IEEE TRANSACTIONS ON MAGNETICS VOL - PDF document

IEEETRANSACTIONSONMAGNETICS,VOL.36,NO.5,SEPTEMBER2000LowDensityParityCheckCodesforMagneticRecordingChannelsHongxinSong,RichardM.Todd,andJ.R.CruzWeproposeasystemformagneticrecording,usingalowdensityparitycheck(LDPC)codeastheerror-correcting-code,inconjunctionwitharate16/17quasi-maximum-transition-runchannelcodeandamodifiedEPR4-equalizedchannel.IterativedecodingbetweenthepartialresponsechannelandtheLDPCcodeisperformed.Simulationsshowthatthissystemcanachievea5.9dBgainoveruncodedEPR4.ThealgorithmsusedtodesignthisLDPCcodearealsodiscussed.IndexTerms—Iterativedecoding,lowdensityparitycheckcodes,magneticrecording.I.INTRODUCTIONURBOdecodingformagneticrecordingchannelshasbeeninvestigatedintwodifferentways:i)usingaclas-sicalturbocodewithatleasttwocomponentcodes[1],[2];orii)usingasingleconvolutionalcodeseriallyconcatenatedwithapartialresponse(PR)channelwhichplaystheroleofasecondconstituentcodeofrateone[3].Bothsystemsperformsignificantlybetterthanuncodedsystems.Inthispaper,insteadofusingasingleweakconvolutionalcode,weinvestigatetheuseofapowerfulblockcode,namelyalowdensityparitycheck(LDPC)code[4]–[6],anditsiterativedecodingwithanefficientdecodingalgorithm.Thepaperisorganizedasfollows.InSectionIIwedescribethebackgroundofseriallyconcatenatedsystemsandLDPCcodes.InSectionIIIwedescribeapracticalLDPCsystemwithturboequalizationformagneticrecording.InSectionIVwepresentthesimulationresultsfortheproposedsystem.InSec-tionVwediscussthedesignoftheLDPCcode.ConclusionsaregiveninSectionVI.II.BACKGROUNDAturbocodeusuallyconsistsoftwoormoreparallelconcate-natedconvolutionalcodes[7].Theapplicationofturbocodestomagneticrecordingchannelshasthepotentialforlargeperfor-mancegainsoveruncodedsystems[1],[2].Turboequalizationisperformedbyfeedingtheinformationfromtheturbodecoderbacktothechanneldecoder.etal.’sserialconcatenationschemesimplifiesthefullturbosystembyreplacingtheturbocodewithasinglecon-volutionalcode[3],andisshowntohaveaboutthesameper-formanceasthefullturbosystem,withlesscomplexity.ManuscriptreceivedFebruary15,2000;revisedMay15,2000.TheauthorsarewiththeSchoolofElectricalandComputerEngineering,TheUniversityofOklahoma,Norman,OK73019USA.PublisherItemIdentifierS0018-9464(00)08404-1.LDPCcodescanbespecifiedbyasparseparitycheckmatrix[4]–[6].Thebeliefpropagation(BP)algorithmcanbeused isdenotedby Thesetofchecksthatbit participatesisdenotedby .TheLog-BPalgorithmisoutlinedbelowusingthenotationof[5].Supposeacodeword istransmittedthroughanAWGNchannelwithsymbols+1and Thereceivedchannelvectoris .Define ,where isthesignand istheabsolutevalue.Similardefinitionsareusedinthefollowing,wherethefirstvariableindicatesthesignofarealvalueandthesecondvariableisitsabsolutevalue. forall and .Iteration: (1) (2)where .Pseudo-posterioriLLR: if III.MAGNETICYSTEMSWITHLDPCCApracticalmagneticrecordingsystemusinganLDPCcodeisshowninFig.1.TheproposedsystemisbasedonamodifiedextendedE PR4(ME PR4)channelwith Ontherecordingend,theuserdatablockisencodedbyarate16/17quasi-maximum-transition-run(QMTR)code[9],andthenfurtherencodedbyanLDPCcode.Thesequence0018–9464/00$10.00©2000IEEE IEEETRANSACTIONSONMAGNETICS,VOL.36,NO.5,SEPTEMBER2000 Fig.1.BlockdiagramofaPRchannelwithanLDPCcode.ofLDPCcheckbitsisinsertedwithguardbitssothattherun-lengthconditionsaresatisfied.Onthereadingend,anprobability(APP)decoder[10],[11]matchestheprecodedME PR4channel.TheAPPdecodertakes andtheapriori ,andcomputestheaposteriori TheLDPCdecodertakestheaposterioriLLRofthechanneldecoderasinput in(1),andoutputstheLLRin(3).TheextrinsicLLR isfedbacktothechannelAPPdecoderastheaprioriThedecodingprocesshastwoiterationloops.OneistheLDPCloopwithintheLDPCdecoder.AftereachiterationofLDPCdecoding,thedecoderchecksthesyndrome .Ifavalidcodewordisfound,theLDPCdecodingisfinished,andthewholedecodingprocessstops.Theotheriterationloopisthechannelloop.ItistheturboequalizationbetweenthePRchannelandtheLDPCcode[12].ThechannelloopiterationtakesplaceonlywhenthemaximumnumberofLDPCloopit-erationsisreachedwithoutfindingavalidcodeword.IV.SIMULATIONESULTSInoursimulation,theLorentzianchannelwithisolatedpulse isassumed.Userdensityisdefinedas ,where istheuserbitduration.Allsimulationsareperformedatuserdensity2.8.ThechannelisequalizedtotheME PR4channelresponse,andadditivewhiteGaussiannoisewithvariance isassumedbeforetheequalizer.Thesignal-to-noiseratio(SNR)isproportionaltotheratiooftheamplitudeoftheisolatedpulseand WeinvestigatetwoLDPCcodes.LDPC1isarate0.9358codewithblocklength4376givenin[13],withcolumnweight3.WedesignedLDPC2,withrate0.9402,4352informationbits,alsowithcolumnweight3.ThecodewasconstructedusingthemethoddiscussedinSectionV.TheproposedsystemwithLDPC2hasoverallcoderate0.8674anduserblocksize4096bits,whereasthesystemwithLDPC1hascoderate0.8622anduserblocksize3854bits.In Fig.2.PerformanceofLDPCcodesonPRchannels.Fig.1,themaximumnumberofiterationsissetat50and100fortheLDPCandchanneliterations,respectively.SimulationresultsarepresentedinFig.2.Alsoplottedinthefigurearetheperformanceoftherate16/17run-length-limited(RLL)codedPR4channel,theRLLcodedEPR4channel,theQMTRcoded PR4channel,andtheLDPC1andQMTRcodedME Thesimulationresultsshowthatourproposedsystemachieveda7.5dBgainoveruncodedPR4ora5.9dBgainoveruncodedEPR4atbiterrorrate10 ThedecodingoftheLDPCcodeshasaparticularlyniceprop-erty.ThewholedecodingprocessstopsifavalidLDPCcode-wordisfound,orifthemaximumnumberofchanneliterationsisreachedwithoutfindingavalidcodeword.Thisprovidesanaturalstoppingcriterionfortheiterativedecodingaswellasaflagindicatingthataparticularblockcontainserrors,whichisadistinctadvantageoversystemsusingconvolutionalcodes.AnundetectederroroccurswhenavalidLDPCcodeworddif-ferentfromthecorrectoneisobtainedbytheLDPCdecoder. etal.:LOWDENSITYPARITYCHECKCODESFORMAGNETICRECORDINGCHANNELS2185 Fig.3.Performanceoftheproposedsystemwithfewchanneliterations.Throughoutoursimulation,noundetectederrorswereobserved.ThismaybeduetothelargeminimumdistanceoftheLDPCTheimpactofthemaximumnumberofchanneliterationswasinvestigated.Fig.3showstheperformanceoftheLDPC2codedsystemwith1,2,3and100maximumchanneliterations.Com-paredwith100maximumchanneliterations,theperformancedegradationisabout0.5dBifnoturboequalizationisallowed.ThetotalnumberofLDPCiterationsforablockisthesumofLDPCiterationsineachchanneliteration.Atbiterrorrate andwiththelimitontotalchanneliterationsbeing1,2,or3,theaveragenumberofchanneliterationsis1,1.2and1.5re-spectively.TheaveragenumberofLDPCiterationsundertheseconditionsisabout5,20and35respectively.FromFig.3,itcanbeseenthatatbiterrorrate10 ,thegainforamaximumof3channeliterationsisabout0.3dBoverasinglechanneliterationorinotherwordsnoturboequalization,butittakesafactorofsevenincreaseintotalnumberofLDPCiterations.V.LDPCCWewantedanLDPCcodetobelongenoughtoholdastan-dard4096-bitdisksectorafterpassingthroughtherate16/17QMTRencoderandwithacoderatearound0.94.Anyparitycheckmatrixcanbethoughtofasamany-to-manymappingfromcodewordbitstoparitychecksandviceversa[14].Ifwecreateaset ofthecodewordbits,witheachbit appearinginthesetanumberoftimesequaltotheweightofthatcolumnof,andasimilarset oftheparitychecks,thenanyparitycheckmatrixspondstosomepermutationfromelementsof toelements .ThegoalistofindamappingthatleadstoanLDPCcodematrixsuchthattheresultingcodehasno4-cycles.Ithasbeenfoundthat4-cyclesaredetrimentaltothebiterrorrateperformanceofLDPCcodes[6].Thecodedesignalgorithmisasfollows:1.Computethedesiredcodewordsize andnumberofparitychecks andrandomlygenerateapermutationofthedesiredsize(thetotalnumberofonesin2.Checktoseethatitcorrespondstoavalidmatrix.Ifwefindthatismappingthebit toacheck morethanonce,werandomlyswapthetargetofthatonemappingwithsomeothermappinginandrepeatuntilwegeta3.Checkthepermutationfor4-cycles.Ifwefindnone,weproceedtoStep5.4.Ifwedidfinda4-cycleinvolvingsomecodewordbit werandomlypickanothercodewordbit andexchangethetargetsofthetwochecksthatmapsthesetwobitstoandgobacktoStep2.5.Nowwehaveancorrespondingtoacycle-free,thefinalstageistoreorderthecolumnssuchthattheright- sectionisinvertibleandcanbeconvertedtoageneratormatrix.Whetherthisalgorithmconvergesinanygivensituationisbynomeansobvious.Inpractice,itappearsthatthealgorithmconvergesmuchlessrapidlywithattemptstocreatecodesofratesmuchabove0.95,codesofveryshortlength,andcodeswithcolumnweightlargerthanthree.VI.CAseriallyconcatenatedsystemusingourLDPCcodeandit-erativedecodinghasbeenintroducedforuseonaME equalizedLorentzianchannel.Simulationresultsshowthatagainof5.9dBoveruncodedEPR4atabiterrorrateof10 canbeobtained.ThesesignificantgainsmakeLDPCcodedsys-temsveryattractiveasanalternativetoturbocodedmagneticrecordingsystems.AlthoughthisworkwasdoneindependentlyofFanetal.[15],theauthorswererecentlymadeawareoftheirworkontheperformanceofanLDPCcodedsystemforanide-allyequalizedEPR4channel.[1]W.E.Ryan,“PerformanceofhighrateturbocodesonaPR4-equalizedmagneticrecordingchannel,”inProc.IEEEInt.Conf.Commun.,1998,pp.947–951.[2]W.E.Ryan,L.L.McPheters,andS.W.McLaughlin,“CombinedturbocodingandturboequalizationforPR4-equalizedLorentianchannels,”Proc.Conf.Inform.SciencesandSystems,1998,pp.489–494.[3]T.Souvignier,A.Friedman,M.Oberg,P.H.Siegel,R.E.Swanson,andJ.K.Wolf,“TurbodecodingforPR4:Parallelversusserialconcatena-tion,”inProc.IEEEInt.Conf.Commun.,1999,pp.1638–1642.[4]R.G.Gallager,“Low-densityparity-checkcodes,”IRETrans.Inform.,vol.IT-8,pp.21–28,Jan.1962.[5]D.J.C.MacKay,“NearShannonlimitperformanceoflowdensityparitycheckcodes,”Electron.Lett.,vol.33,pp.457–458,Mar.1997. ,“Gooderror-correctingcodesbasedonverysparsematrices,”IEEETrans.Inform.Theory,vol.46,pp.399–431,Mar.1999.[7]C.Berrou,A.Glavieux,andP.Thitimajshima,“NearShannonlimiterror-correctingcodinganddecoding:turbo-codes,”inProc.IEEEInt.Conf.Commun.,1993,pp.1064–1070.[8]R.J.McEliece,D.J.C.MacKay,andJ.-F.Cheng,“TurbodecodingasaninstanceofPearl’s‘beliefpropagation’algorithm,”IEEEJ.Select.AreasCommun.,vol.16,pp.140–152,Feb.1998.[9]T.Nishiya,K.Tsukano,T.Hirai,T.Nara,andS.Mita,“Turbo-EEPRML:AnEEPR4channelwithanerror-correctingpost-processordesignedfor16/17ratequasi-MTRcode,”inProc.IEEEGlobalTelecommun.Conf.,1998,pp.2868–2873. IEEETRANSACTIONSONMAGNETICS,VOL.36,NO.5,SEPTEMBER2000[10]L.R.Bahl,J.Cocke,F.Jelinek,andJ.Raviv,“Optimaldecodingoflinearcodesforminimizingsymbolerrorrate,”IEEETrans.Inform.Theoryvol.IT-20,pp.284–287,Mar.1974.[11]P.Robertson,E.Villebrun,andP.Hoeher,“Acomparisonofoptimalandsub-optimalMAPdecodingalgorithmsoperatinginthelogdomain,”inProc.IEEEInt.Conf.Commun.,1995,pp.1009–1013.[12]C.Douillard,M.Jezequel,C.Berrou,A.Picart,P.Didier,andA.Galieux,“Iterativecorrectionofintersymbolinterference:Turbo-equal-EuropeanTrans.Telecommunication,vol.6,pp.507–511,Sept./Oct.1995.[13]D.J.C.MacKay’s,,website.[14]D.J.C.MacKay,S.T.Wilson,andM.C.Davey,“Comparisonofcon-structionsofirregularGallagercodes,”IEEETrans.Commun.,vol.47,pp.1449–1454,Oct.1999.[15]J.Fan,A.Friedmann,E.Kurtas,andS.McLaughlin,“Lowdensityparitycheckcodesformagneticrecording,”inProc.Thirty-SeventhAllertonConf.Commun.,Control,andComputing,1999.