Table Of Content

OCTOBER2005 VOLUME53 NUMBER10 IECMBT (ISSN0090-6778) TRANSACTIONSLETTERS Coding IterativeDecodingandChannelParameterEstimationAlgorithmsforRepeat–AccumulateCodes.... ......... .W.OhandK.Cheun 1597 Fading/Equalization ASoftDecodingSchemeforVectorQuantizationOveraCDMAChannel... ...... ..... ......... ...... ......H.H.Nguyen 1603 CombinedEqualizationforUplinkMC-CDMAinRayleighFadingChannels ........ ...... I.Cosovic,M.Schnell,andA.Springer 1609 Modulation/Detection GeneralizedAPPDetectionofContinuousPhaseModulationOverUnknownISIChannels .. ......... .... A.HanssonandT.Aulin 1615 TransmissionSystems CapacityandCoverageIncreaseWithRepeatersinUMTSUrbanCellularMobileCommunicationEnvironment. ..... ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... .....M.N.Patwary,P.B.Rapajic,andI.Oppermann 1620 NoncooperativePower-ControlGameandThroughputGameOverWirelessNetworks ..... ......... ..... Z.HanandK.J.R.Liu 1625 TRANSACTIONSPAPERS Coding TurboCodesWithRate-m=(m+1)ConstituentConvolutionalCodes ..... ...... ........ ...... .. C.DouillardandC.Berrou 1630 DC-FreeError-ControlBlockCodes ..... ..... ...... ...... ......... ...... ..... ...... ..F.Zhai,Y.Xin,andI.J.Fair 1639 DigitalCommunications JointSource/ChannelCodingforMultipleImages . ...... ...... ......... ...... ..... . Z.Wu,A.Bilgin,andM.W.Marcellin 1648 DiversityCombiningWithImperfectChannelEstimation .. ...... ...... ......... ..... ...... R.You,H.Li,andY.Bar-Ness 1655 Fading/Equalization Rate-AdaptiveTransmissionOverCorrelatedFadingChannels .... ...... ...... ........ ...... ...... ...T.JiandW.Stark 1663 TheExpectation-MaximizationViterbiAlgorithmforBlindAdaptiveChannelEqualization.. ......... ... H.NguyenandB.C.Levy 1671 Modulation/Detection MultirateModulation:ABandwidth-andPower-EfficientModulationScheme...... ..... ......... ...... ...... ..H.B.Peek 1679 ANewPerformanceBoundforPAM-BasedCPMDetectors ...... ...... ........ ...... ...... ..... E.PerrinsandM.Rice 1688 DesignofTurbo-CodedModulationfortheAWGNChannelWithTikhonovPhaseError.... ......... .Y.Zhu,L.Ni,andB.J.Belzer 1697 MultipleAccess AdaptiveOpportunisticFairSchedulingOverMultiuserSpatialChannels ... ...... ...... ........ ...... .. C.LiandX.Wang 1708 SystolicArrayImplementationofaReal-TimeSymbol-OptimumMultiuserDetectionAlgorithm.... ...... ...... ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ..C.-C.Lu,J.-Y.Hsu,andC.-C.Cheng 1718 (ContentsContinuedonBackCover) (ContentsContinuedfromFrontCover) OpticalCommunication AnOverlayArchitectureforManagingLightpathsinOpticallyRoutedNetworks.... ........ ..V.A.VaishampayanandM.D.Feuer 1729 Multiple-SubcarrierOpticalCommunicationSystemsWithSubcarrierSignalPointSequence. ......... .. S.TeramotoandT.Ohtsuki 1738 SpreadSpectrum OntheTradeoffBetweenTwoTypesofProcessingGain... ...... ...... ......... ..... ...... ....E.FishlerandH.V.Poor 1744 TransmissionSystems AnIterativeExtensionofBLASTDecodingAlgorithmforLayeredSpace–TimeSignals. ... ......... .... K.LiuandA.M.Sayeed 1754 BlindCarrierFrequencyTrackingforFilterbankMulticarrierWirelessCommunications .... ...... ...... ...... ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... ....V.Lottici,M.Luise,C.Saccomando,andF.Spalla 1762 AbstractsofForthcomingManuscripts.... ..... ...... ...... ...... ........ ...... ...... ...... ...... ..... .... 1773 InformationforAuthors.. ...... ...... ..... ...... ...... ....... ........ ..... ...... ...... ...... ..... .... 1776 PAPERSSCHEDULEDTOBEINTHENEXTISSUE NOVEMBER2005 TransactionsLetters ImprovingthePerformanceofTurboCodesWithaSimpleProtectionSchemeforError-ProneBitPositions..... .. W.Oh,Y.Kim,andK.Cheun InfiniteSeriesRepresentationsAssociatedWiththeBivariateRicianDistributionandTheirApplications..... ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... ..D.A.ZogasandG.K.Karagiannidis Bit-ErrorProbabilityforOrthogonalSpace–TimeBlockCodesWithDifferentialDetection.. ......... .... T.P.Soh,P.Y.Kam,andC.S.Ng OntheProbabilityofErrorofAntennaSubsetSelectionWithSpace-TimeBlockCodes .... ...... ........ ....... ..... ... D.J.Love CooperativeSpace–TimeCodingforWirelessNetworks... ...... ...... ....... ....... ...... ...... .... A.StefanovandE.Erkip AdaptiveResourceAllocationforMultiaccessMIMO/OFDMSystemsWithMatchedFiltering ..... ......... ..Y.J.ZhangandK.B.Letaief Low-ComplexityMaximum-LikelihoodDecoderforFour-Transmit-AntennaQuasi-OrthogonalSpace–TimeBlockCode .... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... .... M.-T.Le,V.-S.Pham,L.Mai,andG.Yoon TransactionsPapers NonsystematicTurboCodes..... ...... ..... ......... ...... ...... .....A.Banerjee,F.Vatta,B.Scanavino,andD.J.Costello,Jr. ASubbandApproachtoChannelEstimationandEqualizationforDMTandOFDMSystems. ...... ......... ...... .D.MarelliandM.Fu OptimizedDecisionFeedbackEqualizationforConvolutionalCodingWithReducedDelay.. ....... ........ ... J.-T.LiuandS.B.Gelfand IterativeDesignandDetectionofaDFEintheFrequencyDomain.. ...... ...... ........ ...... ...... . N.BenvenutoandS.Tomasin SimulationofRayleighFadedMobile-to-MobileCommunicationChannels.. ...... ........ ......C.S.Patel,G.L.Stüber,andT.G.Pratt Capacity-ApproachingTurboCodingandIterativeDecodingforRelayChannels .... ..... ......... ...... ...Z.ZhangandT.M.Duman WirelessPacketSchedulingBasedontheCumulativeDistributionFunctionofUserTransmissionRates ..... ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ..... D.Park,H.Seo,H.Kwon,andB.G.Lee ExpandingtheSwitchingCapabilitiesofOpticalCrossconnects.... ...... ...... ..... ......... ...... ...... ..... ....C.-T.Lea OntheCombinedInput-CrosspointBufferedSwitchWithRound-RobinArbitration.. ....... ....... R.Rojas-Cessa,E.Oki,andH.J.Chao TimingUltra-WidebandSignalsWithDirtyTemplates .... ...... ...... ......... ..... ...... ...... ..L.YangandG.B.Giannakis AbstractsofForthcomingManuscripts.... ..... ...... ...... ...... ...... ........ ...... ...... ...... ..... ...... .... 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LucentTechnol.BellLabs.,MurrayHill,NJ UNn.iAv.L-oDfHTAeHxIaRs,,SDpaacllea–sT,imTXe,OFDM&Equalization OpticalCommunication UF.nSivA.NoTfUCPCiIs,aW,iPrieslae,ssItSaylsyt.Performance PK.iHelOUEHnEivR.,,CKoideiln,gG&erImtearnaytiveProcessing A.EAstNimASaTtAioSnO,P&OUCLoOdSi,nIgterativeDetection, UI.nAivN.DoOfNSOtVraICth,cOlypdtiec,aGlNlaestgwoowr,ksU&.KD.evices UE.nSivE.RoPfEDLI’NA,qSuyinlcah,rLo’nAizqautiiloan,&ItaSlyensorNetworks AU.niKv..KofHWANaDteArNloI,o,CWodaitnegrl&ooI,nOfoNrm,CataionnadTaheory Univ.ofMichigan,AnnArbor,MI R.HUI,OpticalTransmission&Switching TexasA&MUniv.,CollegeStation,TX K.NARAYANAN,Modulation,Coding,&Equalization STe.xAaRsIYInAsVtIrSuImTAeKnUtsL,,AAlrpehaaEredtittao,rGA UDn.iKv..HofUNKTaEnRs,asP,hLotaownriecnNcee,twKoSrks CA.riTzEoPnEaDSEtLaEteNLUIOnGivL.U,,TSeymncpher,oAniZzation&Equalization TWe.xEas.RAY&AMN,MUondivu.l,aCtioolnl,egCeodSitnagtio&n,ETquXalization SSU.BNAYTABLAuMffaAl,oS,pBreuafdfaSlop,ecNtrYum&Estimation UKn.iKv.IToAfYAEMssAe,x,PChootlocnhiecstNere,twUo.Krk.s& GU.niVv.IToEfTTMA,oEdeqnuaa,lizMatoidoenn&a,FIatadliyngChannels UC.nSivC.HoLfEGAErLiz,oCnoad,inTgucTshoeno,ryA&ZTechniques UNn.iCv..BofEAAUlLbIeErUta,,WEidremleosnstoCno,mAmBu,nC.Tanhaedoary OsFakibaerU-Onipvt.i,cOWsiarkelae,ssJapan NC.a-tLio.nWalATNsGi,nEgqHuaulaizaUtnioivn.,Taiwan UTRnIiEvU.-oKfIAENlbTerRtUa,OENGdm,Coondtoinng,AThBe,oCryan&adTaechniques YU.nFivA.NoGf,FWloirreildeas,sGNaeitnweosrvkislle,FL WHo.fCst.raKWUOnNivG.,,OHpemticpaslteNaedt,wNorYks RKo.yWalILInSsOtN.o,fMTuelcthicnaorlroigeyr,MStoodcuklhaotilomn,Sweden IR-S.Dho.uWUESnEivL.,,CToadiiwnagn&CodedModulation RU.nHivE.AoTfH,TMexIaMs,OATuescthinn,iqTuXes PP.riPnRcUetCoNnALU,nAivr.e,aPErdinitcoerto&n,LNigJht.Networks US.nGiv..WofILCSOalNif,oArnreiaa,ELdoistorA&ngCeloeds,inCgATheory&Appl. 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DigitalObjectIdentifier10.1109/TCOMM.2005.859279 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 1597 Transactions Letters ________________________________________________________________ Iterative Decoding and Channel Parameter Estimation Algorithms for Repeat–Accumulate Codes WangrokOh,Member,IEEE,andKyungwhoon Cheun,Member,IEEE Abstract—Thesensitivityoftheiterativedecoderforrepeat–ac- Thealgorithmsexploitthepropertiesofthelog-likelihoodratios cumulate(RA)codestocarrierphaseandchannelsignal-to-noise (LLRs)generatedwithintheiterativedecoderwhicharesimilar ratioestimationerrorsisinvestigated,andefficientalgorithmsto tothealgorithmsproposedforturbocodes[4],[5].Thecarrier estimateandcorrecttheseerrorsaredeveloped.Thebehaviorof phase offset correction algorithm also exploits the differential RAcodeswithimperfectchannelestimationisdifferentfromthat ofturbocodes,andcorrectionalgorithmsspecifictoRAcodesmust encodingcharacteristicspecifictoRAcodes.Sincetheestima- beformulated.Theproposedalgorithmsusethesoftinformation tion is performed using the soft information already available generatedwithintheiterativedecoder,andthus,arenotonlyhard- within the iterative decoder, the required additional hardware ware-efficient,butalsoofferexcellentperformance. complexityisminimal. Index Terms—Iterative decoding, phase synchronization, TherearetwonotabledifferencesinthebehaviorsofRAand repeat–accumulate (RA) codes, signal-to-noise ratio (SNR) esti- turbocodesunderchannelparameterestimationerrors.Amore mation. subtle difference is that unlike turbo codes [4], [6], RA codes aremuchlesslenienttooverestimationofthechannelSNR.A I. INTRODUCTION more significant difference is that due to inherent differential innerencoding,RAcodesexhibitrelativelygoodperformance PREVIOUS works [1]–[3] have demonstrated that re- undercarrierphaseoffsetinthevicinityof180 .Thisinhibits peat–accumulate (RA) codes, one of the simplest serial straightforward application of stochastic gradient-type carrier concatenatedcodes,canachieveremarkableperformanceover phaseoffsetcorrectionalgorithms,duetothepresenceofalocal additivewhiteGaussiannoisechannelswithiterativedecoding. maximum.Hence,inordertofullyrealizetheexceptionalper- Theseresultswerebasedontheassumptionofperfectestima- formanceofRAcodes,itiscrucialtodevelopchannelparameter tionofchannel parametersatthe decoder,suchas thechannel estimationandcorrectionalgorithmsspecifictoRAcodes. signal-to-noise ratio (SNR) and carrier phase offset. This The remainder of the letter is organized as follows. In Sec- assumption may be unrealistic in practical implementations, tionII,wepresentthesystemmodelandinvestigatethesensi- due to the extremely low operating SNR range of RA codes. tivityofRAcodestochannelSNRandcarrierphaseoffsetesti- Simulation studies indicate that the performance of RA codes mationerrors.InSectionIII,channelparameterestimationand is very sensitive to errors in channel SNR and carrier phase correction algorithms appropriate for RA codes are described offsetestimates. Also, the behavior ofRA codes asa function and their performances are evaluated. Finally, conclusions are of these measurement errors may be quite different from that drawninSectionIV. corresponding to turbo codes. Hence, in order to fully realize the exceptional performance of RA codes, it is crucial to formulateproperestimationandcorrectionalgorithmsspecific II. SYSTEMMODEL to RA codes, giving satisfying performance even at very low SNRs.Naturally,itwouldalsobedesirabletoachievethiswith ThesystemmodelunderconsiderationisshowninFig.1.Bi- minimumhardwareoverhead. naryinformationbits aregroupedintoframesofsize Inthisletter,weinvestigatethesensitivityoftheiterativede- andencodedbyanRAencoder.TheRAencoderconsistsof coderforRAcodestochannelSNRandcarrierphaseoffsetes- aserialconcatenationofa –timesrepeaterandarate–1accu- timationerrors,andproposeverysimplecorrectionalgorithms. mulator, separated by an interleaver of size . The encoded symbols arebinaryphase-shiftkeyingmodulated, andwhiteGaussiannoise withadouble–sidedpowerspec- PaperapprovedbyW.E.Ryan,theEditorforModulation,Coding,andEqual- traldensityof isaddedbeforebeingpresentedtothede- izationoftheIEEECommunicationsSociety.ManuscriptreceivedApril30, 2004;revisedJanuary27,2005andMay4,2005.Thisworkwassupportedby modulator. The corresponding demodulator output with a car- theCenterforBroadbandOFDMMobileAccess(BrOMA)atPOSTECH,sup- rier phase offset of , denoted , is pre- portedbytheITRCProgramoftheMinistryofInformationandCommunica- sentedtotheiterativedecoder,where isazero-meancom- tion,Korea,supervisedbytheInstituteofInformationTechnologyAssessment (IITA). plexGaussianrandomvariable(r.v.)withvariance perdi- TheauthorsarewiththeDivisionofElectricalandComputerEngineering, mension.Let bethechannelSNR,where PohangUniversityofScienceandTechnology(POSTECH),Pohang790-784, is the code rate and is the received energy per information Korea(email:[email protected];[email protected]). DigitalObjectIdentifier10.1109/TCOMM.2005.857150 bit.TheresultingLLRsfor attheendofthe thdecoder 0090-6778/$20.00©2005IEEE 1598 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 Fig.1. Systemmodel.u aretheinputinformationbitsandc aretheRAencodedsymbols,y isthedemodulatoroutputwithacarrierphaseoffset(cid:18),u^ denotes thedecoderoutputforu ,L(u )istheresultingLLRforu attheendofthelthdecoderiteration,(cid:18)~istheinitialcarrierphaseoffsetestimate,and(cid:13)^ and(cid:18)^ are thechannelSNRandcarrierphaseoffsetestimatesusedinthelthdecoderiteration. Fig.2. FactorgraphofanRAcode.L^ and(cid:18)^ arethechannelreliabilityvalueandcarrierphaseoffsetestimatesusedinthelthdecoderiteration. iteration,denoted ,areusedtogeneratetheestimates (3) and of and tobeusedinthe thdecoderiteration. The thiterationoftheiterativedecoderiscarriedoutusing thedemodulatoroutputs compensatedby ,andusing on the factorgraph[7]shown inFig.2, wherethe dependenceof (4) the messages on was suppressed to simplify the figure. The Here iterative decoder computes the messages and using the followingrecursions: (1) [7], denotestherealpartof , istheestimated channelreliabilityvalue[8],and denotesthe thoutputpo- (2) sition of the interleaver (Fig. 2). Also, are the messages IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 1599 Fig.3. BERperformanceoftheiterativedecoderforanRAcodeversusthe Fig.4. BERperformanceoftheiterativedecoderforanRAcodeversusthe channelSNRestimationerror(cid:13)^ (cid:0)(cid:13)dB,assumedtobefixedforalll.Frame carrierphaseoffsetestimationerror(cid:18) ,assumedtobefixedforalll.Framesize sizeis512,q = 3,l = 10,andperfectcarrierphaseoffsetestimationis is512,q=3,l =10,andperfectchannelSNRestimationisassumed. assumed. withturbocodes[4],[6],aslightunderestimationofthechannel passed from the informationnodes to the checknodes, initial- SNRoptimizestheBERperformanceoftheiterativedecoderfor izedtozeroandcomputedas RA codes.2 In Fig. 4, the performance of the standard iterative RA de- (5) coderwith with ,versus ,assumedtobe fixed for all , is shown for and 2 dB. This graph clearlyshowsthedetrimentaleffectofcarrierphaseoffsetesti- where denotestheintegerpartof ,and denotesthe th mationerrors.However,notethattheiterativedecoderforRA outputpositionofthedeinterleaver(Fig.2).Themessages codes shows relatively good performance, with carrier phase arethemessagesoriginatingfromthechecknodesdestinedfor offsetestimationerrorsinthevicinityof180 .Thisisduetothe theinformationnodes,computedusing and as differentialencodingoftheaccumulatorinnercode.Thischar- (6) acteristicisuniquetoRAcodes,andmustbetakenintoaccount whendesigningcarrierphaseoffsetcorrectionalgorithms. with .Finally,theLLRsfortheinformationbits at theendofthe thiterationarethengivenas III. PROPOSED CHANNEL PARAMETER- ESTIMATIONALGORITHMS (7) A. ChannelSNREstimation Similar to the channel SNR estimation algorithms for turbo Thedecoderiterationstopswhenitreachesapresetmaximum codesdevelopedin[4],weinvestigatethebehaviorofthevari- number ,orwhenallbitsaresuccessfullydecoded,which anceoftheabsolutevaluesoftheLLRs,computedasfollows: canbeverifiedusingacyclicredundancycheck[9]. Itisclearthattheestimatedchannelreliabilityvalue plays akeyroleinthedecodingofRAcodes.Thebit-errorrate(BER) (8) performanceofastandarditerativedecoderwith for anRAcodewith and versusthechannelSNR- estimationerrorisshowninFig.3for and2dB.1 where denotestheLLRforthe thbitafterthe thitera- Here,theassumedchannelSNR-estimationerrorwasheldfixed tionofthe thframe.Also,asin[4], givenby(8)maybe throughout the decoder iterations, and the carrier phase offset replacedwiththefollowingsimplifiedversion: estimation error, denoted , is assumed to be zero (9) foralldecoderiterations.Weobservethat,unliketurbocodes, both over- and underestimation of the channel SNR results in The ensemble averaged behavior of and using 100 asizabledegradation.However,itisinterestingtonotethatas simulation runs given by , 1Forallnumericalresults,weassumethatarandominterleaverpatternisgen- eratedpseudorandomlyforeachframe,i.e.,theuniforminterleaver.However, 2This trend may be slightly changed by employing decoder-scheduling properoperationoftheproposedalgorithmswasverifiedundervariousfixed schemesdifferentfromtheonedescribedinSectionII,e.g.,fullflooding[10]. interleavers.Also,1000frameerrorswereobservedforeachBERsimulation However,theproposedalgorithmsaredirectlyapplicabletoiterativedecoders point. employinganydecoder-schedulingscheme. 1600 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 Fig.5. Behaviorofv ,i=1;2versusthechannelSNRestimationerrorfor Fig.6. BERperformanceoftheiterativedecoderusing(cid:13)^ withv and(cid:22)= l = 10.Framesizeis512andq = 3.ThechannelSNRestimationerroris 2:5.Framesizeis512,andq=3,l =10,andperfectcarrierphaseoffset assumetobefixedforalldecoderiterations,andperfectcarrierphaseoffset estimationisassumed. estimationisassumed. simplified measure in place of the more complex power esti- versusthechannelSNRestimationerrorisshowninFig.5for mator: .Here,weassumeperfectcarrierphaseoffsetestimation andthefixedchannelSNRestimationerrorforalldecoderiter- (12) ations.ComparingthiswithFig.3,weobservethattheBERof theiterativedecoderforRAcodesmaybeminimizedbymini- mizing or .Basedonthisobservation,weproposethe Wemayalsoconsiderthefollowingmuchsimplermeasure[5] followingstochasticgradient-typerecursiveupdateequationfor thatmayfurtherreducethehardwarecomplexity,withminimal : lossinperformance: (10) (13) InFig.7,theensembleaveragedbehaviorof and ,using where istheestimatedchannelSNRforthe thiterationof 100 simulation runs given by , the thframe,and istheupdategain.Also,(10)canbefurther ,versusthe carrierphaseoffsetestimationerror areshown simplifiedbyreplacing withitspolarity,resulting for . Here, we assume perfect channel SNR estimation in andthefixedcarrierphaseoffsetestimationerrorforalldecoder iterations.Notetheexistenceofaprominentlocalmaximumat 180 carrier phase offset estimation error. This indicates that (11) unlike turbo codes, straightforward application of a stochastic where if , and , otherwise. The BER gradientupdate algorithm ofthe following type may converge performanceoftheiterativedecoderusingtheproposedchannel to the undesirable local maximum at 180 , depending on the SNRestimate givenby(11),with givenby(9),isshown inFig.6withperfectcarrierphaseoffsetestimation.Weobserve initialcarrierphaseoffset:3 that,aswithturbocodes[4],theiterativedecoderusingthepro- posed channel SNR estimate gives BER performance slightly betterthanthatusingtheexactchannelSNR. (14) Here, istheestimatedcarrierphaseoffsetforthe thiteration B. CarrierPhaseEstimation of the th frame. However, if a coarse initial carrier phase re- Thepresenceofacarrierphaseoffsetestimationerrorresults coveryisavailablewhichguaranteescarrierphaseoffsetinthe in an effective reduction in the received signal power, as seen vicinityof0 ,(14)maysuccessfullybeusedtocompensateand by the matched-filter receiver. This will, in effect, reduce the trackthecarrierphaseoffset.Ifthecoarseinitialcarrierphase average power of the LLRs of the information bits within the iterativedecoder.Thisindicatesthatthemeasuredpowerofthe 3Theupdatetermin(14)haspolarityoppositetothatof(11),duetothefact thats ,j = 1;2areconvexfunctionsofthecarrierphaseoffsetestimation LLRsoftheinformationbitsmaybeusedforcarrierphaseoffset error,whilev ,j=1;2areconcavefunctionofthechannelSNR-estimation estimationandcorrection.Asin[5],weconsiderthefollowing error. IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 1601 Fig.7. Behaviorofs ,i=1;2versusthecarrierphaseoffsetestimationerror Fig.8. BERperformanceoftheproposediterativedecoderusing(cid:18)~and(cid:18)^ forl = 10.Framesizeis512andq = 3.Thecarrierphaseoffsetestimation withs and(cid:22) =0:5,versusthechannelSNRsforp = 1;5.Framesizeis errorisassumedtobefixedforalldecoderiterations,andperfectchannelSNR 512,q =3,l =10,andL^ =4(cid:13). estimationisassumed. isshowninFig.8,assumingperfectchannelSNRestimation.4 recoveryalgorithmisnondataaidedandthuspossessesa180 Note that the proposed iterative decoder under a carrier phase phase ambiguity [11], we may use the following observation offset of 0 and 180 offers performance within 0.03 and 0.1 to easily detect and compensate for the 180 phase ambiguity dBofthestandarditerativedecoder,with at BER within the iterativedecoder beforethe trackingoperationwith with , respectively. The BER performance curve of the (14). proposed iterativedecoder under an initial carrier phaseoffset of45 isalsoincludedinFig.8,andliesbetweenthosecorre- Notethatthemessages generatedunderacarrierphase sponding to and 180 . Although not shown in Fig. 8, offsetof180 areidenticaltothosegeneratedunderazerocar- wehavecarriedoutextensivesimulations,andverifiedthatthe rierphaseoffset,exceptfor ,whichhasareversedpolarity. initial carrier phase offset of 180 results in the largest SNR This is due to the differential encoding inherent in RA codes. loss for the proposed iterative decoder. We have also verified Hence,neglectingtheeffectofnoise,thepolarityof isal- that the increase in the required average number of decoder waysoppositetothatof undera180 carrierphaseoffset. iterations,duetotheincreasein ,isslightlylessthan . Basedonthisobservation,wemaymodifytheiterativedecoder In most packet-based networks, a preamble is usually todetectandcompensateforcarrierphaseoffsetsnear180 by available, and a crude initial carrier phase recovery may be comparingthepolarityofthemessages and afterthe performed using the preamble. In continuous transmission firstdecoderiteration.Let denotetheinitialcarrierphaseoffset systems, the stochastic gradient update algorithm of (14) may estimationwitha180 resolution,givenas first be used to let the residual carrier phase offset converge to the vicinity of either 0 or 180 . A decision on may then if (15) bemadeusing(15),afterwhichthetrackingoperationmaybe otherwise. resumed. If is 0 , the proposed iterative decoder moves on with the remaining decoder iterations. Otherwise, the decoder negates IV. CONCLUSIONS the demodulator outputs and the message , and recal- Inthisletter,weinvestigatedthesensitivityoftheiterativede- culates the messages affected by the negation of before coderforRAcodestocarrierphaseandchannelSNRestimation performing the remaining decoder iterations. Note that only errors,anddevelopedalgorithmstoaccuratelyestimateandcor- outof , ’sneedberecalculatedasaresultofnegating recttheseerrors.ThecharacteristicsspecifictoRAcodeswere . Hence, the additional complexityrequired to detect and identified and fully exploited. The proposed algorithms, using compensateforcarrierphaseoffsetnear180 ismuchlessthan thesoftinformationgeneratedwithintheiterativedecoder,are of that required for one decoder iteration. We have also notonlyveryhardware-efficient,butalsoprovideexcellentper- verified that this trend is maintained and reinforced even after formance. iterations. Hence, in order to increase the reliability of theestimateof ,thedetectionoperationdescribedabovemay beperformedafter decoderiterations. 4Notethatduetothefactthats takesonvaluesmuchlargerthanthoseof The BER performance of the proposed iterative decoder v ,theupdategainusedforthechannelSNRestimationismuchlargerthan under and180 with versusthechannelSNR thatusedforthecarrierphaseoffsetcompensation. 1602 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 REFERENCES [7] F.R.Kschischang,B.J.Frey,andH.Loeliger,“Factorgraphsandthe sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. [1] D.Divsalar,H.Jin,andR.McEliece,“Codingtheoremsfor“turbo-like” 498–519,Feb.2001. codes,”inProc.36thAllertonConf.Commun.,Control,Comput.,Sep. [8] J.Hagenauer,E.Offer,andL.Papke,“Iterativedecodingofbinaryblock 1998,pp.201–210. andconvolutionalcodes,”IEEETrans.Inf.Theory,vol.42,no.3,pp. [2] H.JinandR.J.McEliece,“RAcodesachieveAWGNchannelcapacity,” 429–445,Mar.1996. inProc.13thSymp.Appl.Algebra,AlgebraicAlgorithms,ErrorCor- [9] A.Shibutani,H.Suda,andF.Adachi,“Reducingaveragenumberof rectingCodes,1999,pp.10–18. turbodecodingiterations,”Electron.Lett.,vol.35,pp.701–702,Apr. [3] H.Jin,A.Khandekar,andR.McEliece,“Irregularrepeat-accumulate 1999. codes,”inProc.2ndInt.Symp.TurboCodes,Sep.2000,pp.1–8. [10] F.R.KschischangandB.J.Frey,“Iterativedecodingofcompoundcodes [4] W.OhandK.Cheun,“AdaptivechannelSNRestimationalgorithmfor by probability propagation in graphical models,” IEEE J. Sel. Areas turbodecoder,”IEEECommun.Lett.,vol.4,no.8,pp.255–257,Aug. Commun.,vol.16,no.2,pp.219–230,Feb.1998. 2000. [11] H.Meyr,M.Moeneclaey,andS.A.Fechtel,DigitalCommunication [5] ,“Jointdecodingandcarrierphaserecoveryalgorithmforturbo Receivers. NewYork:Wiley,1998. codes,”IEEECommun.Lett.,vol.5,no.9,pp.375–377,Sep.2001. [6] T.A.SummersandS.G.Wilson,“SNRmismatchandonlineestimation inturbodecoding,”IEEETrans.Commun.,vol.46,no.4,pp.421–423, Apr.1998.

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OCTOBER2005 VOLUME53 NUMBER10 IECMBT (ISSN0090-6778) TRANSACTIONSLETTERS Coding IterativeDecodingandChannelParameterEstimationAlgorithmsforRepeat–AccumulateCodes.... ......... .W.OhandK.Cheun 1597 Fading/Equalization ASoftDecodingSchemeforVectorQuantizationOveraCDMAChannel... ...... ..... ......... ...... ......H.H.Nguyen 1603 CombinedEqualizationforUplinkMC-CDMAinRayleighFadingChannels ........ ...... I.Cosovic,M.Schnell,andA.Springer 1609 Modulation/Detection GeneralizedAPPDetectionofContinuousPhaseModulationOverUnknownISIChannels .. ......... .... A.HanssonandT.Aulin 1615 TransmissionSystems CapacityandCoverageIncreaseWithRepeatersinUMTSUrbanCellularMobileCommunicationEnvironment. ..... ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... .....M.N.Patwary,P.B.Rapajic,andI.Oppermann 1620 NoncooperativePower-ControlGameandThroughputGameOverWirelessNetworks ..... ......... ..... Z.HanandK.J.R.Liu 1625 TRANSACTIONSPAPERS Coding TurboCodesWithRate-m=(m+1)ConstituentConvolutionalCodes ..... ...... ........ ...... .. C.DouillardandC.Berrou 1630 DC-FreeError-ControlBlockCodes ..... ..... ...... ...... ......... ...... ..... ...... ..F.Zhai,Y.Xin,andI.J.Fair 1639 DigitalCommunications JointSource/ChannelCodingforMultipleImages . ...... ...... ......... ...... ..... . Z.Wu,A.Bilgin,andM.W.Marcellin 1648 DiversityCombiningWithImperfectChannelEstimation .. ...... ...... ......... ..... ...... R.You,H.Li,andY.Bar-Ness 1655 Fading/Equalization Rate-AdaptiveTransmissionOverCorrelatedFadingChannels .... ...... ...... ........ ...... ...... ...T.JiandW.Stark 1663 TheExpectation-MaximizationViterbiAlgorithmforBlindAdaptiveChannelEqualization.. ......... ... H.NguyenandB.C.Levy 1671 Modulation/Detection MultirateModulation:ABandwidth-andPower-EfficientModulationScheme...... ..... ......... ...... ...... ..H.B.Peek 1679 ANewPerformanceBoundforPAM-BasedCPMDetectors ...... ...... ........ ...... ...... ..... E.PerrinsandM.Rice 1688 DesignofTurbo-CodedModulationfortheAWGNChannelWithTikhonovPhaseError.... ......... .Y.Zhu,L.Ni,andB.J.Belzer 1697 MultipleAccess AdaptiveOpportunisticFairSchedulingOverMultiuserSpatialChannels ... ...... ...... ........ ...... .. C.LiandX.Wang 1708 SystolicArrayImplementationofaReal-TimeSymbol-OptimumMultiuserDetectionAlgorithm.... ...... ...... ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ..C.-C.Lu,J.-Y.Hsu,andC.-C.Cheng 1718 (ContentsContinuedonBackCover) (ContentsContinuedfromFrontCover) OpticalCommunication AnOverlayArchitectureforManagingLightpathsinOpticallyRoutedNetworks.... ........ ..V.A.VaishampayanandM.D.Feuer 1729 Multiple-SubcarrierOpticalCommunicationSystemsWithSubcarrierSignalPointSequence. ......... .. S.TeramotoandT.Ohtsuki 1738 SpreadSpectrum OntheTradeoffBetweenTwoTypesofProcessingGain... ...... ...... ......... ..... ...... ....E.FishlerandH.V.Poor 1744 TransmissionSystems AnIterativeExtensionofBLASTDecodingAlgorithmforLayeredSpace–TimeSignals. ... ......... .... K.LiuandA.M.Sayeed 1754 BlindCarrierFrequencyTrackingforFilterbankMulticarrierWirelessCommunications .... ...... ...... ...... ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... ....V.Lottici,M.Luise,C.Saccomando,andF.Spalla 1762 AbstractsofForthcomingManuscripts.... ..... ...... ...... ...... ........ ...... ...... ...... ...... ..... .... 1773 InformationforAuthors.. ...... ...... ..... ...... ...... ....... ........ ..... ...... ...... ...... ..... .... 1776 PAPERSSCHEDULEDTOBEINTHENEXTISSUE NOVEMBER2005 TransactionsLetters ImprovingthePerformanceofTurboCodesWithaSimpleProtectionSchemeforError-ProneBitPositions..... .. W.Oh,Y.Kim,andK.Cheun InfiniteSeriesRepresentationsAssociatedWiththeBivariateRicianDistributionandTheirApplications..... ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... ..D.A.ZogasandG.K.Karagiannidis Bit-ErrorProbabilityforOrthogonalSpace–TimeBlockCodesWithDifferentialDetection.. ......... .... T.P.Soh,P.Y.Kam,andC.S.Ng OntheProbabilityofErrorofAntennaSubsetSelectionWithSpace-TimeBlockCodes .... ...... ........ ....... ..... ... D.J.Love CooperativeSpace–TimeCodingforWirelessNetworks... ...... ...... ....... ....... ...... ...... .... A.StefanovandE.Erkip AdaptiveResourceAllocationforMultiaccessMIMO/OFDMSystemsWithMatchedFiltering ..... ......... ..Y.J.ZhangandK.B.Letaief Low-ComplexityMaximum-LikelihoodDecoderforFour-Transmit-AntennaQuasi-OrthogonalSpace–TimeBlockCode .... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... .... M.-T.Le,V.-S.Pham,L.Mai,andG.Yoon TransactionsPapers NonsystematicTurboCodes..... ...... ..... ......... ...... ...... .....A.Banerjee,F.Vatta,B.Scanavino,andD.J.Costello,Jr. ASubbandApproachtoChannelEstimationandEqualizationforDMTandOFDMSystems. ...... ......... ...... .D.MarelliandM.Fu OptimizedDecisionFeedbackEqualizationforConvolutionalCodingWithReducedDelay.. ....... ........ ... J.-T.LiuandS.B.Gelfand IterativeDesignandDetectionofaDFEintheFrequencyDomain.. ...... ...... ........ ...... ...... . N.BenvenutoandS.Tomasin SimulationofRayleighFadedMobile-to-MobileCommunicationChannels.. ...... ........ ......C.S.Patel,G.L.Stüber,andT.G.Pratt Capacity-ApproachingTurboCodingandIterativeDecodingforRelayChannels .... ..... ......... ...... ...Z.ZhangandT.M.Duman WirelessPacketSchedulingBasedontheCumulativeDistributionFunctionofUserTransmissionRates ..... ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ..... D.Park,H.Seo,H.Kwon,andB.G.Lee ExpandingtheSwitchingCapabilitiesofOpticalCrossconnects.... ...... ...... ..... ......... ...... ...... ..... ....C.-T.Lea OntheCombinedInput-CrosspointBufferedSwitchWithRound-RobinArbitration.. ....... ....... R.Rojas-Cessa,E.Oki,andH.J.Chao TimingUltra-WidebandSignalsWithDirtyTemplates .... ...... ...... ......... ..... ...... ...... ..L.YangandG.B.Giannakis AbstractsofForthcomingManuscripts.... ..... ...... ...... ...... ...... ........ ...... ...... ...... ..... ...... .... 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LucentTechnol.BellLabs.,MurrayHill,NJ UNn.iAv.L-oDfHTAeHxIaRs,,SDpaacllea–sT,imTXe,OFDM&Equalization OpticalCommunication UF.nSivA.NoTfUCPCiIs,aW,iPrieslae,ssItSaylsyt.Performance PK.iHelOUEHnEivR.,,CKoideiln,gG&erImtearnaytiveProcessing A.EAstNimASaTtAioSnO,P&OUCLoOdSi,nIgterativeDetection, UI.nAivN.DoOfNSOtVraICth,cOlypdtiec,aGlNlaestgwoowr,ksU&.KD.evices UE.nSivE.RoPfEDLI’NA,qSuyinlcah,rLo’nAizqautiiloan,&ItaSlyensorNetworks AU.niKv..KofHWANaDteArNloI,o,CWodaitnegrl&ooI,nOfoNrm,CataionnadTaheory Univ.ofMichigan,AnnArbor,MI R.HUI,OpticalTransmission&Switching TexasA&MUniv.,CollegeStation,TX K.NARAYANAN,Modulation,Coding,&Equalization STe.xAaRsIYInAsVtIrSuImTAeKnUtsL,,AAlrpehaaEredtittao,rGA UDn.iKv..HofUNKTaEnRs,asP,hLotaownriecnNcee,twKoSrks CA.riTzEoPnEaDSEtLaEteNLUIOnGivL.U,,TSeymncpher,oAniZzation&Equalization TWe.xEas.RAY&AMN,MUondivu.l,aCtioolnl,egCeodSitnagtio&n,ETquXalization SSU.BNAYTABLAuMffaAl,oS,pBreuafdfaSlop,ecNtrYum&Estimation UKn.iKv.IToAfYAEMssAe,x,PChootlocnhiecstNere,twUo.Krk.s& GU.niVv.IToEfTTMA,oEdeqnuaa,lizMatoidoenn&a,FIatadliyngChannels UC.nSivC.HoLfEGAErLiz,oCnoad,inTgucTshoeno,ryA&ZTechniques UNn.iCv..BofEAAUlLbIeErUta,,WEidremleosnstoCno,mAmBu,nC.Tanhaedoary OsFakibaerU-Onipvt.i,cOWsiarkelae,ssJapan NC.a-tLio.nWalATNsGi,nEgqHuaulaizaUtnioivn.,Taiwan UTRnIiEvU.-oKfIAENlbTerRtUa,OENGdm,Coondtoinng,AThBe,oCryan&adTaechniques YU.nFivA.NoGf,FWloirreildeas,sGNaeitnweosrvkislle,FL WHo.fCst.raKWUOnNivG.,,OHpemticpaslteNaedt,wNorYks RKo.yWalILInSsOtN.o,fMTuelcthicnaorlroigeyr,MStoodcuklhaotilomn,Sweden IR-S.Dho.uWUESnEivL.,,CToadiiwnagn&CodedModulation RU.nHivE.AoTfH,TMexIaMs,OATuescthinn,iqTuXes PP.riPnRcUetCoNnALU,nAivr.e,aPErdinitcoerto&n,LNigJht.Networks US.nGiv..WofILCSOalNif,oArnreiaa,ELdoistorA&ngCeloeds,inCgATheory&Appl. 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DigitalObjectIdentifier10.1109/TCOMM.2005.859279 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 1597 Transactions Letters ________________________________________________________________ Iterative Decoding and Channel Parameter Estimation Algorithms for Repeat–Accumulate Codes WangrokOh,Member,IEEE,andKyungwhoon Cheun,Member,IEEE Abstract—Thesensitivityoftheiterativedecoderforrepeat–ac- Thealgorithmsexploitthepropertiesofthelog-likelihoodratios cumulate(RA)codestocarrierphaseandchannelsignal-to-noise (LLRs)generatedwithintheiterativedecoderwhicharesimilar ratioestimationerrorsisinvestigated,andefficientalgorithmsto tothealgorithmsproposedforturbocodes[4],[5].Thecarrier estimateandcorrecttheseerrorsaredeveloped.Thebehaviorof phase offset correction algorithm also exploits the differential RAcodeswithimperfectchannelestimationisdifferentfromthat ofturbocodes,andcorrectionalgorithmsspecifictoRAcodesmust encodingcharacteristicspecifictoRAcodes.Sincetheestima- beformulated.Theproposedalgorithmsusethesoftinformation tion is performed using the soft information already available generatedwithintheiterativedecoder,andthus,arenotonlyhard- within the iterative decoder, the required additional hardware ware-efficient,butalsoofferexcellentperformance. complexityisminimal. Index Terms—Iterative decoding, phase synchronization, TherearetwonotabledifferencesinthebehaviorsofRAand repeat–accumulate (RA) codes, signal-to-noise ratio (SNR) esti- turbocodesunderchannelparameterestimationerrors.Amore mation. subtle difference is that unlike turbo codes [4], [6], RA codes aremuchlesslenienttooverestimationofthechannelSNR.A I. INTRODUCTION more significant difference is that due to inherent differential innerencoding,RAcodesexhibitrelativelygoodperformance PREVIOUS works [1]–[3] have demonstrated that re- undercarrierphaseoffsetinthevicinityof180 .Thisinhibits peat–accumulate (RA) codes, one of the simplest serial straightforward application of stochastic gradient-type carrier concatenatedcodes,canachieveremarkableperformanceover phaseoffsetcorrectionalgorithms,duetothepresenceofalocal additivewhiteGaussiannoisechannelswithiterativedecoding. maximum.Hence,inordertofullyrealizetheexceptionalper- Theseresultswerebasedontheassumptionofperfectestima- formanceofRAcodes,itiscrucialtodevelopchannelparameter tionofchannel parametersatthe decoder,suchas thechannel estimationandcorrectionalgorithmsspecifictoRAcodes. signal-to-noise ratio (SNR) and carrier phase offset. This The remainder of the letter is organized as follows. In Sec- assumption may be unrealistic in practical implementations, tionII,wepresentthesystemmodelandinvestigatethesensi- due to the extremely low operating SNR range of RA codes. tivityofRAcodestochannelSNRandcarrierphaseoffsetesti- Simulation studies indicate that the performance of RA codes mationerrors.InSectionIII,channelparameterestimationand is very sensitive to errors in channel SNR and carrier phase correction algorithms appropriate for RA codes are described offsetestimates. Also, the behavior ofRA codes asa function and their performances are evaluated. Finally, conclusions are of these measurement errors may be quite different from that drawninSectionIV. corresponding to turbo codes. Hence, in order to fully realize the exceptional performance of RA codes, it is crucial to formulateproperestimationandcorrectionalgorithmsspecific II. SYSTEMMODEL to RA codes, giving satisfying performance even at very low SNRs.Naturally,itwouldalsobedesirabletoachievethiswith ThesystemmodelunderconsiderationisshowninFig.1.Bi- minimumhardwareoverhead. naryinformationbits aregroupedintoframesofsize Inthisletter,weinvestigatethesensitivityoftheiterativede- andencodedbyanRAencoder.TheRAencoderconsistsof coderforRAcodestochannelSNRandcarrierphaseoffsetes- aserialconcatenationofa –timesrepeaterandarate–1accu- timationerrors,andproposeverysimplecorrectionalgorithms. mulator, separated by an interleaver of size . The encoded symbols arebinaryphase-shiftkeyingmodulated, andwhiteGaussiannoise withadouble–sidedpowerspec- PaperapprovedbyW.E.Ryan,theEditorforModulation,Coding,andEqual- traldensityof isaddedbeforebeingpresentedtothede- izationoftheIEEECommunicationsSociety.ManuscriptreceivedApril30, 2004;revisedJanuary27,2005andMay4,2005.Thisworkwassupportedby modulator. The corresponding demodulator output with a car- theCenterforBroadbandOFDMMobileAccess(BrOMA)atPOSTECH,sup- rier phase offset of , denoted , is pre- portedbytheITRCProgramoftheMinistryofInformationandCommunica- sentedtotheiterativedecoder,where isazero-meancom- tion,Korea,supervisedbytheInstituteofInformationTechnologyAssessment (IITA). plexGaussianrandomvariable(r.v.)withvariance perdi- TheauthorsarewiththeDivisionofElectricalandComputerEngineering, mension.Let bethechannelSNR,where PohangUniversityofScienceandTechnology(POSTECH),Pohang790-784, is the code rate and is the received energy per information Korea(email:[email protected];[email protected]). DigitalObjectIdentifier10.1109/TCOMM.2005.857150 bit.TheresultingLLRsfor attheendofthe thdecoder 0090-6778/$20.00©2005IEEE 1598 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 Fig.1. Systemmodel.u aretheinputinformationbitsandc aretheRAencodedsymbols,y isthedemodulatoroutputwithacarrierphaseoffset(cid:18),u^ denotes thedecoderoutputforu ,L(u )istheresultingLLRforu attheendofthelthdecoderiteration,(cid:18)~istheinitialcarrierphaseoffsetestimate,and(cid:13)^ and(cid:18)^ are thechannelSNRandcarrierphaseoffsetestimatesusedinthelthdecoderiteration. Fig.2. FactorgraphofanRAcode.L^ and(cid:18)^ arethechannelreliabilityvalueandcarrierphaseoffsetestimatesusedinthelthdecoderiteration. iteration,denoted ,areusedtogeneratetheestimates (3) and of and tobeusedinthe thdecoderiteration. The thiterationoftheiterativedecoderiscarriedoutusing thedemodulatoroutputs compensatedby ,andusing on the factorgraph[7]shown inFig.2, wherethe dependenceof (4) the messages on was suppressed to simplify the figure. The Here iterative decoder computes the messages and using the followingrecursions: (1) [7], denotestherealpartof , istheestimated channelreliabilityvalue[8],and denotesthe thoutputpo- (2) sition of the interleaver (Fig. 2). Also, are the messages IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 1599 Fig.3. BERperformanceoftheiterativedecoderforanRAcodeversusthe Fig.4. BERperformanceoftheiterativedecoderforanRAcodeversusthe channelSNRestimationerror(cid:13)^ (cid:0)(cid:13)dB,assumedtobefixedforalll.Frame carrierphaseoffsetestimationerror(cid:18) ,assumedtobefixedforalll.Framesize sizeis512,q = 3,l = 10,andperfectcarrierphaseoffsetestimationis is512,q=3,l =10,andperfectchannelSNRestimationisassumed. assumed. withturbocodes[4],[6],aslightunderestimationofthechannel passed from the informationnodes to the checknodes, initial- SNRoptimizestheBERperformanceoftheiterativedecoderfor izedtozeroandcomputedas RA codes.2 In Fig. 4, the performance of the standard iterative RA de- (5) coderwith with ,versus ,assumedtobe fixed for all , is shown for and 2 dB. This graph clearlyshowsthedetrimentaleffectofcarrierphaseoffsetesti- where denotestheintegerpartof ,and denotesthe th mationerrors.However,notethattheiterativedecoderforRA outputpositionofthedeinterleaver(Fig.2).Themessages codes shows relatively good performance, with carrier phase arethemessagesoriginatingfromthechecknodesdestinedfor offsetestimationerrorsinthevicinityof180 .Thisisduetothe theinformationnodes,computedusing and as differentialencodingoftheaccumulatorinnercode.Thischar- (6) acteristicisuniquetoRAcodes,andmustbetakenintoaccount whendesigningcarrierphaseoffsetcorrectionalgorithms. with .Finally,theLLRsfortheinformationbits at theendofthe thiterationarethengivenas III. PROPOSED CHANNEL PARAMETER- ESTIMATIONALGORITHMS (7) A. ChannelSNREstimation Similar to the channel SNR estimation algorithms for turbo Thedecoderiterationstopswhenitreachesapresetmaximum codesdevelopedin[4],weinvestigatethebehaviorofthevari- number ,orwhenallbitsaresuccessfullydecoded,which anceoftheabsolutevaluesoftheLLRs,computedasfollows: canbeverifiedusingacyclicredundancycheck[9]. Itisclearthattheestimatedchannelreliabilityvalue plays akeyroleinthedecodingofRAcodes.Thebit-errorrate(BER) (8) performanceofastandarditerativedecoderwith for anRAcodewith and versusthechannelSNR- estimationerrorisshowninFig.3for and2dB.1 where denotestheLLRforthe thbitafterthe thitera- Here,theassumedchannelSNR-estimationerrorwasheldfixed tionofthe thframe.Also,asin[4], givenby(8)maybe throughout the decoder iterations, and the carrier phase offset replacedwiththefollowingsimplifiedversion: estimation error, denoted , is assumed to be zero (9) foralldecoderiterations.Weobservethat,unliketurbocodes, both over- and underestimation of the channel SNR results in The ensemble averaged behavior of and using 100 asizabledegradation.However,itisinterestingtonotethatas simulation runs given by , 1Forallnumericalresults,weassumethatarandominterleaverpatternisgen- eratedpseudorandomlyforeachframe,i.e.,theuniforminterleaver.However, 2This trend may be slightly changed by employing decoder-scheduling properoperationoftheproposedalgorithmswasverifiedundervariousfixed schemesdifferentfromtheonedescribedinSectionII,e.g.,fullflooding[10]. interleavers.Also,1000frameerrorswereobservedforeachBERsimulation However,theproposedalgorithmsaredirectlyapplicabletoiterativedecoders point. employinganydecoder-schedulingscheme. 1600 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 Fig.5. Behaviorofv ,i=1;2versusthechannelSNRestimationerrorfor Fig.6. BERperformanceoftheiterativedecoderusing(cid:13)^ withv and(cid:22)= l = 10.Framesizeis512andq = 3.ThechannelSNRestimationerroris 2:5.Framesizeis512,andq=3,l =10,andperfectcarrierphaseoffset assumetobefixedforalldecoderiterations,andperfectcarrierphaseoffset estimationisassumed. estimationisassumed. simplified measure in place of the more complex power esti- versusthechannelSNRestimationerrorisshowninFig.5for mator: .Here,weassumeperfectcarrierphaseoffsetestimation andthefixedchannelSNRestimationerrorforalldecoderiter- (12) ations.ComparingthiswithFig.3,weobservethattheBERof theiterativedecoderforRAcodesmaybeminimizedbymini- mizing or .Basedonthisobservation,weproposethe Wemayalsoconsiderthefollowingmuchsimplermeasure[5] followingstochasticgradient-typerecursiveupdateequationfor thatmayfurtherreducethehardwarecomplexity,withminimal : lossinperformance: (10) (13) InFig.7,theensembleaveragedbehaviorof and ,using where istheestimatedchannelSNRforthe thiterationof 100 simulation runs given by , the thframe,and istheupdategain.Also,(10)canbefurther ,versusthe carrierphaseoffsetestimationerror areshown simplifiedbyreplacing withitspolarity,resulting for . Here, we assume perfect channel SNR estimation in andthefixedcarrierphaseoffsetestimationerrorforalldecoder iterations.Notetheexistenceofaprominentlocalmaximumat 180 carrier phase offset estimation error. This indicates that (11) unlike turbo codes, straightforward application of a stochastic where if , and , otherwise. The BER gradientupdate algorithm ofthe following type may converge performanceoftheiterativedecoderusingtheproposedchannel to the undesirable local maximum at 180 , depending on the SNRestimate givenby(11),with givenby(9),isshown inFig.6withperfectcarrierphaseoffsetestimation.Weobserve initialcarrierphaseoffset:3 that,aswithturbocodes[4],theiterativedecoderusingthepro- posed channel SNR estimate gives BER performance slightly betterthanthatusingtheexactchannelSNR. (14) Here, istheestimatedcarrierphaseoffsetforthe thiteration B. CarrierPhaseEstimation of the th frame. However, if a coarse initial carrier phase re- Thepresenceofacarrierphaseoffsetestimationerrorresults coveryisavailablewhichguaranteescarrierphaseoffsetinthe in an effective reduction in the received signal power, as seen vicinityof0 ,(14)maysuccessfullybeusedtocompensateand by the matched-filter receiver. This will, in effect, reduce the trackthecarrierphaseoffset.Ifthecoarseinitialcarrierphase average power of the LLRs of the information bits within the iterativedecoder.Thisindicatesthatthemeasuredpowerofthe 3Theupdatetermin(14)haspolarityoppositetothatof(11),duetothefact thats ,j = 1;2areconvexfunctionsofthecarrierphaseoffsetestimation LLRsoftheinformationbitsmaybeusedforcarrierphaseoffset error,whilev ,j=1;2areconcavefunctionofthechannelSNR-estimation estimationandcorrection.Asin[5],weconsiderthefollowing error. IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 1601 Fig.7. Behaviorofs ,i=1;2versusthecarrierphaseoffsetestimationerror Fig.8. BERperformanceoftheproposediterativedecoderusing(cid:18)~and(cid:18)^ forl = 10.Framesizeis512andq = 3.Thecarrierphaseoffsetestimation withs and(cid:22) =0:5,versusthechannelSNRsforp = 1;5.Framesizeis errorisassumedtobefixedforalldecoderiterations,andperfectchannelSNR 512,q =3,l =10,andL^ =4(cid:13). estimationisassumed. isshowninFig.8,assumingperfectchannelSNRestimation.4 recoveryalgorithmisnondataaidedandthuspossessesa180 Note that the proposed iterative decoder under a carrier phase phase ambiguity [11], we may use the following observation offset of 0 and 180 offers performance within 0.03 and 0.1 to easily detect and compensate for the 180 phase ambiguity dBofthestandarditerativedecoder,with at BER within the iterativedecoder beforethe trackingoperationwith with , respectively. The BER performance curve of the (14). proposed iterativedecoder under an initial carrier phaseoffset of45 isalsoincludedinFig.8,andliesbetweenthosecorre- Notethatthemessages generatedunderacarrierphase sponding to and 180 . Although not shown in Fig. 8, offsetof180 areidenticaltothosegeneratedunderazerocar- wehavecarriedoutextensivesimulations,andverifiedthatthe rierphaseoffset,exceptfor ,whichhasareversedpolarity. initial carrier phase offset of 180 results in the largest SNR This is due to the differential encoding inherent in RA codes. loss for the proposed iterative decoder. We have also verified Hence,neglectingtheeffectofnoise,thepolarityof isal- that the increase in the required average number of decoder waysoppositetothatof undera180 carrierphaseoffset. iterations,duetotheincreasein ,isslightlylessthan . Basedonthisobservation,wemaymodifytheiterativedecoder In most packet-based networks, a preamble is usually todetectandcompensateforcarrierphaseoffsetsnear180 by available, and a crude initial carrier phase recovery may be comparingthepolarityofthemessages and afterthe performed using the preamble. In continuous transmission firstdecoderiteration.Let denotetheinitialcarrierphaseoffset systems, the stochastic gradient update algorithm of (14) may estimationwitha180 resolution,givenas first be used to let the residual carrier phase offset converge to the vicinity of either 0 or 180 . A decision on may then if (15) bemadeusing(15),afterwhichthetrackingoperationmaybe otherwise. resumed. If is 0 , the proposed iterative decoder moves on with the remaining decoder iterations. Otherwise, the decoder negates IV. CONCLUSIONS the demodulator outputs and the message , and recal- Inthisletter,weinvestigatedthesensitivityoftheiterativede- culates the messages affected by the negation of before coderforRAcodestocarrierphaseandchannelSNRestimation performing the remaining decoder iterations. Note that only errors,anddevelopedalgorithmstoaccuratelyestimateandcor- outof , ’sneedberecalculatedasaresultofnegating recttheseerrors.ThecharacteristicsspecifictoRAcodeswere . Hence, the additional complexityrequired to detect and identified and fully exploited. The proposed algorithms, using compensateforcarrierphaseoffsetnear180 ismuchlessthan thesoftinformationgeneratedwithintheiterativedecoder,are of that required for one decoder iteration. We have also notonlyveryhardware-efficient,butalsoprovideexcellentper- verified that this trend is maintained and reinforced even after formance. iterations. Hence, in order to increase the reliability of theestimateof ,thedetectionoperationdescribedabovemay beperformedafter decoderiterations. 4Notethatduetothefactthats takesonvaluesmuchlargerthanthoseof The BER performance of the proposed iterative decoder v ,theupdategainusedforthechannelSNRestimationismuchlargerthan under and180 with versusthechannelSNR thatusedforthecarrierphaseoffsetcompensation. 1602 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.53,NO.10,OCTOBER2005 REFERENCES [7] F.R.Kschischang,B.J.Frey,andH.Loeliger,“Factorgraphsandthe sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. [1] D.Divsalar,H.Jin,andR.McEliece,“Codingtheoremsfor“turbo-like” 498–519,Feb.2001. codes,”inProc.36thAllertonConf.Commun.,Control,Comput.,Sep. [8] J.Hagenauer,E.Offer,andL.Papke,“Iterativedecodingofbinaryblock 1998,pp.201–210. andconvolutionalcodes,”IEEETrans.Inf.Theory,vol.42,no.3,pp. [2] H.JinandR.J.McEliece,“RAcodesachieveAWGNchannelcapacity,” 429–445,Mar.1996. inProc.13thSymp.Appl.Algebra,AlgebraicAlgorithms,ErrorCor- [9] A.Shibutani,H.Suda,andF.Adachi,“Reducingaveragenumberof rectingCodes,1999,pp.10–18. turbodecodingiterations,”Electron.Lett.,vol.35,pp.701–702,Apr. [3] H.Jin,A.Khandekar,andR.McEliece,“Irregularrepeat-accumulate 1999. codes,”inProc.2ndInt.Symp.TurboCodes,Sep.2000,pp.1–8. [10] F.R.KschischangandB.J.Frey,“Iterativedecodingofcompoundcodes [4] W.OhandK.Cheun,“AdaptivechannelSNRestimationalgorithmfor by probability propagation in graphical models,” IEEE J. Sel. Areas turbodecoder,”IEEECommun.Lett.,vol.4,no.8,pp.255–257,Aug. Commun.,vol.16,no.2,pp.219–230,Feb.1998. 2000. [11] H.Meyr,M.Moeneclaey,andS.A.Fechtel,DigitalCommunication [5] ,“Jointdecodingandcarrierphaserecoveryalgorithmforturbo Receivers. NewYork:Wiley,1998. codes,”IEEECommun.Lett.,vol.5,no.9,pp.375–377,Sep.2001. [6] T.A.SummersandS.G.Wilson,“SNRmismatchandonlineestimation inturbodecoding,”IEEETrans.Commun.,vol.46,no.4,pp.421–423, Apr.1998.

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