAGAM


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006
































Copyright 2006

by
Arun Avudainayagam


















To my parents, Hema and Nayagam, and my wife, Krithi.
















ACKNOWLEDGMENTS

Numerous individuals have directly and indirectly contributed to my graduate ex-

perience in general and this dissertation in particular. The following few words cannot

fully convey my gratitude for all the help and guidance I have received over the years.

Any acknowledgment should start with my advisor, Dr. John Shea. I have greatly

benefited from his expertise, valuable insights and approach to tackling problems. His

mentoring has made me strive to become a better engineer and researcher. I thank him for

all his help, guidance, and patience.

I would also like to thank Dr. Tan Wong for many fruitful and interesting discussions.

Special thanks go to Dr. Fang for taking me under his wing when I first began my graduate

studies and for encouraging me to continue at UF for my Ph.D. I must also mention my fel-

low graduate students who made my stay at WING a very memorable experience. Abhinav

Roogta, Hanj o Kim and Jangwook Moon deserve a special mention for various successful

and some not-so-successful (but interesting nonetheless) collaborations. I am also grateful

to the Office of Naval Research for sponsoring part of my research.

This dissertation would not have been possible without the support and encouragement

of my parents, Hema and Nayagam. I thank you for all your sacrifices that has moulded

me into this person that I am today.

And to end with a "last but not least" cliche~; my most loving and sincere thanks go to

my wife, Krithi. She was probably the most affected by my schedule during the course of

this work and I cannot express how invaluable her support has been over the years.


















TABLE OF CONTENTS


ACKNOWLEDGMENTS . . iv

LIST OF TABLES . . vii

LIST OFFIGURES . . viii

ABSTRACT. ............................ .. x

CHAPTERS

1 INTRODUCTION . 1

1.1 Objectives and Main Contributions . . 2
1.2 Outline of the Dissertation . . 4

2 BACKGROUND AND RELATED RESEARCH . . 6

2.1 Information-Theoretic Strategies . . 6
2.2 Repetition Based Cooperation . . 9
2.3 Coded Cooperation . . 14

3 SOFT-INPUT SOFT-OUTPUT DECODING. . . 17

3.1 The Log-MAP and Max-log-MAP Algorithms . . 17
3.2 The Density Function of Reliabilities Associated with a Max-Log-MAP
Decoder . ............ . .. .. 19
3.2.1 A High SNR Approximation to the Density Function of Relia-
bilities . . . . ... . 20
3.2.2 On the Correlation Between Output Error Events . . 22
3.3 A Mathematically Tractable Density Function . . . 24
3.4 A Closed-Form Expression for the Bit-Error-Rate of SISO Decoders 25
3.5 Extension to Block-Fading Channels . . 26
3.6 Numerical Results . . 28

4 CODED COOPERATION THROUGH COLLABORATIVE DECODING 34

4.1 Collaborative Decoding through Reliability Exchange . . 36
4.1.1 Collaborative Decoding through the Reliability Exchange of the
Least Reliable Bits . . . . 37
4.1.2 Collaborative Decoding through the Reliability Exchange of the
Most Reliable Bits . . 41












4.2 Guidelines for the Design of Collaborative Decoding Schemes . 47

5 ON CORRELATED BIT ERRORS AT THE OUTPUT OF A MAX-LOG-
MAP DECODER. . . 51

5.1 Terminology and Notation . . . 52
5.2 Revisiting Max-log-MAP Decoding of Convolutional Codes . 52
5.2.1 Obtaining the ML and Competing Path using the BCJR Algorithm 55
5.2.2 On the Utility of Competing Paths in the Design of Collabora-
tive Decoding . . 59

6 IMPROVED LEAST-RELIABLE-BITS COLLAB ORATIVE DECODING FOR
BANDWIDTH-CONSTRAINED SYSTEMS . . 61

6.1 Collaborative Decoding with Constrained Overheads . . 62
6.1.1 Constrained-overhead Incremental MRC . . 64
6.1.2 Overview of Improved Least-Reliable Bits Collaborative De-
coding . . 65
6.2 Estimation of Request Size . . 67
6.3 Estimation of the Request Set . . 71
6.4 Detailed Description of I-LRB Collaborative Decoding . . 72
6.5 Results. . . 74

7 CONCLUSION AND DIRECTIONS FOR FUTURE RESEARCH . 80

7.1 Conclusion . . 80
7.2 Directions for Future Research . . 81

REFERENCES . . 83

BIOGRAPHICAL SKETCH . . 87

















LIST OF TABLES
Table pg

3-1 Error event multiplicity of the (5, 7) convolutional code . . 29

4-1 Overhead of LRB-1 for different number of nodes . . 39

5-1 Notation used in this chapter . . 53

6-1 Instantaneous SNR estimation for trellis sections based on the average of
the instantaneous SNRs of the parity bits in the candidate set . 71

















LIST OF FIGURES
Figure pg

2-1 The relay channel . . 7

2-2 The multiple access channel with cooperating encoders . . 8

2-3 The decode-and-forward cooperation scheme . . 11

2-4 The ampli@S-and-forward cooperation scheme . . 12

2-5 Coded cooperation using rate-compatible punctured convolutional codes. .14

2-6 Coded cooperation using turbo codes . . 15

3-1 The mean of reliabilities as a function of the signal-to-noise ratio when the
correlation between the output error events are ignored . . 28

3-2 The mean of reliabilities as a function of signal-to-noise ratio after taking
into account the correlation between output error events. . . 30

3-3 The PDF of reliabilities of the (5, 7)s CC for two different signal-to-noise
ratios . . 3 1

3-4 The PDF of reliabilities of the (5, 7)s CC obtained using the simpler, math-
ematically tractable expression given in (3.18) . . 32

3-5 The probability of bit error for max-log-MAP decoding of convolutional
codes evaluated using the closed-form approximation given in (3.23) 33

3-6 The mean of reliabilities of the (5, 7) convolutional codes as a function of
signal-to-noise ratio of a block-fading channel . . 33

4-1 System model for collaborative decoding . . 35

4-2 Principle of interactive/collaborative decoding with two nodes. .. .. .. 36

4-3 Performance of two collaborative decoding schemes in which receivers re-
quest information for a set of least-reliable bits . . 40

4-4 Reliability density functions associated with correctly and incorrectly de-
coded bits.. . . . . . ... . . 42

4-5 Performance of two collaborative decoding schemes in which receivers
broadcast information about a set of most-reliable bits . . 43











4-6 Bit indices of reliabilities exchanged as a function of iteration. . 44

4-7 Performance of suboptimal variants two collaborative decoding schemes in
which hard decisions are exchanged instead of soft information. . 46

4-8 Performance of the MRB-2 scheme with eight nodes on a block-fading
channel . . 48

6-1 The code-trellis for the (5, 7) convolutional code with examples of the no-
tation used in this chapter . . 73

6-2 Probability of block error for different number of collaborating nodes when
the overhead constraint is fixed at 5% of the overhead for MRC. . 75

6-3 Throughput for different number of collaborating nodes when the overhead
constraint is fixed at 5% of the overhead for MRC . . 76

6-4 Probability of block error overhead for COI-MRC and I-LRB with eight
cooperating nodes, and different constraints on the overhead. . 77

6-5 Average cooperation overhead for COI-MRC and I-LRB with eight coop-
erating nodes, and different constraints on the overhead. . . 77

6-6 Throughput for COI-MRC and I-LRB with eight cooperating nodes, and
different constraints on the overhead . . 78

6-7 Average number of iterations per collaborative decoding attempt required
by eight receivers . . 79
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COLLABORATIVE DECODING: ACHIEVING COOPERATIVE DIVERSITY IN
WIRELESS NETWORKS USING SOFT-INPUT SOFT-OUTPUT DECODERS

By

Arun Avudainayagam

May 2006

Chair: John M. Shea
Maj or Department: Electrical and Computer Engineering

Spatial diversity techniques use multiple transmit and receive antennas (antenna ar-

rays) to improve performance in wireless environments without requiring additional band-

width or loss in throughput. However, the spacing between antenna elements depends on

the carrier wavelength, and this might often exceed the size of modern mobile radios. Thus,

alternative approaches are required to harness spatial diversity in small terminals. Recently,

cooperation among users has been proposed as an alternative means to achieve diversity in

wireless networks with small radios. In this proposal, we develop collaboration schemes

for scenarios in which radios in a network cooperate to improve performance. Since ra-

dios in a wireless network are typically separated in space, the different nodes can pool

their resources together to form a virtual antenna array. The elements of the antenna array

can then collaborate by exchanging information with each other in order to achieve di-

versity gains. The information exchanged by the collaborating nodes is called cooperation

overhead. Our schemes are targeted towards bandwidth-limited systems in which the coop-

eration overhead should be small. We provide a framework called collaborative decoding

to help design schemes that have low cooperation overhead and still achieve performance

close to that of optimal combining schemes. We present a collaboration technique called










improved least-reliable bits (I-LRB) collaborative decoding that provides a higher level

of adaptation than previously proposed cooperative schemes. The I-LRB scheme utilizes

reliability information and information about competing paths in soft-input soft-output de-

coders to adaptively select the amount of information that is needed to correct a particular

part of a message, as well as which bits should be exchanged. Simulation results show that

the proposed approach offers a significant performance advantage over existing coopera-

tion techniques. For example, I-LRB can provide a 30%-60% improvement in throughput

with respect to traditional cooperation schemes in bandwidth-constrained systems.
















CHAPTER 1
INTRODUCTION

Multipath fading is one of the most common problems associated with wireless com-

munications. Reflections from and refraction through various objects in the channel cause

multiple attenuated and delayed copies of the transmitted signal to constructively or de-

structively combine at the destination. Fading can cause severe fluctuations in the signal-

to-noise ratio (SNR), which in turn affects system performance. Various techniques like

equalization, error-control coding and diversity combining are used independently or in

conjunction to combat fading. Diversity techniques typically improve performance by

making multiple independent copies of the transmitted signal available to the destina-

tion. These multiple copies can then be optimally combined using various techniques like

maximal-ratio combining (MRC) or equal-gain combining (EGC) [1]. Temporal diversity

is typically achieved through error-control coding. Frequency diversity is achieved by us-

ing various physical layer techniques like frequency hopping or multi-carrier modulation.

Recent advances in space-time coding have proven that processing in the spatial do-

main is an efficient approach to achieve diversity in delay-limited and bandwidth-limited

applications. Space-time codes exploit the multipath nature of the wireless medium to

combat the detrimental effects of fading. Spatial diversity can be achieved by using multi-

ple antennas at the transmitter and/or receiver. However, for significant gains the spacing

between the antenna elements should be at least half the wavelength of the RF carrier. This

prohibits the use of antenna arrays in most small, portable radios. This is the reason why

antenna arrays are not used in the cellular downlink or in ad hoc networks.

In recent years, a number of network-a~ssisted diversity techniques have been stud-

ied [2-9]. In these approaches, the users depend on the network to provide diversity at

the physical layer. The broadcast nature of the wireless channel, wherein any node within










range of the transmitter can listen to the transmission, is exploited in this network-based

approach to spatial processing. This property of the wireless medium is referred to as the

wireless brvadca~st advantage (WBA) [10]. Since nodes in a network are spatially sepa-

rated, the different nodes that receive the transmission from a source can be considered

to be the elements of a virtual antenna rrarm. Since the elements are not physically con-

nected, this is referred to as a distributed rrarm. The different users can then collaborate

with each other to achieve diversity gains. Diversity achieved when users in a network

collaborate to improve each other's performance has been termed coopetutive diversity or

multiuser diversity. We will to use the term user cooperation [2] to refer to the process of

collaboration between the various users of the network.

A lot of work on user cooperation (including most information-theoretic and a few

practical schemes) is based on simple repetition coding [2-6]. The basic idea of these

schemes is that any user that listens to the transmission from the source forwards the infor-

mation (either the coded bits after decoding or quantized versions of the received symbol

values) to the destination. The amount of information exchanged by the collaborating

nodes is referred to as the cooperation overhead. The use of repetition codes makes these

techniques inefficient in terms of the overhead. Cooperation through the use of more pow-

erful error correction codes has also been proposed in [7, 8]. The disadvantage of these

schemes is that they do not easily scale to large networks (with more than two cooperating

nodes).

1.1 Obj ectives and Main Contributions

Since the wireless medium is bandwidth-limited, the cooperation overhead is a very

important issue and one that has not been addressed so far in the literature. The obj ective of

this work is to provide a framework to help develop cooperation strategies that are efficient

in terms of the cooperation overhead. It is also important that the collaboration techniques

extend naturally to multiple cooperating nodes. Using this framework, called collabom~-

tive decoding, we develop strategies that provide close-to-optimal performance with only










a fraction of the overhead required by conventional cooperation schemes. Unlike previous

cooperation strategies, collaborative decoding provides a convenient approach to trade per-

formance for overhead, and collaborative decoding scales easily to multiple cooperating

nodes.

Conventional coded cooperation strategies are based on distributed encoding of a mes-

sage among the collaborating nodes. Collaborative decoding is based on a distributed de-

coding of an encoded message among collaborating nodes. All schemes in this dissertation

use soft-input soft-output (SISO) decoders. The magnitude of the output of the SISO de-

coder is called the reliability and is an indication of the correctness of the decoded bit. In all

our schemes, the nodes exchange information for only a fraction of the message bits based

on the reliability information. The design of a collaborative decoding scheme then consists

of the choice of the bits to be exchanged and what information is to be exchanged among

the nodes. In existing cooperation strategies, the messages exchanged by collaborating

nodes is predetermined and fixed. Collaborative decoding adapts the content involved in

cooperation to each channel instantiation. Thus, by tailoring the messages exchanged by

collaborating nodes to the potential bit errors, collaborative decoding aims to lower the

cooperation overhead.

The contributions of this work are two-fold. First the dissertation furthers under-

standing of the fundamental operation of the maximum a posteriori (MAP) convolutional

decoder with the max-log-MAP implementation. In this context the main contributions in

this dissertation are the following:


1. We provide a closed-form approximation to the density and distribution function of
reliabilities at the output of a max-log-MAP SISO decoder. This closed-form esti-
mate is parameterized by a single numerical quantity that can be determined analyt-
ically. The estimates can be can be used to analyze reliability-based systems.

2. Using these closed form approximations we provide an approximation to the bit error
rate of a max-log-MAP decoder in terms of a single Q-function. This is the first such
result in the literature.










3. We investigate the correlated nature of the soft-output of a max-log-MAP decoder.
The max-log-MAP decoder computes the soft-output for a trellis section by con-
sidering the maximum-likelihood (ML) path and a competing path that differs from
the ML path in the input for that trellis section. We show that the time-correlated
reliabilities occur in a max-log-MAP decoder because the same competing path is
considered in computing the soft-output for adj acent bits.

4. We provide an efficient approach to explicitly compute the ML and competing paths
by using computations that are already performed in the decoder.


The second area of contribution in this dissertation is in applying the knowledge

gained about the operation of the decoder to the design of collaborative decoding. The

main contributions in this area are the following:


1. We design a cooperation strategy called improved lea;st-reliable-bits (I-LRB) collab-
orative decoding that has the following features:

Achieves full diversity in the number of cooperating nodes.

Requires a fraction of the overhead involved in full maximal-ratio-combining.

Easily scales to multiple relays.

Offers the ability to easily trade performance for overhead.

2. I-LRB is adaptive in two levels. The trellis sections for which information is com-
bined are adapted to each channel instantiation. For each trellis section, the amount
of information combined is adapted to the reliability of that trellis section.

3. I-LRB exploits correlated bit reliabilities by computing competing paths in the de-
coder, and utilizes knowledge of the competing paths to reduce cooperation over-
head.

1.2 Outline of the Dissertation

This dissertation is organized as follows. In Chapter 2, we summarize various im-

portant results in the literature that pertain to the idea of user cooperation. These results

provide a basis for comparison with our schemes, and makes it easier to emphasize the

distinction between our approach and the existing schemes. We provide an introduction to

soft-input soft-output (SISO) decoders in Chapter 3. A good understand of SISO decoding










is important to understand the operation of our techniques. We also provide a mathematical

characterization/approximation of the statistics of the SISO decoder output. In Chapter 4,

we introduce the concept of collaborative decoding in which various users cooperate in

the decoding process and achieve spatial diversity. Though these schemes are suitable for

AWGN channels, we show that these schemes are not suitable for fading channels. We

develop guidelines to help design cooperative diversity protocols for fading channels. We

also develop various guidelines for the design of cooperative protocols in this chapter. In

Chapter 5, we study the correlated nature of the output of the SISO decoder. We show

that error events encountered in the SISO decoder can be used to capture this correlation.

We also present a technique to efficiently compute these error events with minimal mod-

ifications to the decoder. Based on the design guidelines presented in Chapter 4 and the

technique presented in Chapter 5, in Chapter 6 we design a collaboration scheme that uti-

lizes the correlated output of the SISO decoder to reduce the cooperation overhead. The

dissertation is concluded in Chapter 7.
















CHAPTER 2
BACKGROUND AND RELATED RESEARCH

In this chapter the idea of cooperative communications is introduced and important

references relating to this broad area are summarized. The objective of the chapter is to

familiarize the reader with the different approaches to user cooperation and the various is-

sues involved in the design of such schemes. We start by introducing the very first ideas of

cooperation and some recent techniques that were proposed in the information theory com-

munity. We then highlight some of the more practical approaches that have been studied in

recent years.

2.1 Information-Theoretic Strategies

Studies on the relay channel in the late 1960s can be considered to contain the first

instances of cooperation. The relay channel (shown in Figure 2-1) was first introduced and

studied by van der Meulen in 1968 [l l]. In this setting, an intermediate node, called the

relay, listens to the transmission from the source to the destination, processes this informa-

tion, and transmits additional information about the initial transmission to the destination.

The destination uses the relay transmission to resolve any ambiguity about the original

transmission. The transmission from the relay is done jointly with the source; i.e., relay

transmission for block i is super-imposed on block i + 1 sent by the source. Thus, the relay

cooperates with the source to improve reception at the destination.

In 1979, Cover and El Gamal [12] studied the capacity of relay channels under dif-

ferent scenarios. Cover and El Gamal put forward three different approaches to achieve

user cooperation in a relay channel. In facilitation, the relay passively aids the communi-

cation between a source and destination by not transmitting, thereby reducing interference

to the original communication. In cooperation, the relay decodes the transmission from










Relay








Source Destination


Figure 2-1: The relay channel.


the source and provides additional information about the initial transmission to the desti-

nation to aid in recovering the original message. In observation, the relay just forwards the

observed symbol values to the destination.

The observation scheme was introduced to overcome a problem with the cooperation

scheme. In cooperation, the relay partitions the set of valid codewords into bins using

the Slepian-Wolf partitioning technique [13] and transmits the bin index of the partition

containing the source message. The destination then uses the set of codewords in the cor-

responding partition to resolve any ambiguity about the transmitted message. However, the

computation of the bin index requires correct decoding at the relay, and thus this scheme is

limited by the rate between the source and the relay. The observation scheme can overcome

this problem because it does not require correct decoding at the relay.

Recent studies of the relay channel can be found in [14-18]. The Cover and El

Gamal scheme is extended to multiple nodes in Gupta and Kumar [14]. In Cover and

El Gamal [12] and Gupta and Kumar [14], the nodes cooperate using block-Markov en-

coding and Slepian-Wolf partitioning [13]. The destination decodes the message using two

transmitted blocks; i.e., upon receiving block i, the decoder estimates the message in the

previous block (i 1). The nodes use codebooks of different sizes at the relay and the

source, making it difficult to extend the this strategy to multiple nodes. A technique that

uses codebooks of the same size is proposed in Willems [19] and is extended to multiple

relays in Kramer et al. [15]. The use of equal size codebooks makes it easier to extend










Encoder 2



MULTIPLE
ACCESS
CHANNEL
Destination



Encoder 1


Figure 2-2: The multiple access channel with cooperating encoders.


this scheme to multiple relays. Another advantage of this scheme is that the use of the

complicated Slepian-Wolf inning technique is avoided. The disadvantage manifests itself

as a large decoding delay incurred due to the bacinvards decoding technique used at the

destination. The coding is done over B blocks and the decoding process can start only

after receiving all the B blocks. The decoding is done in the backward direction, starting

with the last block and ending at the first block. A strategy that avoids Slepian-Wolf parti-

tioning and in which all relays use codebooks of the same size but in which the decoder lag

is reduced to just one block (as in the Cover and El Gamal scheme) is proposed in Xie and

Kumar [16]. An extension of this scheme to multiple relays has also been studied [17, 18].

The information theory community also studied cooperation from the view of a mul-

tiple access channel. In the early 1980s user cooperation in the setting of a multiple access

channel was studied [20, 21]. Unlike the relay channel, both the encoders have data to send

to the destination. In Willems [20], the users cooperate over a separate channel before send-

ing their messages. The need for a separate set of channels for cooperation is eliminated

in Willems and van der Meulen [21]. The scenario considered is shown in Figure 2-2. The

cooperative transmission works as follows. Each encoder cooperates with the other and

learns the codeword that the other encoder is going to send in the current transmission.

Thus, in transmission i, each encoder has knowledge of all the previous i 1 codewords of

the other encoder. The encoding process works as follows. Without loss of generality, we










assume that encoder 1 starts the encoding process for transmission i. Encoder 1 forms its

codeword as a function of its own data and the codewords sent by encoder 2 in all previous

transmissions. Encoder 2 learns the current codeword of encoder 1 and forms its own code-

word as a function of its own data and the current and previous codewords of encoder 1.

This strategy requires perfect cooperation between the encoders; i.e., each encoder should

learn the codewords of the other encoder without errors. Note that the model of Willems

et al. reduces to the relay channel if one of the encoders has no data of its own to send.

Then that particular encoder will act as a relay for the other encoder. The authors prove that

encoder cooperation increases the capacity of the multiple-access channel when compared

to non-cooperative transmission.

The techniques described in this section depend on information theoretic concepts like

random inning, typical-set and backwards decoding. These are not viable for practical

implementation. In the next section, we review a few practical cooperation strategies based

on simple repetition coding ideas.

2.2 Repetition Based Cooperation

After the work of Willems, the idea of user cooperation was largely ignored until

the late 90s. The advances in space-time coding [22, 23] proved that exploiting spatial

diversity with the use of multiple transmit and receive antennas can lead to significant

improvements in data rate. However, small portable radios do not permit the use of multiple

antennas. User cooperation is a natural way to achieve spatial diversity by pooling the

resources of many radios, each equipped with a single antenna. User cooperation in a

wireless scenario was first investigated by Sendonaris, Erkip and Aazhang [2, 5, 6, 24].

Sendonaris et al. study the idea of user cooperation in the setting of a cellular CDMA

system. The system model they use is identical to the model shown in Figure 2-2. However,

due to the wireless setting the links between the two encoders (the cooperation channels)

are imperfect channels that experience fading.










The cooperation model is based on the idea of Willems et at, wherein a codeword sent

by one user depends on the codeword sent by the other user. In Sendonaris et at. [2], the

authors consider more practical aspects of user cooperation in this scenario. In particular,

the authors begin by using information theory to evaluate the effects of cooperation on

outage probability, diversity, and cellular coverage. Then the authors propose and analyze

the following practical cooperation scheme.

Let Xi(t) and cs(t) denote the signals transmitted by user i and the spreading code

used by user i at time t. Then the signals transmitted by the two users are


Xl(t) = aiib1 )(t)cl(t), alab 2) (t)C1 1), a13b(2) 1)1 ) 014%) 1~)C

X2 1) = a21b l)(t(t)c2 a) 22b 2) (l)C2 1)1 a23 12) (t)C1 1) 82a4b 2) (t)C2


where b is user i's ft bit and il is the corresponding estimate at the other node. The

parameters {aji} represent how much power is allocated to each bit. Thus, in the first

two periods each user transmits its own bits. In the third period, each encoder sends a

linear combination of its own bits of the second period and its estimate of its partner's bits

of the second period. Since the basis of cooperation is an estimate of the other encoder's

codeword, the authors allocate rate and power to guarantee error-free communication on the

cooperation channels. Thus, each encoder's estimate of the codeword of the other encoder

is perfect. With this system and cooperation model the authors prove that it is possible to

increase the maximum sum-capacity of the network if the transmitter has knowledge of the

channel phase.

Laneman et at. [3, 4], introduce two broad classes of cooperation techniques called

decode-and-forward and ampli@S-and-forward. Their approach is similar to the relay-

channel-based cooperation techniques of Cover and El Gamal. In the decode-and-forward

scheme, the relays first decode the source message and then forward the re-encoded infor-

mation bits to the destination. This is akin to Cover et al.'s cooperation scheme with the










Information bits Coded bits
I EN~CODER I





Phase 1 Phase 2\s




Figure 2-3: The decode-and-forward cooperation scheme.


relay transmission consisting of re-encoded information bits instead of Slepian-Wolf bin

indices.

The operation of the decode-and-forward scheme is shown in Figure 2-3. The scheme

works in two phases as in the conventional relay channel. In the first phase, the source

encodes the information bits (represented by an empty rectangle), and transmits the coded

bits (represented by a solid rectangle). The destination and the relay receive noisy versions

of the coded bits. In the second phase, the relay decodes the information bits, re-encodes

them using the same code used at the source. The re-encoded codeword is then sent to the

destination.

At the end of the second phase, the destination has two independent noisy copies of

the codeword sent by the source (assuming that the relay decoded correctly). These two

independent copies can be combined using various combining schemes like maximal-ratio

combining (MRC) or equal-gain combining (EGC) [1]. This scheme can be considered

as an instance of rate-1/2 repetition coding since the destination receives two independent

copies of the same message. However, the repetition is done by the relay instead of the

source itself.

The effectiveness of cooperative communication schemes is often assessed in terms

of the effects on capacity and on the diversity achieved. The ability to achieve diversity is

quantified in terms of the diversity order, which is defined as the asymptotic slope of the bit

or block error rate curve on log-log scale. If there are M~ elements in an antenna array, then










Information b'its"""nt Co Ided bits
I ENCODER





Phase 1 Phase 2

MW Wr

Figure 2-4: The amplify/-and-fonvard cooperation scheme.

at most M~ independent copies of the message are received. Hence, the maximum diversity

order that can be achieved is M~. Any scheme that achieves diversity order that is equal to

the number of cooperating nodes is said to achieve full diversity.

As in the cooperation scheme proposed by Cover et al., the decode-and-fonvard

scheme depends on correct decoding at the relay. It is proved that the decode-and-fonvard schemes

can provide all of the capacity benefits offered by cooperative transmission, but cannot

achieve full diversity (in the number of collaborating nodes) [3, 4]. The reason is that a

diversity channel is created on the link between the relay and the destination only when

the relay decodes successfully, and hence this scheme is limited by the channel between

the source and the relay. Note that the scheme of Sendonaris et al. also falls under the

decode-and-fonvard class of cooperation schemes since a perfect estimate of the partner's

bits is required for cooperation.

The operation of the ampli@S-and-fonvard scheme is illustrated in Figure 2-4. The first

phase is identical to first phase in the decode-and-fonvard scheme. In the second phase, the

relay does not perform decoding. Instead, the relay amplifies/scales its observations (the

received symbol values) subj ect to a power constraint and forwards it to the destination. If

y is the message received from the source, the transmission of the relay can be expressed


x = y,


(2.1)










where the amplification factor /9 is constrained by


PI (2.2)


Here P is the maximum transmit power of the source (and relay), a~ represents the fading

amplitude between the source and relay, and NVo/2 is the noise variance. On average, the

scaling factor /9 constrains the transmission power of the relay to its maximum allowed

value P. Unlike the decode-and-forwald scheme, the relay also amplifies its own receiver

noise. This is identical to Cover et al.'s observation scheme if the amplification factor

is set to unity i.e., if the relays have no power constraint. As in the decode-and-forwald

scheme, the destination has two independent noisy versions of the original codeword that

can be optimally combined. Thus, from the perspective of the destination, it still appears

as though a rate-1/2 repetition code is used at the source. It is shown that anpliS-an2d-

forwald schemes can achieve full diversity in the number of cooperating nodes [3, 4] .

In the decode-an2d-forwald scheme the relay just transmits the binary codeword. In the

anmpliS-and-forwald scheme, the relay must amplify the received symbols, and retransmit

these amplified soft values. This soft-amplification process will not be practical in many

real systems. Instead, the relay would have to quantize the received symbol values and

then transmit the quantized bits to the receiver. The information exchanged by the relays

in order to improve performance is referred to as the cooperation overhead. Then if B

bits are used for quantization, then the cooperation overhead of the anmp@iC~-and-forwald

schemes is B times the decode-and-forwald cooperation overhead. However, the anp@iC-

and-forwald scheme does not depend on correct decoding at any of the relays. Thus, the

anmpliS-and-forwald scheme achieves full diversity at the cost of overhead.

The schemes introduced in this section are based on simple repetition coding ideas.

The relays just repeat their estimate of the original codeword or their received symbol

values. In the next section we review a few cooperation strategies that are based on better

error-correction codes.










Information bits Coded bits
~ECDR I I IBC~R I11

O


W
Phase 1 Phase 2




Figure 2-5: Coded cooperation using rate-compatible punctured convolutional codes.

2.3 Coded Cooperation

Cooperative diversity through the use of better error-correction codes is called coded

cooperation [7]. Other schemes for coded cooperation have also been proposed [8, 9].

Coded cooperation schemes can be divided into two main classes. In the first class of

techniques, distributed encoding is performed among the cooperating nodes, and decoding

takes place only at the destination. In the second class, encoding is performed only at the

source, and decoding takes place in a collaborative manner among the cooperating nodes.

In this section, we review two coded cooperation schemes that belong to the former class.

Techniques belonging to the latter category will be introduced in the following chapters.

Hunter and Nosratinia study the idea of user cooperation using rate-compatible punc-

tured convolutional (RCPC) codes [7, 25, 26]. RCPC codes were introduced by Hage-

nauer [27] as a means to achieve incremental redundancy in ARQ schemes. The operation

of the RCPC-based coded cooperation scheme is illustrated in Figure 2-5. For this exam-

ple, we have shown the use of a rate-1/3 convolutional mother code. The data is encoded

with the mother code in the source. Then a part of the codeword (the center part in the

example) is punctured out and the remaining code bits are transmitted. Thus, the relay and

the destination receive noisy versions of a rate-1/2 codeword. The relay decodes this high-

rate transmission and then re-encodes the information bits using the mother code. Then

the relay punctures those sections of the codeword that were transmitted by the source in









TURBO ENCODER


Information bits ENCODER

PERMUTER 4 ENCODER






Phase 1 Phase 2
**~
I I I C_ -M1__fTr

Figure 2-6: Coded cooperation using turbo codes.

the first phase (the first and last parts in the example). The relay then transmits the remain-

ing part of the codeword (the part that was punctured in the first phase) to the destination.

Thus, the destination effectively has all parts of the original mother code. Thus, the relay
transmission helps transform the initial high rate code (rate-1/2) into a decodable lower

rate (rate-1/3) transmission thereby improving performance at the destination.

Zhao and Valenti [8] investigate cooperation using turbo codes [28, 29]. Their scheme

is shown in Figure 2-6. A turbo encoder (also shown in Figure 2-6) consists of two recursive

systematic convolutional (RSC) encoders. The information bits are fed directly into one
of the encoders and a permuted version of the information bits is fed into the other. The

two sets of parity bits along with the systematic (information) bits form the codeword.

The cooperation scheme works as follows. The source encodes the information bits with

an RSC encoder and transmits the parity bits along with the systematic bits. The relay

decodes this transmission using a convolutional decoder. It permutes its estimate of the
information bits and re-encodes it with the same RSC code used at the transmitter. This

produces the second set of parity bits that make up the turbo code. This second set of

parity bits is sent to the destination by the relay. Thus, the destination has in effect received










a codeword encoded by a turbo code. It can make use of the powerful, iterative turbo

decoding algorithm [28, 29] to improve performance.

Note that both the schemes relies on correct decoding at the relay and thus fall in the

decode-and-forward category. Thus, like any of the decode-and-forward schemes, these

techniques are not guaranteed to achieve full diversity. It is proved that the RCPC-based

cooperation scheme is capable of achieving full diversity only when the relay is able to

decode correctly [26].

Another big drawback of the coded cooperation schemes that utilize distributed en-

coding is scalability. These schemes do not scale easily to multiple relays. When there is

more than one relay, it is not immediately obvious on how the distributed encoding should

be done. In the sequel, we present collaborative decoding, which is a coded cooperation

scheme that is based on distributed decoding that scales naturally to any number of coop-

erating nodes.















CHAPTER 3
SOFT-INPUT SOFT-OUTPUT DECODINTG

This chapter presents a brief overview of soft-input soft-output (SISO) decoding. A

good grasp of SISO decoding concepts is required to understand our collaborative decoding

scheme that is presented in the following chapters. A mathematical characterization of

the output of a particular implementation of the maximum a posteriori (MAP) decoder is

also presented. This characterization aids in the analysis of one of our approaches to user

cooperation.

3.1 The Log-MAP and Max-log-MAP Algorithms

Decoders that operate on floating point (soft) inputs and produce floating point outputs

are called SISO decoders. The sign of the soft-output is the hard-decision and the magni-

tude of the soft-output is called the reliability of the hard-decision [30]. The reliability

is an indication of the correctness of the hard-decision; i.e., a high value of the reliability

implies a high probability of the decision being correct and vice-versa. In this proposal,

we restrict our attention to SISO MAP decoders. It is well known that bit-by-bit MAP

decoding produces the minimum probability of bit error among all decoding algorithms.

The inputs to a typical MAP decoder are a priori probabilities of the information bits and

channel symbols. The a priori probabilities are usually initialized to equally likely values.

The soft-output of a MAP decoder corresponds to the a posteriori probability (APP) of an

information bit ui being 0 (or 1), P(ui = 0|r) (or P(ui = 1|r)). Due to reasons of speed

and numerical stability, MAP decoders are typically implemented in the log-domain (Log-

MAP decoders). The output of a log-MAP decoder corresponds to log-likelihood ratios

(LLRs) of the APPs. The LLR for each information bit ui is computed as follows

P(u, = 0|r) CEcc P(c r)
L (ui Ir) = In = In (3.1)
P(u, = 1|r) ccPcr)










where r is the received codeword, C+ is the set of all codewords with ui = 0 and C_

is the set of all codewords with ui = 1. Note that ck E {+1, -1}. The output LLR is

also referred to as the soft information. Assuming that all the codewords are equally likely

and using Baye's rule, the soft information for codewords transmitted on a additive white

Gaussian channel (AWGN) with noise variance a2 0 V/2 can be written as




L;(usir) = In P(r c) -In P(r c),
cEC- cEC

In exp I c12i2 In exp I 2e2 'i
cEC+ cEC'
(3.2)


A suboptimal implementation of the Log-MAP decoder, called the Max-Log-MAP

decoder, is obtained by using the approximation In(C xi) = max(In(xi)) to evaluate the

LLR in (3.2). Thus, for a Max-Log-MAP decoder the soft-output is given by,


L(ui Ir) = mmn Ir-c1 mmn Ir-c1 (3.3)
cEC 2a2 cEc- 2a2

Since the union of C+ and C_ spans the space of all valid codewords, one of the terms in

(3.3) corresponds to the Euclidean distance between r and the maximum-likelihood (ML)

decoding solution. Thus, the reliability for bit i (As) can be expressed as



As ^ L(usr)| = mnun ||r I c | CML2 (3.4)

where c63 is a codeword corresponding to an input sequence that differs from the ML in-

put sequence in the ith bit. Since the distance between r and the ML codeword is smaller

than the distance between r and any other codeword, the difference in (3.4) is always posi-

tive. Thus, the Max-Log-MAP decoder associates with the ith bit, the minimum difference

between the metric associated with the ML path and the best path that differs from the

ML path in the input of the ith trellis section [31i]. A high value of reliability implies that










the ML path and the next best path are far apart, and hence there is a lower probability

of choosing the other path and making a bit error. It has also been shown via simulation

in [32, 33] that reliability is a measure of the correctness of the bit decision. Thus, a bit

with high reliability is more likely to have decoded correctly than a bit with low reliability.

Note that the scaling of the reliability by the noise variance in (3.4) does not affect the

performance of the Max-Log-MAP decoder and is just an implementation consideration.

If channel estimates are available to the decoder, the scaling can be performed.

3.2 The Density Function of Reliabilities Associated with a Max-Log-MAP Decoder

Reggiani and Tartara [34] provided the first characterization of the soft information in

terms of its probability density function (PDF). Reggiani and Tartara [34] examine the pro-

jection of noise in the direction corresponding to an error event and interpret this random

variable as a distance in Euclidean space to derive the PDF. Here an error event denotes a

sequence that translates one codeword into another, where the path through the code trellis

that is induced by the error sequence is only in the same state as the original codeword at

the endpoints of the sequence. For the rest of the paper, the random variable resulting from

the projection of noise onto a direction specified by an error event will be referred to as

the projection random variable (PRV). Reggiani and Tartara [34] present two approaches

to obtain the PDF. In the first approach, the PDF is derived based on the assumption that

different PRVs (projection of noise onto directions specified by different error events) are

independent. The PDF obtained using the independence assumption results in conserva-

tive reliability estimates that are lower than the actual values. The authors suggest incor-

porating the correlation between the PRVs into the PDF to avoid conservative estimates.

In the second approach, the authors obtain a covariance matrix involving the correlation

between different PRVs and use it in a joint multivariate distribution to obtain the PDF.

Though the PDF obtained using the second approach produces good reliability estimates,

the expression for the density function is very complicated. Even with the independence

assumption, the PDF obtained using this technique cannot be expressed in closed-form and










involves products and summations that depend on the enumeration of all possible error

events. Thus, the PDF given in Reggiani and Tartara [34] is not attractive for use in the

analysis of reliability-based techniques.

We now present a streamlined derivation of the densities of reliabilities at the output

of a max-log-MAP decoder by working in the conventional Hamming space (Reggiani and

Tartara [34] work in Euclidean space) and use a high signal-to-noise ratio (SNR) approx-

imation to obtain the PDF. A simple technique to account for the correlation between the

PRVs is also presented. Using this technique, we can avoid the use of complicated joint

multivariate distributions. Though this approach produces good estimates of the density

function and other statistics of the reliability, the expression is still complicated to be of

further use in analysis. To this end, we also present an ad hoc estimate of the PDF that

is mathematically tractable. This closed-form estimate of the PDF is parameterized by a

single quantity that can be numerically evaluated. We show that our technique produces an

accurate approximation of the true PDF.

3.2.1 A High SNR Approximation to the Density Function of Reliabilities

The reliability of the output of a max-log-MAP decoder is given in (3.4). Since cji is

a codeword corresponding to an input sequence that differs from the ML input sequence in

the ith bit, c ) can be expressed as,



C =CM/L + e (3.5)

where el" is an error event generated by an input sequence with bit i equal to 1. Since the

symbols of c~" and cML take on values in {+1, -1} and the error event transforms one

codeword into another, the components of e~i take on values in {+2, 0, -2}. Using (3.5)

in (3.4), we get

Asi = mninl e||e ||2 2(r CM1LT 3) }.!> (3.6)

Note that we have dropped the scaling by the noise variance (1/2o.2) in (3.6). After deriving

the density and distribution functions using (3.6), a simple transformation can be used










to account for the scaling in (3.4). At high SNRs, the ML decoder will find the correct

codeword (input sequence). Thus for high SNRs we can express the received sequence as


r = cMIL + e, (3.7)

where e ~ N(0, I).OT This assu~mption is similar to the appmroah in [34], in which the

authors obtain the conditional density function given correct decoding of a bit. Using (3.7)

in (3.6) we get

Asi = min (|| ||i!2 + 2eT 3) }. (3.8)

Note that according to our terminology, e(i is an error event, whereas eT i) is the PRV
i~ne.,; the projection of; the noise o~ntth drectio, +n of,, the ero vnt e Let


Zj = ||ea3 ||2 +t 2eT 3).i (3.9)

Since Zj is just a linear combination of Gaussian noise samples, Zj is also a Gaussian

random variable. It is easy to see that


ZMy ~ N 4d, 1d'2 (3.10)

where dj is the Hamming weight (number of non-zero elements) of ea~

Thus, the reliability can be expressed as the minimum over a sequence of Gaussian

random variables with distributions given by (3.10). Assuming that all the Zjs are inde-

pendently distributed, the cumulative density function (CDF) of A can be written as


FA(A) = 1 Prob(Zy > A)



d= dmin

where a(d) is the multiplicity of error events of weight d and Q(x) represents the Gaussian

complementary distribution function. The PDF can be obtained by differentiating the CDF.










Using the product rule of differentiation, the PDF is obtained as

fA, (A)j =X ex d)


diyddmin 4( (2xdge2) 32d e



(3.12)


Thus, even under the simplifying assumption of independent PRVs, the density function

obtained from first principles is very complicated and not suited for use in the analysis of

techniques involving reliabilities. For the Max-Log-MAP decoder with the noise scaling

implemented (as in (3.4)), the CDF and PDF of the reliability can be obtained by a simple
transformation as


FA.,( ) E F (2e2"X /Aa()=2a2/A~ (2La2). (3.13)

The subscript a is used in the above expressions to indicate that the soft-information is

scaled by the noise variance in the Max-Log-MAP decoder. Since A is non-negative and

continuous, the mean of the reliability can then be evaluated numerically as




oodmax 2a2X a4 d
ix~I (dA.16a (3.14)
d=dmin




3.2.2 On the Correlation Between Output Error Events

In Section 3.2, we model the reliability as the minimum of a number of Gaussian

random variables that are assumed to be independent. This assumption is valid only if all

the PRVs are independent. However, this is not a valid assumption. It is possible that the

different error events (e ") associated with the PRVs (eT ij)) Share the same path through










the trellis at certain time instants. At each of these trellis sections, the PRVs share com-

mon noise samples from the vector e and thus, the PRVs are correlated. Because of this

correlation, the expressions for the PDF and mean of the reliabilities given by (3.13) and

(3.14) can significantly differ from the simulation results, as will be shown in Section 3.6.

Thus, the correlations among the output error events should be considered in order for the

analytical expressions to agree with the simulation results. In [34], the authors account for

this correlation by obtaining the j oint multivariate distribution of Zj and using this distribu-

tion to compute the density function. However, this approach would involve computing a

covariance matrix involving different pairs of error events and using this covariance matrix

in the density function. This approach results in a very complicated expression. Even with

the independence assumption, the density function in (3.12) is complicated. Further, the

approach using the multivariate distribution offers no further insight into the behavior of

the reliabilities.

Note that the correlation between different PRVs arise because they share common

noise samples, which is a consequence of the associated error events differing from the

correct codeword in a common set of symbols. We introduce a simple approach to ac-

count for the correlation between PRVs by computing the correlation among different error

events. We first define the correlation between two error events el and e2 Of lengths 11 and

12 TOSpectively as


minm(ll ,12)
C Ei=1 er,si ( e2,i,(.5
ex~e"max(li, 12

where ej~i refers to the ith bit of error event ej and the '0' operator denotes the XNOR

operation. For example, 'll 10 10 11' and 'll 10 10 00 01 11' are two error events of length

8 and 12 respectively, and the correlation between the two events can be computed using

(3.15) to be 0.5. We account for the correlation between output error events by eliminating

some of the error events that are highly correlated and using the reduced set of error events

to compute the PDF/CDF of reliabilities. We define a correlation threshold Too,,, and










whenever two error events have a correlation value greater than Tcorr, the longer of the

two events is eliminated from the event set. The longer of the two events is removed from

the event set because performance is usually dominated by the low weight error events.

This process is continued until all remaining pairs of error events have correlation less

than Tcorr. We normalize the correlation by the longer of the two error event lengths to

ensure that events with very dissimilar lengths have a low value of the correlation. This

eliminates the possibility of discarding a long event which may share a common initial path

through the trellis with a small error event. Thus, a condensed event set is obtained within

which the events have low correlation value. We expect the small correlation between the

events in the condensed set to have a negligible effect on the independence assumption

used in deriving the PDF. It will be shown in Section 3.6 that if the summation in (3.12)

is performed over the condensed event set, the resulting values are strikingly close to the

simulation results for properly chosen values of Tcorr. Thus, the need for joint multivariate

distributions involving the covariance matrix of output error events is avoided using this

technique.

3.3 A Mathematically Tractable Density Function

The expressions for the density function of the reliability given by (3. 12) and (3. 13) are

complicated and not convenient for use in mathematical analysis of reliability-based tech-

niques. We address this issue with an ad hoc estimate of the PDF based on the following

observations:

The mean of the reliabilities obtained from (3.14) is very close to the simulation

results. (This fact will be substantiated in Section 3.6).

Given the correct decoder output, the conditional distribution of the soft output for a

bit is approximately Gaussian with variance approximately equal to twice the mean

(cf. [35, 36]).










Thus, we suggest modeling the reliability as the absolute value of a Gaussian random

variable that satisfies the symmetry condition, i.e.,


A =|X| X ~ N(p, 2p)1, (3.16)

where p is the mean obtained by numerically evaluating (3.14). The cumulative distribution

function (CDF) can easily found to be



FA X) Q ( ) Q ( ), A > 0 (3.17)
0, otherwise.

Differentiating with respect to A, the PDF of the reliability is,

exp (- exX2> p (- (LX2. A >
fA~x 2,a~ (3.18)
0, otherwise.

Unlike the expression in (3.13), the density function in (3.18) does not involve summations

and products and is expressed in closed-form. Indeed, we have to resort to numerical

computation to obtain the mean, p, but for problems involving explicit probabilities of

reliabilities, (3.18) is mathematically more tractable than (3.13). In Section 3.6 we provide

results that show this Gaussian approximation is extremely accurate.

3.4 A Closed-Form Expression for the Bit-Error-Rate of SISO Decoders

In this section, we demonstrate one application of the approximate density function of

reliabilities presented in the previous chapter (see 3.18). We will use the density function

to derive a closed form expression for the bit-error-rate of max-log-MAP decoding. The

probability of a bit decoding incorrectly conditioned on its reliability is given by


Fb |A=l c.


)91 3 (










The probability of bit error can be obtained by integrating Pb|x over the density of A\

given in (3.18).

Pb AI ( A) (3.202)

1 exp, (- (- )2 + e xp ( (+)2
4~L 4~L(3.21)


4~L (3.22)
2/Nr~ o cl L+)1 eX>'I"

=( Q = Q .0 (3.23)

Thus, the bit error rate can be expressed as a Q-functionl with the argument depending

solely on the mean of reliability. The mean of the reliability can be calculated using (3.14).

The probability of bit error depends on the weight distribution of the error events of a

code. The mean of the reliability encapsulates all properties of the code into a single

quantity thereby leading to a convenient expression for the bit-error-rate. The expression

of the bit error rate given in (3.23) can be used in various receiver-driven strategies that

rely on the receiver having an estimate of the bit-error-rate. For example, consider a real-

time streaming audio/video streaming application. These applications are loss-tolerant but

delay-intolerant. Thus, an ARQ scheme for such a scenario can be designed as explained

below. After decoding, the receiver estimates the number of bits that have decoded in error

by using (3.23). Since these applications can tolerate some loss, the receiver requests for a

re-transmission only when the number of bits in error exceeds a certain threshold.

3.5 Extension to Block-Fading Channels

We now extend the results presented to block-fading (quasi-static fading) channels.

In a block-fading environment, all bits in a packet experience the same channel gain. For



1 we consider this to be closed form since the Q-function is widely used in communica-
tion theory and can be computed accurately and efficiently










such a scenario, we characterize the reliability at the output of a max-log-MAP decoder

conditioned on a particular realization of the channel. Assuming coherent detection (the

channel gains are known to the decoder), the entire code space is rotated and scaled by the

channel gain. Then ML or MAP decoding is performed on the received vector in the new

code space. Thus, the reliability conditioned on the block-fading channel gain a~ can be

expressed using (3.4) as


Asl ~ a2~ = II nun, i) || -~~ ac|2- CM (3.24)

Note that aci" and acMC1L aTO COdewords in the rotated code space. Using the same ap-

proach as before, we can assume that the ML codeword is the true transmitted codeword.
Thus we have,

r = acMC L +e, (,e

= a(CML + e'), e'~ (02) (3.25)

Using (3.25) and (3.5) in (3.24), the reliability conditioned on the fading coefficient is

Asi a = nn { ||e ||2 2e /T (j) } (3.26)

Proceeding as in (3.8)-(3.13), the conditional CDF and PDF of reliabilities can be obtained



FA;,,/ ) = FA 2 Ai,/ 2 A2 2 A, 3.7

where FA (A) and fA (A) are defined in (3.11) and (3.12) respectively. The conditional mean
of soft information can then be obtained as


E[Aa = p~ 2 (3.28)


where p(x) is defined in (3.14).













-e (5,7) CC, Simulation
-*- (5.7) CC. Analysis
60~ (554,624,764) CC, Simulation
(554,624,764) CC, Analysis /
50-


= 40-







~ 0
0 13 46 7
Ebl (dB)

Fiue31:Temano eibiiisasafntono h sga-onos aiowe h
correatio bewe h uptero vnsaeinrd

3. NueiclReut
Intissciote nltia epesin ervdinte rvou etin recm


coes(C) ih boc iz o 10 bits are usd. The Ma-o-A imlmettino



obtained ~ ~ ~ ~ (5462,74 usin (314 are cmaewihteculmanobied frmsiuato. h



mial 1 +Dan1+D+D2 Or (5 7) in oca noain h osran egh7C a



diuces31 etimaes thato aresmabllerthate actual values.n As texpligaonedin Section 3.22 nd in

RregganiadTraa[4,teasmtion that all the output error events are independent



lead to overcoutiong hccue the analytical epsio driesults to produce coservtives results.










Table 3-1: Error event multiplicity of the (5, 7) convolutional code

d a(d) a(d)
All Events Tco,=0.7
dmin= 5 1 1
6 2 2
7 4 4
8 8 8
9 16 9
10 32 5
11 64 11
12 128 5
13 256 14
14 512 13
dmaz=15 1024 20


At low SNRs, there is a larger gap between analytically obtained values and the simulation

results when compared to high SNRs. This is because the performance at low SNRs is

dominated by blocks that decode incorrectly and hence the assumption (3.7) is violated.

To tighten the gap between the analytical and simulation results, it is required to con-

sider the correlation between error events. The number of error events with weight d (event

multiplicity) is shown in Table 3-1 for the (5, 7)s CC It is seen that eliminating events that

have a correlation value higher than the correlation threshold (Toorr = 0.7 in this case)

results in a condensed set of error events. We expect that using this condensed set of events

with low correlation will reduce the over-counting problem caused by the independence

assumption.

The mean of the reliabilities after accounting for the correlation between output error

events (as explained in Section 3.2.2) is shown in Figure 3-2. If the summation in (3.12) is

performed over a condensed set of error events (as shown in Table 3-1), and the mean then

computed using (3.14), it can be seen from Figure 3-2 that the analytical results are very

close to the true values even at low SNRs.

The PDF of reliabilities (eqn. (3.13)) for the (5, 7)s code is compared with the true

PDF in Figure 3-3. The true PDF was obtained experimentally by simulating the decoding
















LI. 50-


40-


S30-


S20-

10 ~8- (5,7) CC: d 5, d =15, T =0.7
4 mm max Corr
(554,624,764) CC: dmi=15, dmax30, T =0.7
mm maxCorr
0 1 2 3 4 5 6 7

E blN (d B)

Figure 3-2: The mean of reliabilities as a function of signal-to-noise ratio after taking into
account the correlation between output error events.


of a number of blocks that were transmitted over an AWGN channel. The reliability of

each bit was recorded and the true PDF was estimated from the histogram of the recorded

reliabilities. It can be seen from Figure 3-3 that results close to the true PDF can be obtained

when the correlation between output error events is considered in the computation of the

density function. The correlation is considered by evaluating the PDF in (3.12) over the

condensed set of error events shown in Table 3-1. Note that the analytical PDF is much

closer to the true PDF at higher SNRs.

The PDF obtained using the simple, ad hoc estimate in (3.18) is shown in Figure 3-4.

The mean, ft, that specifies the PDF is obtained numerically from (3.14). It is observed

that this ad hoc expression produces results that are closer to the true PDF when compared

to the expression in (3.12). Unlike the PDF given in (3.12), the ad hoc estimate produces

results that are very close to the true PDF even at low SNRs. The correlation between error

events can be accounted for in the ad hoc PDF estimate by evaluating p over a condensed

set of low-correlation error-events (as shown in Table 3-1). As before, accounting for the





















Eb oN=5 dB
L 0.08- ,





0.02

0 5 10 15 20 25 30 35 40 45 50



Figure 3-3: The PDF of reliabilities of the (5, 7)s CC for two different signal-to-noise
ratios.


correlation produces better results when compared to treating the PRVs as independent

random variables.

The closed-form approximation for the probability of bit error (Pb) given in (3.23) is

compared to simulation results in Figure 3-5. Results are shown for both the memory-2

and memory-6 codes. Since Pb depends solely on p, results are shown for two approaches

to compute p. In the first approach, p is calculated analytically using the expression given

in (3.14). In the second approach, p is obtained through simulation. This approach shows

the utility of the approximation to Pb in applications which require the receiver to have an

estimate of its bit-error-rate. For both scenarios it is seen that the closed-form approxi-

mation is very close to the simulation results. For the results in Figure 3-5, all the error

events were considered (correlation between error events were ignored) when the mean of

the reliabilities was calculated analytically using (3.14). If the mean is computed after ac-

counting for the correlation between error events, the closed-form approximation produces




















< 0.08 -i
L.
Eb No=5 dB
0.06-



0 "'


0g 5 10 15 20 25 30 35 40 45 50


Figure 3-4: The PDF of reliabilities of the (5, 7)s CC obtained using the simpler, mathe-
matically tractable expression given in (3.18).


better results (when compared to ignoring the correlation). This is not shown in Figure 3-5

for the sake of clarity.

The mean of reliabilities obtained when the coded bits are transmitted over a Rayleigh

block-fading channel is shown in Figure 3-6 for the (5, 7) convolutional codes. The ana-

lytical values are obtained by integrating the conditional mean in (3.28) over the density of

the fading amplitudes given by f (a) = 21e-o2n(a) where u(a) is the unit step function.
















10



10
O

.. 1 0



,X10


-*-- Analytic mean (E9)
SExperimental mean (E9)
- Simulation


Figure 3-5: The probability of bit error for max-log-MAP decoding of convolutional codes
evaluated using the closed-form approximation given in (3.23)


(5,7) code


















Simulation
e_ Analysis, Tcorr=0.7


Ca,
r
Wr
v ~J
v,~C
cuo
Ers,
r
n
coU
a,
rr~
~cO
O
11

cur
yo


78 9


Figure 3-6: The mean of reliabilities of the (5, 7) convolutional codes
signal-to-noise ratio of a block-fading channel.


as a function of


0 1 2 3 4 5N~dB 6 7 8


0 12 34 5 6

EblN, (dB)
















CHAPTER 4
CODED COOPERATION THROUGH COLLABORATIVE DECODING

The coded cooperation schemes summarized in Section 2.3 exploit the encoder struc-

ture of the error control code that is used; i.e., the nodes cooperatively encode the message

to form a more powerful codeword at the receiver. In this chapter we introduce a coopera-

tive strategy that involves collaboration in the decoding process; i.e, the destination and the

relay(s) cooperatively decode the message from the source. The system model is shown in

Figure 4-1. A distant transmitter broadcasts a message to a cluster of receiving nodes, one

(or many) of which could be the intended destination. If any of the other nodes decodes

correctly, it can use of one of the traditional decode-and-forward schemes to forward the

message to the destination. The more interesting scenario is when none of the nodes de-

codes correctly. In this case, we cannot use any scheme that depends on correct decoding at

the relays. The ampli@S-and-forward scheme could be used, but the cooperation overhead

is very high. Thus, alternative techniques are required to minimize the overhead.

The fundamental drawback of the decode-and-forward based approach is that each

relay forms its transmission based on its own decoding decisions and not on the decoding

decisions at the other nodes. For example, if the intended destination or one of the other

nodes in Figure 4-1 has decoded all but two of the bits correctly, then it is not necessary for

the other relays to forward information as in the previously described decode-and-forward

schemes. If the relays have some information about the decisions made at the other nodes,

they can accordingly schedule their transmissions to minimize the collaboration overhead.

Thus, in the model shown in Figure 4-1, there is no distinction made between the relay

and the destination. The nodes cooperate with each other to obtain some information about

the decisions made at the other nodes. This information helps minimize the cooperation

overhead required for correct decoding at one of the nodes.





















Distant Transmitter Cluster of receiving nodes

Figure 4-1: System model for collaborative decoding.


The system model also represents a broadcast scenario wherein all nodes are interested

in the message from the transmitter. After collaboration, the node that decodes correctly

can pass this information onto other nodes. An iterative technique to achieve cooperation

in this broadcast scenario with two cooperating nodes was proposed by Wong et al. [9, 38].

The basic principle of this technique with two cooperating nodes is shown in Figure 4-2.

The scheme proceeds in two stages as in the previous cooperation schemes (see Chapter 2).

In the first stage, the distant transmitter broadcasts its message to the cooperating cluster.

Stage 2 proceeds in multiple iterations with the iterations continuing until one of the nodes

decodes correctly or a fixed number of iterations elapse. In each iteration, the nodes first

exchange some coordination information. The coordination information is shown as the

solid diamond in Figure 4-2. The coordination information gives each node some infor-

mation about decoding at the other node. Based on this information, each node transmits

some information about the initial message sent by the source to the other node. On re-

ceiving this message, each node performs decoding and if either node decodes correctly,

cooperation is terminated. If not, the cooperation procedure is repeated as shown in Fig-

ure 4-2. The process of iterating between decoding and information exchange is referred

to as collaborative decoding.

An information-theoretic study of the system model shown in Figure 4-1 with two

cooperating nodes was studied by Draper et al. [39]. The nodes start collaborating after










Phase 1:Phase 2:
Transmission Collaborative Decoding








Iteration 1 Iteration 2
A: Coordination
B: Diversity Combining + Decoding


Figure 4-2: Principle of interactive/collaborative decoding with two nodes.


they receive the entire block sent by the transmitter. A round of conversation is defined as

a message exchange between the nodes with one message transmitted from each node to

the other. The sum of the transmission rates (sum-rate) between the nodes required for cor-

rect decoding is used as the performance criteria. A low sum-rate implies low cooperation

overhead. The authors prove that collaboration with multiple rounds of conversation be-

tween the two nodes can guarantee correct decoding (in the Shannon sense: arbitrarily low

error probability as the block length goes to infinity) with a lower sum-rate (overhead) than

collaboration consisting of one round of conversation. This is because information sent

in an earlier iteration serves as side-information at the receiving node and the transmitting

node can use more efficient coding techniques that use side-information at the transmitter

and receiver to encode future conversations. With reference to Figure 4-2, note that there

is no specific coordination information in this scheme. The information transmitted by a

node in one iteration serves as coordination information for the next iteration. Thus, each

node tailors its transmission based on information about decoding at the other node that it

received in the previous iteration.

4.1 Collaborative Decoding through Reliability Exchange

Wong et al. [9, 38] present an iterative approach to cooperation for the scenario de-

picted in Figure 4-1. The basic principle of their approach is as follows. On receiving










the message from the transmitter, the nodes perform MAP decoding on their received sym-

bols. Each node uses the bit reliabilities to determine which bits are unreliable and requests

additional information about these bits from other nodes. The other nodes transmit their

estimates of the a posteriori LLR for these bits. The original requester uses this informa-

tion as the corresponding a priori information in its MAP decoder and performs decoding

again. This process of information request and decoding is repeated for a few iterations.

For simplicity, the process of exchanging soft-information will be henceforth referred

to as reliability exchange. The information that passes between the different nodes will be

referred to as the overhead' in collaborative decoding. We build on the scheme of Wong

et al. [9, 38] by investigating the performance and overhead for two different classes of

reliability exchange schemes for multiple nodes (greater than two).

4.1.1 Collaborative Decoding through the Reliability Exchange of the Least Reliable Bits

In this section, we provide results for an extension of the scheme proposed in [9].

For all the results in this chapter, a rate 1/2 nonrecursive convolutional code with generator

polynomials 1+D02 and 1+ D+D02 is used to encode the information sequence. For conve-

nience, we refer to this code as the (5, 7) code, where 5 and 7 are the octal representations

of the generator polynomials. The encoded messages are transmitted over additive white

Gaussian noise channels using binary antipodal signaling and are coherently demodulated.

Each receiver decodes the received message using the BCJR [37] algorithm. Each node

then requests reliability information for a certain percentage of the least reliable informa-

tion bits by broadcasting the bit indices of those bits. Each node that receives the bit indices

replies with its estimate of the soft information for those bits. The node that requested the

information then uses these reliabilities as a priori information and runs the BCJR algo-

rithm again. In Wong et al. [9] it is shown that for a packet size of approximately 1000

bits encoded with a (5, 7) convolutional code, collaborative decoding with two receivers

provides performance very close to MRC at values of Eb N~O greater than 5 dB. Three iter-

ations of collaborative decoding was performed by requesting soft information for 7.5% of










the least reliable information bits in each iteration. The reason for requesting the least reli-

able bits (LRBs) is that most of the bits that decode incorrectly have low reliability values.

Using MRC would require exchanging all of the received coded symbols. The overhead in

bits, denoted by 8M/RC can be calculated as


8M/Rc =, x q, (4.1)


where NV is the size of the information message in bits, R, is the code rate and q is the

number of bits required to represent a (floating point) channel symbol. Note that (4.1)

represents the overhead contribution of a single node. Using the collaborative decoding

scheme mentioned above, the overhead contribution of a single receiver can be split into

two parts. The first part consists of the bit indices that a receiver broadcasts to request the

soft information of the LRBs, and the second part consists of the soft information that a

node transmits each time it receives an LRB request from another node. Thus, the overhead

for this scheme can be expressed as


Brus = N~ x a x NVx ( [log2 1V 1R 1) x q), (4.2)


where NJI is the number of iterations of collaborative decoding, a is the fraction of in-

formation about which reliability information is requested, NsR is the number of receivers

involved in collaborative decoding and [x] is the smallest integer greater than or equal

x. The first term in the summation on the R.H.S of (4.2) accounts for the bit indices that

need to be transmitted to request soft information, and the second term in the summation

accounts for the bits required to send out the soft information (each node receives NaR 1

LRB requests). Note that the size of the requests can be further reduced through source cod-

ing or by exploiting the time-correlation between the reliabilities of the bits in error [33].

Both (4.1) and (4.2) refer to the overhead per receiver. All the schemes in this paper will

be compared using the overhead contribution per receiver.










Table 4-1: Overhead of LRB-1 for different number of nodes.

Number of nodes Overhead (bits) % reduction
(relative to MRC)
2 2025 77.5 %
5 4050 55.0 %
10 7425 17.5 %


Generally five bits are enough to represent a (floating point) channel symbol accu-

rately [40], [41]. For a packet size of 900 bits, the overhead for MRC can be calculated

using (4.1) as 9000 bits. Ten bits are required to represent each bit index in packet of 900

bits, and if we perform three iterations of collaborative decoding with soft information of

5 % of the LRBs being requested, the overhead is 2025 bits for two nodes (using (4.2)).

Thus we see that performing collaborative decoding reduces the overhead by 77.5 % when

compared to the MRC overhead.

Performing collaborative decoding with three iterations of 5% LRB exchange will be

henceforth referred to as scheme LRB-1. LRB-1 has collaborative decoding overhead of

22.5 % of MRC overhead for a packet size of 900 bits and a cluster size of two nodes.

For the rest of the chapter, the overhead for collaborative decoding will be reported as

a percentage with reference to the MRC decoding overhead. Note that the overhead per

receiver for LRB-1 increases with the number of receivers. The overhead for LRB-1 for

different number of nodes is shown in Table 4-1. The reduction in overhead relative to

MRC decreases with an increase in the number of nodes.

In Figure 4-3, the performance of LRB-1 is shown for different number of collaborat-

ing receivers. We note that the performance saturates for more than four receivers. This

indicates that biasing the least reliable bits with a lot of apriori information from too many

receivers will not improve the performance significantly. This is because there are some

incorrectly decoded bits that may have relatively high reliabilities. This is again substan-

tiated in the next section. When least-reliable bits are exchanged, the incorrect bits with











10
1 Rx.
\ 2 Rx., LRB-1
s' s- 4 Rx., LRB-1
10- \ 6 Rx., LRB-1
\ \ t6 Rx., LRB-1 (10% LRB)
\ -r- 4 Rx., LRB-2
-3 -A- 6 Rx., LRB-2
S10 -e- I 8 Rx., LRB-2







10-7





0 1 2 3 4 5 6
EblNo (dB)

Figure 4-3: Performance of two collaborative decoding schemes in which receivers request
information for a set of least-reliable bits.


high reliabilities may never be corrected regardless of how much information is provided

for the LRBs.

An obvious method to improve the performance of LRB-1 is to increase the percentage

of LRBs requested. From Figure 4-3, we see that requesting 10% of LRB reliabilities

instead of 5% gives an improvement in performance of approximately 1 dB for a cluster

of six collaborating nodes. However, this increases the collaborative decoding overhead,

and our simulations show that the performance saturates for more than four receivers even

in this case. Another disadvantage of requesting more information is that as the number

of receivers increases, the time for information exchange also increases. Each receiver has

to send out a set of bit indices requesting reliability information, and then all the other

receivers have to respond. To coordinate this information exchange, a good MAC protocol

will have to be designed. This latency would not be acceptable in certain applications.

A simple extension to LRB-1 is to transmit all the soft information via a broadcast

channel and to have each node use all the received soft information, even if that node was










not the original requester. Since the nodes other than the one that requested information

also receive the soft information, they can make use of it as a priori information in their

next round of SISO decoding. Thus, for the nodes that did not request the information,

reliability information for a set of bits with random reliabilities is obtained. This scheme

with 5' request and three iterations will be referred to as LRB-2. LRB-2 has the same

overhead as LRB-1. The results in Figure 4-3 show that LRB-2 even outperforms LRB-

1 with 10% LRB exchange. Further, LRB-2 does not suffer from the saturation problem

like LRB-1. Hence, LRB-2 would be a better choice if exchanging LRBs was the scheme

chosen to perform collaborative decoding.

The biggest disadvantage of this scheme is that the per-receiver overhead grows lin-

early with the number of receivers (cf. (4.2)). Thus, if the number of nodes is large, even

requesting a very small percentage of LRB soft information might cause the overhead to

become larger than the MRC overhead. In the next section, an exchange scheme is pre-

sented that has an overhead which is independent of the number of receivers.

4. 1.2 Collaborative Decoding through the Reliability Exchange of the Most Reliable Bits

One way to significantly reduce the overhead is to prevent a node from transmitting

soft information more than once. From (4.2), we see that for block sizes of approximately

1000 and more than ten receivers, multiple transmissions of soft information contributes

towards more than 82% of the overhead per receiver. Suppose that, after SISO decoding,

each receiver selects a certain set of bits and broadcasts the reliabilities of these bits to

the other nodes. It is important to ensure that the nodes broadcast "good" reliability in-

formation, i.e., reliability information about bits that are decoded correctly. The critical

step in this scheme is to determine the set of bits for which a node will broadcast the soft

information. Since each node only sends out soft-information only once, the collaborative

decoding overhead per receiver is given by


OnMRB = Iv X X EX (r0g2 N] + q).


(4.3)













0.4 E IN =3 dB
co 0.3
S0.2-
a- 0.1-

0 2 4 Reliability 6 8 10
Density Function Of The Reliability Of Bits Decoded Correctly

SEb 0,=0 dB
E IN =3 dB
.0 b 0
co0.1-


-0 0.05-


0 5 10 15 20 25 30 35 40
Reliability

Figure 4-4: Reliability density functions associated with correctly and incorrectly decoded
bits.


Note that for this scheme, unlike the LRB-based schemes, the overhead per receiver is

independent of the number of receivers.

If a node broadcasts the soft information for a bit that was decoded incorrectly, using

this value as a priori information would degrade the performance of the other nodes. The

motivation behind our approach for selecting bits comes from observing the density func-

tions of the reliabilities associated with correctly and erroneously decoded bits. Figure 4-4

shows the density function of the reliabilities for a (5, 7) convolutional code with a block

size of 900 bits.

We note that the trend followed by the density functions is as expected; i.e., most of

the incorrectly decoded bits have low reliabilities and the correctly decoded bits have a

relatively high reliability. We observe that at an Eb N~o of 0 dB, the maximum value of

the reliability of a bit that decodes incorrectly is about half of the maximum value of the

reliability of a bit that decodes correctly. For values of Eb N~o greater than 3 dB, more than

50% of the bits that decode correctly have reliabilities greater than the maximum reliability











10-
1 Rx.
S10 Rx., MRB-1
2 Rx., MRB-2
10 o-"
-A- 4 Rx., MRB-2
6 Rx., MRB-2
3 1.~ 8 Rx., MRB-2
10-


10-4


.0 10-s


10-6 h


10-7
-2 -1 0 1 2 3 4 5 6
EblNo (dB)

Figure 4-5: Performance of two collaborative decoding schemes in which receivers broad-
cast information about a set of most-reliable bits.


of the incorrectly decoded bits. Hence, if a node broadcasts a small percentage of its most

reliable bits (MRBs), it is very likely to send out "good" soft information. These bits

will correspond to a set of bits with random reliabilities at the other nodes. Performing

three iterations of 10% MRB reliability exchange will be referred to as scheme MRB-1.

The collaborative decoding overhead (per receiver) of MRB-1 is calculated, using (4.3),

as 45% that of MRC. Though reliability information is exchanged for more bits than in

LRB-1 and LRB-2, the overhead is still smaller than in LRB-1 and LRB-2 for more than

five nodes.

Our simulations showed that the performance improvement is less than 2 dB even

with ten nodes when compared to the performance of a single receiver. This is shown in

Figure 4-5. The smaller performance improvement can be attributed to the set of bits that

are broadcast in each iteration. At the end of the first SISO decoding, reliabilities of 10%

of the most reliable bits are broadcast. Since we are biasing certain bits with "good" a

priori information, at the end of the next SISO decoding, the reliabilities for these bits will











LRB-1 Number of r~ee~ated *bit indices =1
8 3 i *i ****
12-* t**** ***
19 ** f* *** **

0 200 400 600 800 1000
r MRB-1 Number of repeated bit indices =102
0~j 3 *** **r *rr ** ** ***e ** -
213 rr** x*** ** *~ ****** *x- *
- 1* *** **ll *t ***t

0 200 400 600 800 1000
MRB-ll Number of repeated bit indices =0
*3- ***-***W*-tt*-*f -*X** ***
12 *i *f =* -** *f ** *f***f* *
t1) *** f** t* ***

0 200 400 00lci: 800 1000


Figure 4-6: Bit indices of reliabilities exchanged as a function of iteration.


become large, and it is very likely that these bits will lie in the 10% MRB set. So reliabilities

of these bits will be broadcast in the next iteration. But since the other nodes have already

received the information about these bits, biasing them with more a priori information will

not improve the performance significantly. This can be observed in Figure 4-6 in which

we show the bit indices broadcast in each iteration for one packet of 900 bits. An asterisk

on a bit position implies that reliability information about that bit was either requested

(LRB-1) or transmitted (MRB schemes). We see that for MRB-1, a very large portion of

bits are broadcast again in every iteration. In three iterations, the reliabilities of 102 bits

are broadcast again among the total of 270 bits transmitted. This constitutes around 37%

of the total bits sent. As the value of Eb,/No increases, there are fewer bits in error and in

order to improve the performance, these erroneously decoded bits need to be biased with

reliable a priori information. If a good percentage of the bits are repeated, there will be a

low probability that a priori information will be received for all of the bits that are in error.

A simple method to eliminate this problem is to give the nodes memory to remember

the set of bits for which soft information is transmitted or received. This ensures that bits










that are already likely to have good reliabilities after one iteration do not get biased with

more a priori information in the next iteration. Other bits are now given an opportunity to

receive reliability information. This scheme, which is just MRB-1 with memory, will be

referred to as MRB-2. In MRB-2, each node sorts its bits in ascending order of reliability

after the first SISO decoding. Then each receiver broadcasts 10% of the MRBs for which

soft information was not transmitted by any node in the previous iterations. Thus, in each

iteration a new set of bits get reliability information. This is illustrated in Figure 4-6.

In MRB-2, there are no bits for which soft information is transmitted in more than one

iteration. The performance of MRB-2 is shown in Figure 4-5. If in any of the iterations,

a node is not able to find a bit about which a priori information has not been transmitted

earlier, it does not send out any reliabilities. Thus, the overhead in MRB-2 is less than or

equal to the overhead in MRB-1, but the performance of MRB-2 is much better than that

of MRB-1 .

Note that adding memory to LRB-1 will not improve the performance significantly.

This is because in each iteration, a priori information biases the least reliable bits and their

reliability increases after SISO decoding. Thus, in the next iteration a new set of bits will

constitute the set of LRBs. Hence, there is only a negligible overlap in the set of LRBs in

each iteration. This can be observed in Figure 4-6, in which LRB-1 has just one bit that is

repeated in three iterations.

A good suboptimal variant of MRB-2 sends hard-decisions of the MRBs instead of

the soft decisions. This reduces the overhead for transmitting soft information from q (cf.

(4.3)) bits per information bit to one bit per information bit. Thus, for MRB-2 with three

iterations of 10% reliability exchange, the collaborative decoding overhead is only 33% for

a packet of 900 bits. For a reasonably large number of receivers, the hard decisions from

different receivers form a priori information that is sufficient to bias the information bits

to produce correct decisions at the output of the SISO decoder. The performance of this

scheme for six receivers is illustrated in Figure 4-7.











10-1 10-
Floating point
Hard-decision





10- 10-


1-4 10-


1 MRB-ll '1_ LRB-ll
10- 10-


10-7
-2 0 24 0 1 2 3
Eb/No (dB) Eb/No (dB)

Figure 4-7: Performance of suboptimal variants two collaborative decoding schemes in
which hard decisions are exchanged instead of soft information.


Note that this technique can be extended to any of the schemes discussed earlier. The

performance of LRB-2 with hard-decisions is also shown in Figure 4-7. We see that a loss

of approximately 0.5 dB can be expected for the suboptimal scheme when compared to the

original scheme.

By comparing the performance of the LRB and MRB schemes in Figure 4-3 and

Figure 4-5, it is seen that LRB-2 outperforms MRB-2 for a small number of receivers and

MRB-2 performs better when the number of cooperating nodes is large. This is because an

improvement in performance is obtained when bits that are decoded incorrectly get good a

priori information. In the LRB schemes, the bits that are likely to be decoded incorrectly

are specifically targeted leading to a improvement in performance. The MRB schemes

are more optimistic in nature. Each node broadcasts reliable information that may or may

not be useful to the other nodes. Thus, at one of the other nodes, good reliabilities are

received for a set of bits with random reliabilities. There may or may not be an incorrectly

decoded bit in these bit positions. When a few receivers collaborate it is not likely that all










the unreliable bits receive a priori information from other nodes. Thus, the performance of

MRB-2 is relatively worse for a small number of cooperating users. However, for a large

number of receivers, it is likely that many of the bits in error are covered. For example,

with 10% of MRB exchange and more than eight receivers, it is likely that information will

be exchanged for almost all the bits in a block of 900 bits, and hence the performance of

MRB-2 is better than LRB-1 for a large number of receivers.

The schemes that work with the MRBs also require less-complex channel access tech-

niques. If the number of nodes are fixed, a simple round robin of all the nodes can be used

to allow them to broadcast reliabilities of a certain percentage of their MRBs. For dynam-

ically formed ad hoc networks, a cluster head could be chosen that assigns the order in

which the nodes broadcast the reliabilities. When coupled with the fact that the overhead

of LRB schemes can become prohibitive for large cooperating groups, the simplicity of

MRB-2 and its performance in reasonably big cooperating groups makes it more suited for

practical implementation.

4.2 Guidelines for the Design of Collaborative Decoding Schemes

Note that the reliability exchange schemes described earlier can be considered to lie in

the realm of the decode-and-forward schemes with the relays transmissions consisting of

the soft information. However, unlike the decode-and-forward schemes mentioned earlier,

collaborative decoding does not depend on correct decoding at the cooperating nodes. The

SISO decoders in our schemes use bit-by-bit MAP decoding like the BCJR [37] algorithm,

and hence correct decoding is not needed to extract useful information for a small subset

of bits. Thus, our reliability exchange schemes are an improvement over the decode-an2d-

forward schemes since they are not limited by the capacity between the source and the

cooperating nodes.

All the results shown in this chapter correspond to transmission over AWGN chan-

nels. There is no diversity in its true sense on AWGN channels since all the channels are

equivalent. Thus, in order to study the diversity benefits of collaborative decoding using











100
-e- MRB-2
-MRC





10-2
3 All soft information
a F-~ exchange









10-s
U 0.5 1 1 .5 2 2.5 3 3.5 4
Eb /NO (

Figure 4-8: Performance of the MRB-2 scheme with eight nodes on a block-fading channel.


reliability exchange, we need to study the performance of the reliability exchange schemes

over fading channels. The probability of packet error for collaboration using MRB-2 with

eight receivers experiencing block-fading is shown in Figure 4-8. It is seen from the simu-

lation results that the MRB-2 scheme does not achieve full diversity (the curves for MRC

and MRB-2 are not parallel). It is observed that the performance of MRB-2 scheme with

eight receivers is over 5 dB worse when compared to MRC. For the sake of comparison,

the performance of a scheme that exchanges soft-information for all the information bits is

also shown. This is the best performance that any of the MRB schemes can achieve. The

performance of this scheme is around 3 dB worse than MRC. Thus, the best MRB scheme

requires more than twice the SNR to achieve the same performance as MRC. The reason

for the poor performance of the MRB scheme is as follows. If all nodes experience severe

fading, then it is difficult to extract useful soft information for use in the MRB scheme.

The reason for not achieving full diversity can be attributed to the fact that reliability ex-

change falls under the realm of the decode-and-forward schemes. Laneman [3]i proved that










the decode-and-forward schemes are not capable of providing all the diversity advantages

associated with cooperative schemes.

However, ampliS-and-forward schemes are guaranteed to achieve full diversity ad-

vantages [3]. Thus, it seems necessary to exchange information for the coded bits as in

the ampliS-and-forward schemes (information is exchanged for the information bits in re-

liability exchange) in order to achieve full diversity. MRC, which is an instance of this

category, also exchanges received symbol values. However, MRC combines information

in an inefficient manner with respect to overhead. Because of the use of error correction

codes, there are certain bits (trellis sections) about which reliable decisions can be made

without the exchange of information. The LRB schemes make use of this observation to

request for information for only those trellis sections that are likely to be in error. But the

LRB schemes request for the same amount of information for all the trellis sections. When

the index of a trellis section is transmitted by the node requesting information, all the other

nodes in the cooperating cluster will respond. Thus, all trellis sections in the set of LRBs

receive the same amount of information. However, it is not clear if a trellis section that

decodes incorrectly with a high reliability requires the same amount of information in or-

der to correct the decision as a trellis section that decodes incorrectly with a low reliability.

Ideally, the amount of information requested should be adapted to the reliability.

In fading channels, some nodes have better channels to the original transmitter and

hence have made a greater number of correct bit decisions. Such nodes should share more

information with other nodes when compared to the relays with bad channels. All the nodes

transmit an equal amount of information if MRC or one of the LRB/MRB schemes is used.

These observations lead to three principles that should be kept in mind while design-

ing cooperative protocols:


Pl. In order to obtain full diversity advantages, it is necessary to exchange information
closest to the RF front end, i.e., the received symbol values (soft demodulator out-
puts) .










P2. The information exchanged in the cooperating cluster should be adapted to the errors
in each packet. If reliabilities are used to adapt the collaboration content, the amount
of information requested for each trellis section should be based on the reliability.

P3. Nodes with good channels should share more information than nodes with bad chan-
nels.


Note that MRC and the reliability exchange schemes described in this chapter violate

all of these principles. In Chapter 6, we present an improved-LRB (I-LRB) scheme that

is based on these principles. It will be shown that I-LRB achieves full diversity (same

as MRC) in the number of cooperating nodes. The performance of I-LRB is better than

comparable ampli@i-and-forward based collaborative approaches while still achieved per-

formance close to that of MRC with a fraction of the collaboration overhead. The I-LRB

scheme exploits the correlated reliabilities at the output of a SISO decoder in order to re-

duce the overhead. In the next chapter, we investigate the underlying decoder mechanism

that leads to correlated bit reliabilities. This understanding will prove useful in the design

of the I-LRB scheme.
















CHAPTER 5
ON CORRELATED BIT ERRORS AT THE OUTPUT OF A MAX-LOG-MAP
DECODER

We begin this chapter by first motivating the need to understand the correlated na-

ture of reliabilities. We will use LRB as an example to show how correlated reliabilities

can help in decreasing the overhead associated with collaborative decoding. In the LRB

schemes, the output of the SISO decoder is used to identify the bits with low reliabilities,

and information is requested for such bits because these bits are more likely to be in error.

However, as noted in [33], errors at the output of a decoder are typically time-correlated.

Since the LRBs are more likely to be in error, the LRBs are also correlated i.e., if a bit has

decoded with a low reliability it is likely that the adjacent bits also have a low reliability.

This can also be observed in Figure 4-6 where it is seen that in each iteration, information

is requested for sets of consecutive bits. Thus, in LRB a node will request for information

about a set of consecutive trellis sections with low reliabilities. This is a conservative ap-

proach because correcting one LRB will have an effect on the neighboring LRBs due to

the correlated nature of the output of the decoder. In other words, if two bits are strongly

correlated, it is likely that combining information for one bit will influence the decision at

the other bit. Thus, it is not necessary to request for additional information for the entire

set of consecutive LRBs. In order to decrease the cooperation overhead associated with

collaborative decoding, it is necessary to understand the interaction between decoded bit

reliabilities. In this chapter, we show that the error event that separates the ML codeword

and a competing codeword in the max-log-MAP decoder can succinctly capture the corre-

lated nature of bit errors. We also show how this error event can be efficiently computed

using computations that are already performed in the BCJR max-log-MAP decoder. In the










next chapter, we use this error event that separates the ML and competing path to design a

collaborative decoding scheme that improves on the performance of LRB.

5.1 Terminology and Notation

The terminology and notation introduced here are specific to rate 1/2 convolutional

codes. It is straight-forward to generalize these to rate k/n codes.

input and' output labels: An input label is used to indicate the input that causes a

particular state transition in the code-trellis, and an output label is used to indicate

the corresponding output caused by that state transition.

path and' event: A sequence of valid state transitions in the trellis is called a path

through the trellis. Note that every codeword represents a path through the trel-

lis. Because the code is linear, the difference between any two codewords is a path

through the trellis. Such a path is also called an event.

valid state: A valid state lies on any path through the trellis. Because the trellis starts

and stops in the all-zeros state, not every state is a valid state near the ends of the

trellis.

metric: The Euclidean distance between the received vector r and any codeword c,

||r c||2, iS referred to as the metric2 Note that the metric is a maximum-likelihood

(ML) decision statistic for additive white Gaussian noise (AWGN) channels.

The notation used in this Chapter is given in Table 5-1.

5.2 Revisiting Max-log-MAP Decoding of Convolutional Codes

The soft-output of a max-log-MAP decoder for codewords transmitted on an AWGN

channel with noise variance a2 can be written as (see Chapter 3, eq. (3.3 ))


L(ui Ir) = min Il 12 mmn .l c (5.1)
cEC' 2(T2 cECZ 2(T2




2 Note that a metric is associated with a particular codeword. In other words, each
codeword has a different metric.













Table 5-1: Notation used in this chapter


Block-size (the number of sections in the code-trellis).
The input to the encoder at time i; i.e., the input label for trellis section i.
For binary codes considered in this paper ui E {0, 1}. We will refer to ui as
the information bit.
The output of the encoder at time i. This is a two-dimensional vector con-
sisting of the two parity bits output by the encoder at each time. If BPSK is
used for modulation then c( E {-1, 1}, Vj E {0, 1}. Since every c, corre-
sponds to a particular branch in the trellis ci will used as the output labels
for the branches in the trellis at time i. We will use parity bits or coded bits
to refer to the output labels at any particular time in the trellis.
A valid codeword (output labels on a path through the trellis). Appropriate
subscripts will be used to indicate the codeword being considered.
The received vector corresponding to ci.
The received vector corresponding to c.
Cz, Ca+1. ,b-1) Abl bs ISSimilarly defined.
Input label at trellis section I in codeword c.
Component I in codeword c. Note that this refers to a particular bit in the
corresponding output label.
{c : ui(c) 0}O i.e., the set of all codewords with input label 0 at trellis
section i.
{c : ui(c) = 1} i.e., the set of all codewords with input label 1 at trellis
section i.
The set of all valid codewords. c ci Uc .
Set of states in the trellis. For the memory-two code considered in this paper,
there are four states.
Therefore, S = {, 1, 2, 3}.
State of the encoder at time k. Note Sk E S.
The set of valid states at time k~ 1 that have branches leading into state S
at time k.
The set of valid states at time k + 1 that have branches emerging from state
S at time k~.
The state that codeword c passes through at time k~.
log(P(si s), rf)
log(P(si S, & Si-1 = S ))
log(P(rN+1|Si S))
TepfeSents a Gaussian distribution with mean p~ and variance O.2


[c ci]


c = [c7, 9,4

ri = [r4 r



c(1)


C








S(~ s)

S(s ~)

Sk~(c)
as(s)
Y (S', S)
Pits)
nl(p, O.2)










Note that the maximum-likelihood (M~L) cI, Ick~I
the received vector r,

cMLt = arglinlll$ C 12
cEC

It is possible that there is more than one ML codeword (although this occurs with proba-

bility zero for the unquantized AWGN channel), in which case we arbitrarily choose one

of the paths as the ML codeword.

O Definition2 1. Comp~eting c. rck'n
i is the path that is closest to the received vector among all paths that differ from the ML

path in the input label for trellis section i,


ccomp ar~gml? Ir -Cl 2. (5.2)
{cec:ui(c)qui(CML)

As in the case of the ML codeword, there may be more than one codeword that satis-

fies (5.2), in which case the tie is broken by choosing one of the codewords arbitrarily.

Note that although there is only one CML,~ there may be many different clomp FOr different

values of i.

Then the reliability for bit i, which is the magnitude of the soft information in (5.1i),

can be expressed as

As 1 Lu~) | ccmp 2 -7- CML2 (5.3)


Note that ccm is referred to as c~j in Chapter 3. We have replaced the notation to simplify

exposition, and to stress the fact that clomp is competing with cMLt for the hard-decision on
trellis section i. Since the distance between r and the ML codeword is smaller than the

distance between r and any other codeword, the difference in (5.3) is always positive. A

high value of reliability implies that the ML path and the next best path with the opposite

input label for bit i are far apart, and hence there is a lower probability that the decoder

chose the wrong path and made a bit error. Thus, reliability is a measure of the correctness










of the bit decision. This has also been shown via simulation results in [32, 33]. A bit with

high reliability is more likely to have decoded correctly than a bit with low reliability.
The I-LRB scheme that is described in Section 6.4 utilizes both the bit reliabilities and

knoledvlge of CMVL and Compp in determining which information should be exchanged in the
collaborativep decodingr mprocss In the next section ,r we dtail howr cML~ andl com can be

determined for a particular trellis section.

5.2.1 Obtaining the ML and Competing Path using the BCJR Algorithm

Following the development in [42], the soft information in (5.1) can be expressed as


L (ui Ir) = max _(s)+is,) s)-mxa_()+iss+i),(54

where ak 8s), yk(s', S), and Pk(s) are defined in Table 5-1.

It can also be shown that (see [42])


as(s) =max (agi_i(s') + yi(s', S)) (5.5)



yi(S', S) oc -|r ,|,(5.7)

where S' E S(~ S) and S' E S(S ~) are defined in Table 5-1, Qo(0) = 0 and PN(0) = 0.

Thus, it is seen from (5.7) that yi(S', S) is proportional to the branch metric (cf. [43]),

P(rslc4), used in the Viterbi algorithm (where the constant of proportionality depends on

only the channel coefficient and signal-to-noise ratio).

Let the ordered pair of states (Si_l, aS) that maximizes the first term in (5.4) be (sf_7, s+).

Let (s _7, a ) be the ordered pair of states that maximizes the second term. By compar-
ingr1 (51) andl (54) it is se eln that o~ne ofC the o~rdemred pairsl ofC saes+~ (s_7 a) or (S _l, a )

corresponds to cMLt, while the ot~the orderedT pair correpondsrl to clmp For example, if










then ag_1(CML. St-1 Si CM 8,~ and as1ciomp) 3-, iC m) Thus, when

computing soft-output for trellis section i, it is possible to identify the branches through

the trellis at time i that correspond to the ML path and the competing path.

We now introduce two theorems that will enable us to obtain CMLt and com in a

straight-forward manner using the computations performed by the decoder.

Theorem 1: The branch selection theorem

Given the state in the code trellis at time k, Sk, = s' and the vector of received symbols r,

the following statements are true:

(a) Trace-back: The state-transition s* s', where Sk-1 = s* = argmaX Gk-1 8~) 7k ~s,

is a branch on the codeword c* given by, c* = argmin ||r~ c~ ||2
{cec:sk(c)=s }

(b) Trace-forward: The state-transition s' s*, where Sk+1 = s* = argmaX (yk+1 s)

is a branch on the codeword c* given by c* = argmin|r -c |2


s )n>,


k~+1 8 ,


Proof- To prove the Trace-back procedure in Theorem 1, we

Lemma.

Lemma: ak~(s) oc mmn ||rf c ||l for any state s at time
c6C:sk(c)=s
valid codeword

Proofi By mathematical induction.

Note that to(0) = 0. Then ai1(0) is computed using (5.5) as


first prove the following



k that is on the path of a


at (0) = 0 + y1(0, 0)


(5.8)


because there is only one valid state leading into state 0 at time 1. Similarly, az1(2) =

0 + y1(0, 2). The lemma does not apply to the other states at time 1 because they are not

valid states for a rate 1/2 convolutional code initialized to state 0 at time 0. So using (5.7),

the lemma holds for k = 1.










Assume that the lemma holds for time k 1. Then


COk S )= maX Gk~-1~s + rk (s, s*)) (5.9)


OCsES(->s*) cEC:s l(c)=s (; 5.0

oc mm | c|2 (5.11)
c6C:sk(c)=s*

where (5.10) follows from the assumption about the claim, and the last equation follows

from (5.7). Thus, the claim is true for time k. The principle of induction completes the

proof.

Remark: From the lemma, ck (s) is proportional to the partial-path metric (log P(rf |ct))

[43] of the surviving path at state s at time k in the Viterbi algorithm when the branch met-

ric is the Euclidean distance.

Proof of the trace-back theorem:

Compare the trace-back theorem and (5.5). The trace-back theorem chooses the previous

state (si_l) that corresponds to the branch involved in computing the alpha for the current

state (as). Since ck (s) is proportional to the partial path metric of the surviving path leading

tO Sk = s, the branch involved in computing cOk(s) is part of the corresponding surviving

path.

Thus, conditioned on the current state, the trace-back theorem chooses the previous

state as the state at time k 1 on the surviving path at time k. The proof of the trace-back

procedure follows because the surviving path has the best partial-path metric (min ||r~ -
c(||2) among all pathsc that pass thrmouh Sk~ =

The trace-forward theorem can be proved in a similar manner by comparing the trace-

forward theorem with (5.6).

Theorem 2: The conditional path selection theorem.

Given a state transition at time i, i.e., siea = s' and as = s*, let C, represent the set of all

paths through the trellis (codewords) passing through this transition at time i. That is,

C, = {c EC : aS_l(c) = S', si(c) = S*}. Then the sequence of state transitions










(So, S1, S-a S' S* -2 1, S* } given by


Ski = argmax {ak-_i 8) +k-i+1 8 sk-+),i=2 ,.. (5.12)

Si, argmax {/7k+i 8) + k+i 8 +i-1, s) }, i = 1, 2, .., NV k (5.13)


corresponds to the codeword c* that is closest to the received vector r among all the code-

words in C,, c* = argmin||r c||2
c 6C'r


ProofI The proof follows by repeated application of the trace-back and trace-forward

theorems.

As mentioned earlier, the state transitions from time i 1 to i that correspond to the

ML path and the competing path can be obtained during the computation of the soft-output

for bit i. Given the states ag_l(CMlL), and Si(cMlL), the ML codeword cMLt can be obtained

using the conditional path selection theorem. The codeword output by the conditional path

selection theorem is closest in Euclidean distance to the received vector among all paths

that pass through ag _1(cMlL), and si(cMlL), and is thus the ML path. Similarly, the competing

path can be obtained using the conditional path selection theorem given ag_1l(ciomp), and

se (ciomp -
As noted in the Lemma, the trace-back theorem always chooses the previous state

(asi_) that corresponds to the branch involved in computing the alpha for the current state

(as). Similarly, the trace-forward theorem always chooses the next state (si41) that cor-

responds to the branch involved in computing the beta for the current state(si). This ob-

servation enables an efficient modification of the BCJR algorithm that enables computing

cMLt and clomp FOr any trellis section i. During the computation of the asi(s), s E S, i s

{1,..., N}, record the state si_l = s' that maximizes as_1l(s') + yi(s', s) as the previous

state for s. Similarly, during the computation of the /94(S), S E S, is { 1,..., N}), record

the state siay = s' that maximizes /9441(s') +yi(s, s') as the next state for s. Then given the

branch in the code-trellis corresponding to cMLr Of Comllp for at time i, the entire codeword










can be obtained by recursively following the states recorded in this way. Thus no additional

computations are required to compute CMLt and ccomp. By recording information about the
states that lead to the maximum values in (5.5) and (5.6) during the BCJR algorithm, CMLt

and clomp can easily be obtained through a series of table-lookups.

During the trace-back (or trace-forward) procedure, if Si- CML) Si- Comp) foT

some k, then the sequence of state-transitions obtained for any time before k will be the

same for CMLt and ccomp. Similarly, if or Si+k CML) Si+k Ccomp), then the sequence of

state-transitions will be the same for CMLt and clomp FOr any time after k. It will be shown

in Chapter 6 that I-LRB only requires knowledge of trellis sections where clomp and cMLt

differ. Thus, it is sufficient to execute the trace-back and trace-forward procedures until

Sifk CML; Silk Co~mp -

It is well known that the soft-output/reliabilities of adjacent bits in a convolutional

code are correlated [34]. For max-log-MAP decoders, it was found through simulations

that groups of neighboring bits have the same reliability. Since r and cMLt are the same

for all trellis sections, (5.3) implies that bits decoding with the same reliability should

have the same competing path. By using the technique described above, and observing the

competing paths for adjacent bits that decoded with the same reliability it is verified that

the competing paths are indeed the same for those bits. Thus, for max-log-MAP decoders,

the strong correlation between the reliabilities of adj acent bits is reflected in the choice of

the same competing path in the code-trellis for those bits.

5.2.2 On the Utility of Competing Paths in the Design of Collaborative Decoding

In this section, we provide a brief outline of how explicit knowledge of these compet-

ing paths can help reduce cooperation overhead. It is well known that errors at the output

of a convolutional code are bursty, and similarly the soft-output/reliabilities are tempo-

rally correlated [34, 33]. It was shown that the reason for this correlation is that bits that

are close to each other in the trellis may often share the same competing codeword/path.

For max-log-MAP decoding, such bits have exactly the same reliability, as can be seen

from (5.3).










A particular bit decodes incorrectly if the ML path is not the transmitted codeword.

Thus, all the neighboring bits that also choose the same ML path will also decode in error

leading to time-correlated errors. The fact that errors in convolutional codes occur in bursts

was also noted in [33].

Consider the use of the LRB scheme. Consider a LRB (say bit i) that decoded in error

with chfL and clomp as the competing paths. This means that bit i chose the wrong path

through the trellis as the ML path. In this case, it is likely that the next closest path (with

respect to the received vector) with the opposite bit decision corresponds to the true trans-

mitted codeword i.e., it is likely that clomp is the true codeword. Now suppose that receiving
additional information about bit i from other nodes is able to correct the decision. This im-

plies that the additional information changed the ML path to be clomp. Assume that there
were adj acent trellis sections (i + k), that had originally decoded with the same reliability

(same choice of ChfL and clomp) before requesting additional information for bit i. Since

additional information changed the choice of the ML path to clomp FOr bit i, then it is likely

that all the adjacent trellis sections that originally decoded with the same reliability as bit

i will also choose clomp as the ML path. This will also correct the errors at these adjacent

trellis sections. Thus, by combining information for one trellis section it is possible to cor-

rect bit decisions at other trellis sections also. The I-LRB technique introduced in the next

chapter uses this idea to reduce overhead by not requesting information for all the trellis

sections that decode with the same competing path. I-LRB requests minimal additional

information that will flip the decision from ChfL to clomp, thereby correcting all bit errors
associated with the incorrect choice of ciop
















CHAPTER 6
IMPROVED LEAST-RELIABLE-BITS COLLABORATIVE DECODING FOR
BANDWIDTH-CONSTRAINED SYSTEMS

In this section we describe the Improved LRB (I-LRB) collaborative decoding scheme

for convolutionally encoded communications. We modify the LRB scheme described in

Chapter 4 to satisfy the design guidelines mentioned in Section 4.2. We use LRB as our

baseline scheme because the information exchanged in the LRB schemes targets the trellis

sections that are most likely to have decoded in error. In addition to conforming with the

design guidelines of Section 4.2, the I-LRB scheme also exploits correlated reliabilities to

reduce the cooperation overhead.

The system model for collaborative decoding is shown in Figure 4-1. A distant trans-

mitter broadcasts a packet to a cluster of receiving nodes. ARQ is not possible because

of the power limitations of the mobiles and the distance to the transmitter. Cooperation

overhead is critical in bandwidth-constrained systems. In these systems, the cooperation

overhead is upper bounded by a maximum value. This constraint may be necessary in or-

der to provide a minimum throughput guarantee to the distant transmitter. Since there is no

feedback channel to the distant transmitter, it will continue to transmit messages at a certain

rate. In such a scenario, cooperation (exchange of messages) cannot continue indefinitely.

If the collaboration proceeds for a long time, then the process of collaborative decoding

will interfere with additional transmissions from the source. Therefore it is necessary to

constrain the cooperation overhead in order to ensure that collaboration does not conflict

with transmissions from the source.

We first begin with a broad perspective of collaborative decoding in bandwidth-constrained

environments. After introducing an baseline ampli@S-and-forward scheme (MRC variant)

for such systems, we develop the I-LRB scheme.










6.1 Collaborative Decoding with Constrained Overheads

In collaborative decoding, the message at the source is packetized and encoded with a

code that permits SISO decoding. The codeword is then broadcast to a cluster of receiving

nodes that will attempt to decode the message. The received message for symbol i at node j

can be modeled as

rigj = ajxi + us,j, (6.1)

where xi is the transmitted symbol at time i; aj is the channel coefficient at receiving

node j, which we assume is fixed over each packet; and us~j is white Gaussian noise. In all

that follows, we consider rate R = 1/2 codes, but it is straight-forward to generalize the

work to other code rates.

If any node in the cluster decodes the message correctly, then we consider the mes-

sage to be successfully received. If none of the nodes decodes the packet correctly, then

the nodes begin the process of cooperating to receive the message. In the collaborative

decoding schemes presented in Chapter 4 the nodes use the outputs of the SISO decoders

to select which information should be exchanged and which nodes should transmit that

information. The a posteriori probability (APP) log likelihood ratio (LLR) at the output

of a SISO decoder is a real number and is commonly referred as the soft output. The sign

and magnitude of the soft output for an information bit represent the hard decision and the

reliability of that decision, respectively [30]. The sample mean of the reliabilities at node

j, pyl, is a measurement of the overall reliability of the decoder's decision. We assume that

the nodes exchange the pys after the first decoder iteration and that combining occurs at the

node with the largest py, which we refer to as the "best" node.

The nodes then broadcast information about a selected set of the received symbols

(as in A-F) to the best node. The cooperative process can go through several iterations,

each of which consists of three parts. In the first part of the iteration, the nodes identify

information to be exchanged. In the second part, a selected group of nodes will transmit

that information to the best node. In the final part of each iteration, the nodes decode the










message and check whether it has decoded correctly. The process stops if any of the nodes

has decoded the message correctly or if the limit on the number of iterations is reached.

In each iteration, we constrain the maximum number of bits that can be transmitted in

the cooperative process. This may be necessary in many systems to ensure that the cooper-

ative process does not conflict with the transmission of additional packets from the source.

We specify the constraint as a portion of the total information exchanged in maximal ratio

combining (MRC). Let NV be the information block size, R be the code rate, Nrm, be the

number of receivers, and q be the number of bits used to quantize the channel observations.

Then the cooperation overhead for MRC is OM/RC = NGnirz/R bits. The large OM/RC Will

be not acceptable for many applications. Hence, we constrain the amount of information

that can exchanged in the cooperating cluster to be a fraction p of Omre. Note that this places

a limit on the maximum amount of information exchange in the cooperative process for a

particular packet; however, the actual amount of information exchange for any particular

packet may be much less because we allow the cooperative process to terminate whenever

the packet is decoded correctly. In each iteration, we constrain the overhead to pemre/Neiter,

where NViter is the total number of iterations allowed.

Note that there are three main differences between the collaborative decoding scheme

described in Chapter 4 and the collaborative decoding scheme described above for bandwidth-

constrained systems.

In the collaborative decoding scheme of Chapter 4, combining is performed at all

nodes. For example, all nodes in the LRB scheme request for additional informa-

tion about their LRBs. In collaborative decoding for bandwidth-constrained systems,

combining is performed only at the best node. This solves the problem of overhead

being proportional to the size of the cooperating cluster in the LRB schemes.

In collaborative decoding scheme of Chapter 4, information from all other nodes is

combined in each iteration. For example, all nodes broadcast the APPs for the LRBs










requested by a node in the LRB scheme. In collaborative decoding for bandwidth-

constrained systems, information is carefully chosen from a select subset of nodes.

This is keeping in accordance with design principle P3 given in Section 4.2.

*Unlike collaborative decoding of Chapter 4, the amount of information combined

in each iteration is constrained in collaborative decoding for bandwidth-limited sys-

tems.

We next describe the two main cooperative schemes that will be compared in this

chapter. The first, which we call constrained-overhead incremental MRC (COI-MRC), is

an iterative form of maximal-ratio combining in which the overhead is constrained as ex-

plained above. The second scheme is a collaborative decoding scheme called the improved

least-reliable bits (I-LRB) scheme. Because of the complexity of this scheme, we first

provide an overview of it in Section 6. 1.2, and contrast it with the LRB scheme described

in Section 4.1.1. We then develop the tools required for I-LRB, and provide a detailed

description of I-LRB in Section 6.4.

6.1.1 Constrained-overhead Incremental MRC

Consider first an implementation of full MRC in a group of collaborating radios. Each

node (other than the best node) scales its received symbols by the fading gain, quantizes

them, and transmits them to the best node. As mentioned above, this would result in a

large overhead. A variant of this scheme that can offer even better performance than MRC

with lower overhead is incremental MRC (I-MRC). In incremental MRC, the cooperation

is done over several iterations.l In iteration i, the node with the i + Ith largest pi trans-

mits information about all of its received symbols to the best node2 Then the best node




SWe thank an anonymous reviewer of a previous paper for proposing this cooperative
scheme.

2 Note that for quasi-static fading channels the value of pi is generally dominated by the
fading coefficient. If two nodes have similar fading coefficients, this approach allows us to
choose the one whose received information provides more confidence in decoding.










combines that information with its own received symbols and any previously received in-

formation, decodes the message, and checks whether the message has decoded correctly.

If the message decodes correctly, the cooperative procedure terminates, and thus the aver-

age overhead of I-MRC is typically much less than MRC. In addition, because decoding

is performed after each information exchange, I-MRC can achieve a slightly lower error

probability than MRC.

Although I-MRC has a lower average overhead than MRC, the overhead in each iter-

ation consists of all of the received symbols from one node, and the maximum overhead is

the same as MRC. As explained above, it may be necessary to constrain the maximum over-

head. Thus, we introduce a constrained-overhead I-MRC (COI-MRC) scheme. In COI-

MRC, the overhead is constrained to pl~qlVz/R bits. We allow a total of Nviter = New, 1

iterations, so in each iteration, pl~qlVz/(RI~iter) bits are exchanged. The information in

each iteration represents a set of pNV1rz/(RI~iter) received symbols from the best node that

has not previously transmitted all of its received symbols. The set of symbols is uniformly

selected from the remaining set of symbols at that node. Once all of the symbols at a node

have been transmitted, then the next best node (in terms of ps) will transmit information

for its received symbols.

After each round of information exchange, the best node uses MRC to combine the

new information with its previously received information. The best node then decodes the

message. If the message decodes correctly or if the maximum number of iterations has

been reached, collaboration ends. Otherwise, another iteration of information exchange is

performed.

6. 1.2 Overview of Improved Least-Reliable Bits Collaborative Decoding

The MRC-based schemes are effective approaches for cooperation. However, these

schemes are "dumb" schemes in the sense that they do not utilize information that is avail-

able that could improve the performance for the same constraint on the collaborative over-

head. SISO decoders offer the ability to assess which bit decisions are reliable and which










are unreliable. By first exchanging information that can improve the unreliable bit deci-

sions, we may be able to achieve a better tradeoff between overhead and performance.

The scheme that we propose is based on the least-reliable bits (LRB) schemes that

were described in Section 4.1.1. In these LRB schemes, each node identifies the set of

bits with the least reliabilities (i.e., smallest magnitude of the APP LLR) and requests

information for these bits from every other node. Our technique improves on the prior

LRB schemes in several ways:

1. We request information at only the best node, so that the overhead from the informa-

tion requests is reduced.

2. We utilize the fact that the set of LRBs is often correlated, and we develop techniques

to avoid requesting too much information because of this correlation.

3. The set of nodes that respond to a request sent by the best nodes transmit quantized

values of their received symbols and not the APPs as in the LRB scheme. This

satisfies P1 of the design guidelines in Section 4.2.

4. The amount of information required to correct a bit depends on its reliability, so we

present a technique to adapt the amount of information based on a bit's reliability.

This satisfies P2 of the design guidelines in Section 4.2.

5. All nodes do not respond to a request sent by the best node. When the best node

request for additional information about a trellis section, only the next best node

that has not already transmitted information about that trellis section transmits its

received symbols. Since coded symbols are combined starting with the second best

node, it is likely that the second best node will transmit more coded symbols than

another node. This satisfies P3 of the design guidelines in Section 4.2.

6. Not all bits that surround an unreliable bit will necessarily help to correct that bit,

so we present a technique to select the set of bits which are most likely to correct an

unreliable bit.










7. LRB attempts to reduce cooperation overhead by targeting individual trellis sections

that decoded incorrectly. I-LRB reduces cooperation overhead by targeting compet-

ing paths through the trellis that potentially decoding incorrectly. By targeting the

competing paths, I-LRB has the potential to correct all the bit errors associated with

this path.

We refer to the new approach as the improved LRB (I-LRB) scheme. In this paper, we

demonstrate how the goals of the I-LRB scheme can be achieved for convolutionally en-

coded communications by utilizing information generated in the max-log-MAP implemen-

tation of the BCJR decoding algorithm. The details of I-LRB with convolutional codes are

given in Section 6.4.

Recall that in I-LRB, the best receiver sorts the trelli s sections according to the reliabil-

ities, and requests information from the other collaborating nodes to improve the decoding

of some set of least reliable bits. The LRBs will often occur in groups because they are

caused by the same error event, and thus it is only necessary to provide enough informa-

tion to correct the error event to correct all of the bit errors caused by that event. Moreover,

we show that some of the received symbols corresponding to a LRB may not be useful in

resolving the most likely error event. In the rest of this chapter, we first propose a sim-

ple analytical technique that can be used to determine how much information needs to be

transmitted for each least reliable bit. We then describe how the decoder can use informa-

tion about the ML and competing paths to decide which information can most efficiently

correct any bit errors in the LRBs. Finally, we provide a detailed description of the I-LRB

scheme for convolutionally encoded communications.

6.2 Estimation of Request Size

During the collaborative decoding process, the decoder must act under the assumption

that any LRB is in error, when in fact the error probability for even the least reliable bit

is generally less than 0.5 (otherwise, we would just invert that bit decision). Given the

reliability of a LRB, the decoder needs to estimate the amount of information that should










be requested to correct the bit. The most likely error event for bit i is the event that separates

CMLt and ccomp, which is given by


.s = CML 6H Ccomp >


where a represents the XOR (addition or subtraction in a binary field) operator. For linear

convolutional codes, as considered in this paper, ei is a codeword.

The reliability in (5.3) can be further simplified as


As, = -gr'T (CML Ccomp). (6.2)

If the channel from the distant transmitter to the collaborating cluster in Figure 4-1 does

not have unit channel gains, then the reliability at the jth receiver can be expressed as


Ag~ = -ga r:IT (CML -cop,(6.3)

where we have suppressed the dependence of CMLt and clomp On the particular receiver num-

ber, j.

The decoder tries to estimate the amount of information required to change the deci-

sion from the ML path to the competing path (assuming that this will correct the error). Let

CML(k) and ccomp (k) denote the kth parity bit on the ML path and competing path for infor-

mation bit i, respectively. If CMlL(k) = clomp(k), then that parity bit does not provide any

distinction between the two paths in the trellis. Thus, requesting information about such

parity bits from the other collaborating nodes will not be helpful in resolving between these

two paths. In the most likely case, in which either CML. Of Com~p is the correct codeword, the
decoder will only improve its decision if additional information is received for those parity

bits for which the decisions of ML and competing codeword are different.

O Definition 2. Candidate set ofparity bits Si for trellis section i: The set of parity

bits for which the decisions of the ML codeword (cML) and competing codeword (ccomp,,

are different.












Si = {k : cMlL(k) / ccomp(k)} = {k : e"(k) = 1} (6.4)

Once the candidate set of parity bits is obtained, the decoder tries to estimate the number

of parity bits from the candidate set Si that have to be requested from other nodes in order
for the decoder to decide in favor of com inStead of cML-

Let r* be the received vector after requesting a coded symbols from another receiver"

say receiver 2. The decoder estimates the minimum number of additional coded symbols

(h:) that will change the decision from CML to ceomp with probability greater than some
threshold. That is, after receiving the additional information, we desire a high probability

that


||r* clomp 12 < -l* CML 12 (6.5)

2r*T (CML C omp) < 0 (6.6)

2rT (CML C omp) +t C 2r) (1)(CML 1) C omp(1)) < 0, (6.7)


where rl is the subset of the candidate set that has been transmitted in this iteration, and r/

corresponds to the symbols received due to those transmissions; i.e., r* = al*r + a2*r'

(a*~ is the conjugate of the fading coefficient at receiver 2). Using (6.3), we obtain 2a~rT

(cML C omp) = 2o.2As, where As is the reliability of trellis section i before combining.

Note that in the above equationS CML and ccomp, refer to the ML path and competing path

encountered in computing the soft-output for trellis section i before receiving additional

coded symbols from receiver 2.

As previously mentioned, the decoder assumes that the parity bits in the candidate set

are in error. Then we can calculate the required value of a under the assumption that the



3 r* is obtained by combining the original received vector r and the additional symbols
using maximal-ratio combining.










all-zeros CW has been transmitted, in which case ceomp(1)= 1 and cML.(1) = -, VI E Si.

Since the all-zeros CW is the true transmitted codeword, r'(1) ~ Ni(a2, 2). Thus,


X, i- > 2a r 1 CL1 comp())~ (-a n, 1 6;a s2)



Thus the decoder estimates that after the first-retransmission, correct decoding is made if

Xi < --2a2Az.

The decoder estimates the number of coded bits a for which information is required

from another receiver as follows,


min P(Xi < -2a2Az) > 8 (6.8)

min Q a~i-2~62 j> 8, (6.9)


where a is the number of parity bits retransmitted and 8 is a predefined threshold. Thus,

the decoder estimates the number of bits to be retransmitted as the minimum number that

would cause the decoder to decide in favor of clomp inStead of cMLt with a probability that

is at least 8. This provides the minimum number of bits that is most likely to correct

bit i if it is in error. P(Xi < -2a2Az) will be referred to as the correction probability

after combining (P,). Thus, the receiver requests the minimum number of coded bits such

that Pc exceeds 8. Therefore, by request coded symbols for a parity bits, I-LRB has the

potential to correct all the bits that chose the same competing path (i.e, decoded with the

same reliability). For example, assume a = 1, and that there are 3 bits that decoded

with the same competing path. Suppose the additional copies of this 1 coded symbol is

enough to flip the decision from CMLr~ to clomp, then the bit decision at the 3 trellis sections

that originally chose coomp as the competing path will also change. Thus, I-LRB has the

potential to correct 3 bit errors by requesting only 1 coded symbol. Note that LRB would

have requested for 3 reliability values.








71

Table 6-1: Instantaneous SNR estimation for trellis sections based on the average of the
instantaneous SNRs of the parity bits in the candidate set

Output label Output label Estimate
on trellis section i 1 on trellis section i of the instantaneous SNR
fo htfor cML_ FO Cm for trellis section i
11 -1-1 (|r41 |+|r |)/2
-1 1 1 -1 "I
-1 1 1 1 |r |
--1 1 --1 -1 |rf


6.3 Estimation of the Request Set

After the decoder estimates a from the candidate set, it needs to select the subset of a

parity bits in Si for which information will be requested from another receiver. We estimate

an instantaneous SNR for each trellis section involved in the error event ei that separates

cML; and clomp to decide the candidate set for collaborative exchange. The receiver sorts the

trellis sections in the error-event according to the instantaneous SNRs, and requests for a

parity bits from the trellis sections with low SNRs.

The concept of instantaneous SNR was proposed in [44] for use in selecting which

symbols should be retransmitted in an ARQ scenario. Several different schemes were

considered in [44], and the one described here was found to offer the best performance.

If for a particular trellis section i, cML and Miomp differ in only one parity bit, then the

instantaneous SNR of that section is equal to the absolute value of the received symbol

corresponding to that parity bit. If for a particular trellis section i, cML and ciomp differ

in both parity bits, then the instantaneous SNR of the trellis section is the average of the

instantaneous SNRs of the two parity bits. The receiver selects a parity bits corresponding

to trellis sections with the lowest SNRs from the candidate set. The instantaneous SNR of
a particular trellis sectionn for di~fferent output l abels on CML and lomp IS given in Table 6-1.

Note that all possible output labels can be obtained by interchanging the output labels on

the ML and competing paths in each row of Table 6-1.










6.4 Detailed Description of I-LRB Collaborative Decoding

With the above approaches to estimate the request size and the request set, we can

describe I-LRB collaborative decoding in detail. Upon initiation of collaboration, the nodes

broadcast their ps to determine the best receiver. Starting with the best receiver, let the

receivers be numbered RX1 to RXNT.. The second best receiver RX2, transmits its fading

coefficient a2 to RX1. RX1 needs the fading coefficient to estimate the number of coded

symbols that have to be requested.

Let the number of iterations in collaborative decoding be denoted by Ivter. For the

results presented in this paper, we set Ivier = NVrz 1. Given the overhead constraint,

RX1 limits the number of bits that can be exchanged in each iteration to penMR~C Niter. In

each iteration, RX1 sorts the information bits according to the reliabilities, and obtains the

competing path for each LRB using the technique described in Chapter 5.2.1. Then for

each LRB,

1. RX1 estimates a using (6.9).

2. RX1 obtains the candidate set and the set of parities to be requested based on the

instantaneous SNRs.

3. RX1 broadcasts n, and the indices of the parity bits that need coded symbols from

another node.

4. For each bit index, the best node that has not previously transmitted information for

that bit will transmit information for that bit. Each node scales it received symbols

by the channel coefficient and broadcasts that information for a bit. If a > |Si | (the

number of coded symbols required is more than the size of the candidate set), then

coded symbols are obtained from the next best receiver until a total of a symbols are

transmitted.

Consider an example to illustrate to illustrate step 4 above. Assume that the codeword

shown in bold in Figure 6-1 is the competing path for bit i and that the ML path is the all-

zeros path. Assume that thi s i s the first iteration in which bits in thi s candidate set have been











c=[- 00,...,~~00, 11.101100.., 0
1 i-1i i i+1 N
1/10
3 - u i = 0, ci = 1 0

liolX u (c=1

110~c (2(2 1) + 1) =1


1/11 0/ S(1 ~)= {0, 2}
0 --- ---g g s,(c) =1
0/00

time: ... i-2 i-1 i i+1 .


Figure 6-1: The code-trellis for the (5, 7) convolutional code with examples of the notation
used in this chapter.

selected to receive information from the collaborating nodes. For the sake of exposition,

assume that trellis sections i 1, i, and i + 1 have increasing instantaneous SNRs in that

order. If a = 2, information about ci_, will be obtained from RX2. If K = 3, infOrmation

about ci_, and c) will be obtained from RX2. If K = 7, COded symbols for all the parity bits

in the candidate set are obtained from RX2, and coded symbols for ai_, are obtained from

RX3. Once the appropriate number of coded symbols are combined for the LRB, RXI

requests for coded symbols for the next LRB that has a different competing path. There

may be other adjacent trellis section with the same reliability. But by requesting for a

parity bits, all the bits that had originally decoded with the same ML and competing paths

are corrected with a probability that is greater than 8. Hence, if multiple trellis sections

have the same competing path (and hence the same reliability), it is enough to consider

only one of them to compute the request set and request size. If the information received

is able to change the decision from CMLt to ccomp, then all the trellis section that chose the

same competing path will be corrected. Thus, I-LRB exploits time-correlated reliabilities

by not requesting information for all adj acent bits that decoded with the same reliability.










As previously described, coded symbols for a particular trellis section from a particu-

lar collaborating nodes are only transmitted once. Using the previous example, assume that

the branch from state 2 to state 1 has already received coded symbols from RX2 because

this branch was part of a different competing path for some other bit that had a reliability

less than that of bit i. Then when n = 3, information for for ci_, and c}+2 will be obtained

from RX2 (aSSuming that coded symbols for these bits have not been obtained from RX2

earlier). Also, if coded symbols for c) is required in the next iteration, it should be ob-

tained from RX3, and not RX2. This procedure is repeated until a total of I'he :c/Newer,

bits are exchanged within the cluster. Note that this includes the bits required to index the

parity bits requested by RX1. In practice, all of the information requests can be performed

at the beginning of an iteration, followed by each receiver's response starting from RX2

to RXN, RX1 combines all of the received information with its previously received in-

formation using MRC (on a bit-by-bit basis). If RX1 is able to decode correctly or the

maximum number of iterations has been reached, then the collaborative decoding process

terminates. Otherwise, another iteration of collaborating decoding is performed.

6.5 Results

In this section, we present the performance of our collaborative decoding scheme.

For all the results in this chapter, a rate 1/2, memory-three, non-recursive, non-systematic

convolutional code with generator polynomials 1 + D2 and 1 + D + D2 ((5, 7) in octal

notation) is used for encoding at the distant transmitter. The message consists of NV = 900-

bit packets. For all the results, the channel between the distant transmitter and the cluster

of cooperating nodes is assumed to be a quasi-static Rayleigh fading channel, where the

fading is constant over each packet. For all results, the number of collaborating iterations

Nviter = N,., 1.

The block error rate for I-LRB and COI-MRC is shown in Figure 6-2 for different

number of collaborating nodes. For these results, a 5% overhead constraint with respect to












-Q Solid Lines: COI-MRC
-- Dashed Lines: I-LRB








-2 0
Ebl

Fiue -: rbaiit f bok ro for difee t nme fclaoaigndswe h
ovred osrin sfxe t5%o h oeha frMC
the vereadreqire fo MR wa imose. I isobsrve tht ILRBouterfrmsCOI
MRC~ for alszsothcoprtnclseshw.Iisaosen ta teganofeedb
I-R nressa tenme of col bortn oe nrae oreape ihatre
blc ro at f1-,ILR upromsCIM Cb ppoiaey2 Bwe hr
ar 8cllboa innodes The pefrac of= onyoend n ooeain sas
show fo th aeo oprsn igercie civsabokerrrt f1-
ataon 3dBE O ece oprto uigILBprvdsagino rud2









signa-t-nie 62 roabtio(SR of 2lc ro dB, adwith reigt ub o collaborating receiver I-LRB inceae





thke troughpute by almost 30%R withfrespett COI-MRC, n by 350%xmael witBhrespc to a

singe n ode.rtn oe.Th efrac fonyoend n ooeain sas















035-





0x 15 -i
01;


0 05



-2 1 02 35





The blok erro rate f COI-RCandI-LRB scompard Lins FOigure o ifrn


cooertin vehed s sow i Fgue -5 I is see tahatLis I-LRB prom etrta






ofeight -.Trugptfrdfern ubro collaborating nodes ise shw nFgue66fr ifrn overheadcosrit.I


cofsran I-MR (CI-MRC wi5% th n overhead consrait anhtofasnlercieraeas


shw.ThouhIMChstebs block error rate afCIMCad -R cmongallted scees(e Figure 6-4 o ifrn



4), irthas lowerha throuhput whincompared65 t sse toa I-LRB or OIMR. hu, t s lear that


I-RC achieves g oodr block error rate perfa corner at the ot of ige overhead. Ihet is also




obevdta h hogptof I-LRB decreases when the overhead constraint a tt snl rcise are laxed






This implies that the gain in block error rate is not significant as more combining is al-

lowed in the cooperating cluster. The increase in overhead caused by relaxing the overhead













100






10
I-MRC =

Or
-8 g-M C5


S-e-CR-MC 15


P -- COl-MRC 25% 1

10-2H -+- I-LRB 5% 1 'sm
-o- I-LRB 15%
-a- I-LRB 25%


-5 -4 -3 -2 -1 0 1 2 345




Figure 6-4: Probability of block error overhead for COI-MRC and I-LRB with eight coop-
erating nodes, and different constraints on the overhead.


C2-RC5
COl-MRC 15%
\- COl-MRC 25%
.2 20- \ -+- I-LRB5%
I-LRB 15%
R,- I-LRB 2 5%










-5 4 -3 1
PrE 0,

Fiue65:Aeae oprto n ovrha fo O-R n -R wt ihoprtn
noe, n dfern cntait o heoered

cosran ovrsaostedces nbokerrrtlaigt oe hogpt

Thus th -R ceei aal o rvdn ag nras ntruhu ihavr



















,"0.25
g 0.25// '~I-MRC



0.05 '


0.1Eb 0





Figure 6-6: Throughput for COI-MRC and I-LRB with eight cooperating nodes, and dif-
ferent constraints on the overhead.


small overhead. This is because I-LRB targets the trellis-sections which are likely to be

in error, and adapts the amount of information combined for these sections based on their

reliabilities.

The average number of iterations required by the COI-MRC and I-LRB schemes is

shown in Figure 6-7. It is seen that collaborative decoding is terminated faster in I-LRB

than in COI-MRC. Since the amount of information combined in each iteration is the same

in I-LRB and COI-MRC, and since I-LRB requires fewer iterations, the overhead of I-LRB

is smaller than that of COI-MRC (as shown in Figure 6-5). For example, at an SNR of 0

dB and a 5% overhead constraint, I-LRB requires fewer than half the number of iterations

required by COI-MRC. It can be verified from Figure 6-5 that the overhead of I-LRB is

indeed around 50% of COI-MRC at 0 dB (for the 5% constraint).










































-e COl-MRC 15%
.5.5 RCOl-MRC 15%
E E- I-LRB 5%
s$ e -o 1-LRB 15%
.t 5 I-LRB 25%








1O .5C

Eb

Fiue67 vrg ume fieain e colaoatiedcdn tepeurdb
eight reeves
















CHAPTER 7
CONCLUSION AND DIRECTIONS FOR FUTURE RESEARCH

7.1 Conclusion

In this work, we have studied the idea of user cooperation from a decoding perspec-

tive. Our objective is to achieve good performance with low collaboration overhead. Our

schemes are based on a very simple idea. If the cooperating nodes have some information

about decoding at other nodes, the cooperation overhead can be significantly reduced. We

introduced a framework called collaborative decoding to help develop cooperation strate-

gies that are efficient in terms of the cooperation overhead. In any collaborative decoding

scheme, the cooperating nodes iterate between a process of information exchange and de-

coding. The information exchange portion of collaboration provides information about

decoding at a particular node to other nodes. The other nodes use this information to de-

cide what information to transmit. Collaborative decoding relies on the use of soft-input

soft-output (SISO) decoders. The magnitude of the output of the SISO decoder is called

the reliability and is an indication of the correctness of the decoded bit. In collaborative de-

coding, the nodes use reliability information from the SISO decoder to adapt the messages

that are exchanged among the cooperating nodes. This is the biggest difference between

collaborative decoding and conventional cooperation strategies wherein the information

exchanged during collaboration is predetermined and fixed. Unlike previous cooperation

strategies, collaborative decoding provides a convenient approach to trade performance for

overhead, and collaborative decoding scales easily to multiple cooperating nodes.

We also develop guidelines for the design of collaborative decoding strategies. We use

these guidelines and design a novel approach called improved lea~st-reliable-bit (I-LRB)

collaborative decoding for user-cooperation in bandwidth-limited scenarios. The I-LRB

scheme has the advantage over previously proposed cooperation strategies in that it adapts










the information exchanged in collaborative process based on the a posteriori probabilities

at the decoding node. There are two levels of adaptation in I-LRB. First, I-LRB adapts

the set of bits for which information is requested based on the reliabilities. Second, for

each chosen trellis section, I-LRB adapts the number of coded-symbols exchanged based

on the reliability. I-LRB reduces the overhead by not combining coded symbols for all of

the trellis sections that correspond to a single error event.

The advantages of the I-LRB scheme come from exploiting information generated

in the BCJR decoder. We show that temporal correlation in reliabilities arise due to the

same choice of competing paths for different trellis sections. We show that the competing

paths can be explicitly calculated using computations that are already performed in the

decoder. By observing competing paths that occur in the decoder, I-LRB can request for

the minimum number of coded symbols that can correct all the trellis sections that choose

that competing path in their reliability computation. Simulation results show that I-LRB

achieves a lower probability of block error with a lower average collaborative information

exchange than the COI-MRC scheme. The results show that I-LRB can provide a 30%-60%

improvement in throughput with respect to traditional cooperation schemes. The overhead

required for this improvement is less than 5% of the overhead of traditional combining

schemes like MRC. Thus, I-LRB offers an efficient approach for collaboration when the

maximum collaborating overhead is constrained.

7.2 Directions for Future Research

We now present areas of potential research that can be pursued using the ideas present

in this dissertation.

*The performance of collaborative decoding can be studied in multiple access wireless

networks by abstracting the results in this dissertation into a network simulator. Since

collaboration among a group of nodes introduces interference in the network, it is

not clear if collaborative decoding can actually improve the throughput of the entire

network.










* So far we have used fixed convolutional codes that are guaranteed to achieve the

minimum bit error rate among all the convolutional codes with the same constraint

length. We have not considered the issue of code design. It is not clear if a convo-

lutional code that provides the best error-performance in a single-user scenario will

also provide the best error performance in a cooperative setting. For example, we

found that a recursive systematic convolutional (RSC) code provides a lower prob-

ability of block error than an equivalent non-recursive convolutional code for the

same cooperation overhead. It will be interesting to study the reason behind this

observation.

* We have studied user-cooperation in a bandwidth-constrained setting. User-cooperation

can also be studied in an error-costrained system. In these systems, the nodes should

achieve a certain bit/block error rate through cooperation. We can compare collabo-

rative decoding to conventional cooperation schemes to see which technique achieves

the required error rate with the lowest overhead. One way to design a collaborative

decoding scheme for this system is compute the number of trellis sections for which

information is to be requested in order to achieve the target error rate. The closed

form expression for the bit error rate can be used to compute this. Using the closed

form approximation it is seen that a target bit error rate translates to a target mean

of the reliabilities. Thus, after a round of decoding, each node can compute its mean

of the reliabilities and then estimate how much information to request in order to

achieve the target mean.
















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BIOGRAPHICAL SKETCH

Arun Avudainayagam received the B.E degree in electronics and communication engi-

neering in 2000 from Anna University, India, and the M. S degree in electrical and computer

engineering in 2001 from the University of Florida. Since January 2002, he has been work-

ing towards his Ph.D. degree at the University of Florida. He interned with the WCDMA

systems test group at Qualcomm, Inc. from May 2005- Dec. 2005, where he was involved

in access stratum performance evaluation of Qualcomm's WCDMA modem chips. His

research interests include wireless communications, codes on graphs and belief propaga-

tion, iterative decoding techniques, cooperative communication, and applied error control

coding.