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IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.19,NO.4,APRIL2001649ReducedComplexitySequenceDetectionforHigh-OrderPartialResponseChannelsMichaelLeung,BorivojeNikolic,Member,IEEE,LeoKi-ChunFu,andTaehyunJeonDetectorhardwarecomplexityofhigh-orderpartialresponsemagneticreadchannelsisamajorobstacletohighdatarateoperationandreducedareaandpowerconsumption.Themethodpresentedherereducesthecomplexityofsingle-stepandtwo-stepimplementationsoftheViterbidetectorbyapplyingadistance-enhancingcodethateliminatessomestatesfromthecodetrellis.Thecomplexityofthedetectorisfurtherreducedbyeliminatingless-probablebranchesfromthetrellis.Thisisaccomplishedbyasimplecontrolmechanismthatusesthesignsoftheconsecutiveinputsamples.Thereducedsetofadd-com-pare-select(ACS)unitsisdynamicallyassignedtothedetectorstates,decreasingthecomplexityoftheViterbidetectorbyroughly50%.Thismethodisdemonstratedonhigh-orderpartialresponsesystemswiththeEPR4targetandan11-level/32-statetarget.Thesimulationresultsshownegligiblebiterrorrate(BER)degradationforsignal-to-noiseratiosintherangeofoperationofcontemporarydiskdrivereadchannels.IndexTerms—Magneticrecording,partialresponsesignalling, (1)where denotesaunitsampledelay.Afrequentlyusedsetoftargetsisrepresentedbyaclassofpolynomials[6] (2)When ,thechannelisknownaspartialresponseclass-4 correspondstoextendedPR4(EPR4); usuallydenotedasE PR4. requiresa -stateimplementation.Itisgenerallyreal-izedasanarrayof channel,andSectionVIdealswiththereducedcomplexityde- 650IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.19,NO.4,APRIL2001TABLEIIZESOFACSU TABLEIIRRORVENTSIN PR4.SEQUENCESINRACKETSNEOR thecomplexityofthetwo-stepdetectorallowsfortradingareaandpowerforspeed.B.Time-VaryingMaximumTransitionRunlength(TMTR)CodesandQuasi-MTR(QMTR)CodeRecently,severaltrelliscodesthateliminatethemostcommonerroreventsbyusingcodingconstraintshavebeenproposedforPRsignaling[8]–[18].MostcommonerrorsinhigherorderPRchannelsarecausedbysequencesofthreeormoreconsecutiverecordedtransitions.Therefore,restrictingthemaximumtransitionrunlength(MTR)intherecordingcodecaneliminatedominanterrorevents.ThelistofshortdistanceerroreventsfortheE PR4channelissummarizedinTableII.Errorevents(sequences)canbedefinedasdifferencesbetweenrecordedanddetecteddataUserdata: Channelinput: Channeloutput: Bliss[12]–[14],Karabedetal.[11]andKnudsen–Fitzpatricketal.[17]introducedatthesametimea8/9blockcodethatmapseightinputbitsintoninecodebitswithtime-varyingMTR(TMTR)constraints.ThemajordisadvantageofthiscodeisthatitsrestrictiononallowedtribitsrequiresanimplementationoftheViterbidetectorthatisvariableintime,whichsignificantlyincreasesthedetector’scomplexityandreducesthespeed.TheTMTRblockcodeisconstructedbydeletingallcode-wordsthatcontainquadbitsandbyrestrictingtribitstocertainpositions[11].Becausethebeginningofthetribitisallowedatpositions2,4,6,and9,withthetribitstartingatposition9beingwrappedupinthenextword,thisimplementationresultsinatime-varyingtrellis.WhenthecodeisappliedtoanE trellis,theresultingdetectorstillhas16states.Twostates,0101and1010,havetobeeliminatedfromthetrellisincycles1,3,5,7,and8,andthetrellischangeseveryninecyclestoconformtocodeconstraints.Therealignmentofthedetectoris,thus,re-quiredeveryninecyclestoaccommodatethesechanges.TheTMTRblockcoderesultsinmaximumzerorunlengthof [14].HigherratecodesarepossiblebyrelaxingtheMTRcon-straints.In[17],aconstructionofarate9/10code,whichalsorequiresatime-varyingdetector,isreported.Toavoidcode-ratelossintheTMTRcode,arate16/17quasi-MTR(QMTR)codewasreportedin[19].Thecodeeliminatesquadbitsbutleavesalltribits,whichresultsinahighercoderate.TheapplicationoftheQMTRcodeingeneraldoesnotincreasetheminimumerrordistance,butitreducesthenumberofpossibleerrorevents.IntheE PR4channel,theQMTReliminatesthedominanterroreventsoftypes2,3,and4fromTableII,leavingonlytworemainingerrorpatternstobecorrectedbyapostprocessor.C.Turbo-PostprocessingandParityBitsAnalternativewayofcopingwithdominanterroreventsusesapostprocessortocorrectthem.AsoriginallyproposedbyWood[20],Turbo-PRMLenhancestheperformanceofthePR4channel.ATurbo-PRMLdetectorconsistsofaconventionalPRdetectorfollowedbyapostprocessorcontrolledbyanerrorfilterblock.ThefiltercorrelatestheidealPRMLsampleswithactualequalizedPRMLsamples.Ifthecorrelationexceedsaspecifiedthresholdvalue,thepossibleerroriscorrectedinthepostprocessor.Toenhancetheperformanceoferrorpostprocessing,paritybitscanbeappendedtochanneldatablocksconsistingofseveralcodewords[21].Usually,oneormoreparitybitscanbeusedtoidentifyoccurrenceofanerrorinablock.Errorfilters,matchedtoaselectedsetofdominanterrorevents,areusedtopinpointtheerrorlocationandtocorrecttheerrorifthedetectedparityisincorrect.D.ReducedComplexitySequenceDetectorIthasbeendemonstratedthatthesequencecouldbeestimatedusingreduced-stateestimationwithasmalllossintheerrorrate[22].Theperformancelossislargerthan1dBfortheEPR4channelandisunacceptableforthemagneticrecordingapplica-tions.Near-optimalperformancewasachievedinEPR4-equal-izedmagneticreadchannels,withsignificantlyreducedcompu-tationalrequirements[23].Amethodproposedin[24],basedontheeliminationofpathswithhighererrordistances,resultedinacapabilitytosharehardwarebetweenstatesofthedetector.ThecomplexityoftheViterbidetectorcanbereducedbydy-namicallyeliminatingimprobablebranchesfromthetrellis[23].ShafieeandMoonproposedamethodofreducingthedetec-tioncomplexityoftheEPR4equalizedsequencebydividingtherangeofdataintosix“ambiguityzones.”Ambiguityzonedetec-tionwasfirstproposedbyKobayashiin1971[25].EqualizationtargetsinEPR4are and andtheyrepre-senttheboundariesbetweensixambiguityzones: and .This etal.:REDUCEDCOMPLEXITYFORHIGH-ORDERPRCHANNELSeliminationoftrellisbranchesisdependentonthevaluesofchannelsamples,andamaximumofonlytwovaluesareal-lowedastheoutput ofthePRchannel.Twovaluesareal-lowedforzones onlyonevalueisallowedforzones and Theprobabilityofnoiseadded—suchthatitbringsthesampleoutsideofthezone—issmall,assumingahighsignal-to-noiseratio(SNR)operatingrange.ThisleadstoasimplificationoftheEPR4trellisbyeliminationofrestrictedbranches.Thesizeofthedetectoris,thus,significantlyreduced,butitsimplemen-tationiscomplicatedbyaddingtimevariance.OtherapproachestoreducingthecomplexityoftheViterbidetectorhavebeenproposedin[26]and[27].Insteadofelim-inatingimprobablebranchesfromthetrellis,theseapproachesfocusonmergingtrellisstates.Usingsimilarconceptsasinsetpartitioning[28],symmetrictrellisstatesarepairedandmergedtoshareACSresourcesbycreatingparallelbranches.Incomingsamplevalues[27]orpreliminarydecisionsinthesurvivalreg-isters[26]areusedtodiscriminatethepathswithintheparallelbranches.In[27],acompare-select-add(CSA)structureisused[29],whichrequiresthecomparison,selection,andadditionop-erationstobecarriedoutsequentiallytoreducethehardware.Thetrellissimplificationmethodin[27]isnotsystematicandproducesnonoptimalCSAsharing.MultiplexersarerequiredattheinputandoutputofACSunitsandinbranchmetricsaddi-tions.Usingpreliminarydecisions[26]reducesthenumberoftrellisstatesbyhalf,butitintroducesanadditionaltightfeed-backloop.DecisionsfromtheACSunitsofthecurrentcompu-tationcycleareneededtogeneratebranchmetricsforthenextcycle.Atypicalimplementationrequiresparallelbranchmetricshardware,andtheACSdecisionswouldbeusedtomultiplexinthecorrectbranchmetricvaluesattheendofthecomputationcycle.III.TEDUCTIONBYONSTRAINTSANDATE8/9TMTRCODEWITHTATIONARYETECTORApplicationofcodingcanaffectthetrellisandthecomplexityofthedetectorbydefiningconstraintsthatrestricttheoccur-renceofcertainpatternsinthecode.OneofthecommonlyusedcodesthatreducesthecomplexityofthematchingdetectorintheE PR4channelistherate2/3(1,7)runlength-limited(RLL)code[30].Theresultingtrellishas10states,comparedwith16statesforafulltrellis.Thecodehasdistance-enhancingprop-ertiessimilartotheTMTRcode,butthecoderateislow.How-ever,arate8/9TMTRcoderequiresatime-varyingdetector.Atime-invarianttrelliswithsomestatespermanentlyeliminatedallowsfortheuseofalowercomplexitysequencedetectortomatchthecodeconstraints.ToexploretheTMTRcodestructureforpossibletrelliscom-plexityreduction,anewrate16:18TMTRcodeisderived.Thecodehasamaximumzerorunlengthof andthesamedis-tancepropertiesastherate8/9TMTRcode.Theresultingtrellishasaperiodof2,ratherthanof9.Theapplicationofthiscodekeepsthetwo-stepimplementationofthedetectorstationary.Thenewcodeeliminatesthesamedistanceerroreventsasthe8/9TMTRcode,bynotallowingquadbitsincodewordsandbylimitingtheoccurrenceoftribitstocertainbitpositionsin- Fig.1.Rate16:18TMTRcodewithanillustrationofresolvingboundaryquadbitsandtribits.sidethecodeword.Expansionoftheconstraintsto18-bitcode-words[12]andcreationofa16:18blockcoderesultinasta-tionarytrellis.However,direct16-to18-bitmappingwouldrequirelargeencoding/decodinglogicandtheresultingcodewouldhavelongbyteerrorproperties.Thenewcodeisconstructedbyaninitialencodingofoddandevenuserdatabytesseparately.Theoddcodewordsallowtribitstostartatevenbitpositions,whereasevencodewordsallowtribitstostartatoddbitpositions,thus,resultinginpossiblecon-catenation.IfaftertheinitialencodingeitheraquadbitoratribitatapositionthatviolatestheMTRconstraintisdetectedattheboundarybetweentheoddandevencodewords,theevencode-wordisremappedtoasetofreservedcodewords.Informationabouttheremappingisencodedintheadjacentoddcodeword,asillustratedinFig.1[15].Practicalerrorpropagationislim-itedtotwobytes.Three-byteerrorpropagationispossibleonlywhenanerroreventof5bitsinlengthhitsbothremappedbitsintheoddcodeword.Theencoderanddecoderaredesignedusing“gated”parti-tions[16],whichresultsinabout2.5morecomplexitythanaTMTRencoder/decoder.However,thisincreasedcomplexityisstillaverysmallportionofthereadchannelchip.Thiscode,whenappliedtotheE PR4trellis,requiresonlya14-state,two-step,stationaryViterbidetectorinsteadofa16-statedetector.These14statesrequireeightfour-wayandsixthree-wayACSunitsforimplementation,asshowninFig.2.IV.IMPROVEDETHODOFThecomplexityoftheViterbidetectorcanbereducedbydynamicallyeliminatingimprobablebranchesfromthetrellis,basedonthevaluesofchannelsamples[23].TheschedulingofdynamicallyassignedACSunitsiscomplicated,especiallywhenitisappliedtohigh-orderPRtargets.Whenpostpro-cessingtechniquesareusedtocorrectdominanterrorsintheViterbidetector,thenumberofprunedbrancheshastobecarefullydeterminedsothatthereducedtrellisoperationwillnotaffecttheoverallbiterrorrate(BER).Anexampleofan PR4trellisisusedasademonstrationoftheimprovedtrellisreductionscheme.WiththeapplicationofatrelliscodetotheE PR4channel,theminimumsquareddistanceforanerroreventis10.Thiscor-respondstoasingle-bitchannel-inputerrorevent, orequivalently, attheoutputofthe 652IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.19,NO.4,APRIL2001 Fig.2.Two-stepE PR4trelliswithstates1010and0101eliminatedbycodingchannel.Forsample-by-samplethresholddetection,theequiv-alentsampleerroreventforanoiselesssampleofvalue segmentedintotheregions[ ]or[ ]is .Therefore,foranobservedsampleofvalue ,pruningoffalltrellisbranchesexceptthosethatcorrespondto[ ],where and aretheceilingandfloorof willhavelittleeffectontheoverallsystemBER.Theeffectiveerrordistanceofpruningoutthetruebranchis dBlargerthanthatofthedominanterroreventoftype .Moreover,theerrordistanceofcommontrellis-coded PR4errorevents,e.g., willfur-therbereducedbycorrelationbetweenthenoisesamples.Theprobabilityofmisdetectingthetruesignalsequence asan-othersignalsequence (assuming isalsoavalidsequence)is (6)where and isthevarianceofthenoiseprojectedintotheerroreventsubspacealongthedirectionoftheerrorevent (7) Fig.3.RelativeerroreventSNRforpruningerrorandthreedominantregularerrorevents.SNRisnormalizedsuchthattherelativeSNRfor[+]zeroatuserdensityof3.5. isthevectorrepresentationof ,and istheauto-correlationmatrixofthenoiseattheoutputoftheequalizer.Fortheerrorevent thecor- ’sare and .Tocomparetheprobability ofpruningthetruebranch(ey=[4])tovariousdominanterroreventsintrellis-codedE PR4systems,wedenote theerroreventSNRfor Fig.3showstherelativeerroreventSNRforaLorentzianpulsewithwhiteGaussiannoise.FourerroreventSNRcurvesarecharted,includingthetrellispruningerrorevent, andthethreemostdominanterroreventsinatrellis-coded PR4system.AsshowninFig.3,theSNRforthepruningerroreventisatleast3dBlargerthanfortheotherdominanterrorevents.Itisaround4dBbetterthan (ratherthanonly2.04dB),duetotheeffectofnoisesamplecorrelation.Furthermore,theSNRpenaltyforpruningthiserroreventissignificantlylargerthanforthenexttworegulardominanterrorevents.Therefore,theeffectofbranchpruningonsystemBERshouldbeinsignificantevenifpostprocessingisusedtofurthereliminatemajordominanterrorevents.ThiseffectwillbefurtherdemonstratedbysimulationresultsforapracticalTheseparationofallpossibleinputcasesthatcorrespondto foreachsamplevaluewouldresultincomplicatedcontrolandselectionlogic.Inthispaper,asimplemethodologyoftrellisreductionistraced.Asinglethresholdatzeroisused.Themethodusesthesignofashortsequenceofdatasamplestoconfigurethedetector:1)Iftheinputsignalisgreaterthan0,mostofthenegativesignallevelsaredisallowed.Whentheinputsignalisneg-ative,mostofthepositivesignallevelsaredisallowed.2)Furthertrellisreductionispossiblebyobservingthesignsofprecedingandfollowinginputsamples.Becausecer-tainstateswouldbedisallowedbythepolarityofprevioussamplesandbecauseotherstateswillbedisallowedbythepolarityofthefollowingsamples,additionalbranchescanbeprunedfromthetrellisinthecurrentstate.3)AdditionaltrellisbranchesareaddedbacktotheprunedtrellistosimplifytheACSunitassignment. etal.:REDUCEDCOMPLEXITYFORHIGH-ORDERPRCHANNELS (a)(b)(c)(d) (e)(f)(g)(h)Fig.4.Reducedtrellisesfortheincomingsignalsamples,previousone,currenttwo,andnextone,markedaspositive()ornegative():(a)����,(b),(c),(d),(e),(f),(g),and(h)+++Theeffectofthesetrellis-pruningstepscanbemoreeasilyun-derstoodbyexaminingtheirapplicationtotheE PR4trellisinthenextsection.V.RETECTORMPLEMENTATIONIN PR4CWiththeapplicationofthenewTMTRcode,thestates0101and1010arepermanentlyeliminatedandtheresultingtwo-step PR4trellishasonly14states,asshowninFig.2.ThemethodoutlinedinSectionIVisappliedtotheresultingtrellis.1)Iftheinputsignallevelisgreaterthan0,theallowedinputlevelsare andwhentheinputsignallevelislessthan0,theallowedinputlevelsare 2)Asequenceoffourincomingsamplesisanalyzed,andtheresultingtwo-steptrellisisformed.Theselectionisbasedontheprecedingsample,twocurrentsamples,andthenextsample.Basedontheabovecriteria,thereducedtrellisesinFig.4areformed.Onlyeightoutof16reducedtrellisesareshowninFig.4,becausetheremainingeightaresymmetricaltotheonesshown.Fortheimplementationofthistypeofsequencedetector,asmallernumberofACSunitsisneeded.Aminimalimplemen-tationofthe14-stateE PR4trelliscontainsthefollowing:•fourfour-wayACSunits; 654IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.19,NO.4,APRIL2001 (a) Fig.5.(a)Exampleofsharingtwo-wayACSbetweenstates0and15.(b)Exampleofsharingtwo-wayACSbyusingonlyinputmultiplexer.•onethree-wayACSunit;•fourtwo-wayACSunits;•twoadders(one-wayACS).The14stateswillbedynamicallyassignedto11ACSunits,whichrequireadditionalmultiplexersinthecriticalpathandcontrollogicoutsideofthecriticalpath.ThisapproachreducesthehardwarecomplexityoftheViterbidetectorbyapproximately50%.Forexample,thetotalnumberofaddersinACSunitsisreducedto29,comparedwith64inthefulltrellisimplementation.Fordifferentsequencesofinputs,differentdetectorstructuresareneeded,asseeninFig.4.TherequirementsforthenumberandsizeofACSunitsforeachstatevarywithdifferentsamplesequences.ThedirectassignmentplanforthestatesthatsharethereducedsetofACSunitswouldleadtoacomplicatedcon-trol,resultinginsignificantoverheadinmultiplexerlogic.Themultiplexersofdifferentsizes,fromtwo-waytofive-way,wouldbeneeded.A.SimplifiedACSAssignmentAsimplesolutionistoaddmoreresourcesbesidestheinitialfourfour-way,onethree-way,fourtwo-wayACSunits,andtwoadders.Thestraightforwardassignmentplancanbederivedifthesymmetricalstates(0and15,1and14,2and13,3and12,4and11,6and9,7and8)sharethesameresources.Inthiscase,afewbranchesareaddedbacktotheprunedtrellisandsomeoftheACSunitsareextendedtoacceptmoreinputs.Theminimumsolutionrequiresfourfour-way,threethree-way,threetwo-wayACSunits,andoneadder.Intheproposedimplementation,thebranchmetrics(BM)andACSunitswouldbeassignedtogether.Thisimplementa-tioneliminatestheBMmultiplexersinfrontoftheACSunits. Fig.6.Reducedcomplexitydetectorstructure.SharingofACSunitscanbedonebyaddingthemultiplexersbothattheinputsandattheoutputsoftheACSunits.Becausetheassignmentissimple,onlytwocontrolsignalsareneededattheinputandattheoutput,whichwillcontrol11two-waymultiplexersattheinputand11two-waydemultiplexersattheoutput.Becausethestates{0,15},{1,14},{2,13},{3,12},{4,11},{6,9},and{7,8}sharethesameresources,thecontrollogicisbasedonlyonasinglesamplesign[Fig.5(a)].Moreover,thedemultiplexersafterACSunitscanbeelimi-natediftheimplementationisACS-based,orassignmentdriven,asopposedtobeingbasedonconventionalstates[Fig.5(b)][31].Inthiscase,wearebasingourdetectiononACSunits,notonactualstates.ThisisthereasonwhyACSunitsarenamed:0/15,15/0,1/14,2/13,13/2,3/12,4/11,11/4,6/9,9/6,7/8.Thisresultsinaslightlymorecomplicatedcontrolof11two-waymultiplexersattheinput.TheresultingACSassignmentplanisshowninTableIII,andthedetectorarchitectureisshowninFig.6.Aproblemthatarisesbyaddingextrabranchesinthetrellisisthatsomeofthestatesthatgeneratethemdonotexist.Suchstatesshouldnotbeconsideredinthecalculationofnewstates.Thiscanbedonebyaddingasinglevaliditytagbittoeverystate. etal.:REDUCEDCOMPLEXITYFORHIGH-ORDERPRCHANNELSTABLEIIIACSUSAGEEPENDINGONTHEIGNSWITHESOURCESANDNDICATETHEIZEOFTHEACSU InvalidstatescouldexistincasesinwhichtwostatesshareonlyoneACSresource(states1and14,3and12,7and8).Thereducedcomplexitydetectorhasonlyfourtimingcriticalfour-wayACSunits,whichresultsinamuchshorterintercon-nectlength.ThespeedpenaltyassociatedwithaddingthegatethatevaluatesthetagbitandmultiplexersiswellcompensatedforbyreducedinterconnectlengthintheACSarray.Thedetectorstructureresultingfromtheassignmentplanusingsymmetricstatesissimilartothestructurereportedin[27].AsdescribedinSectionII-D,themethodin[27]producesnonoptimalACSsharing,whichrequires12trellisstateswhenappliedtotheTMTR-codedtrellis.TheadditionofextrabranchesbacktotheprunedtrellisresultsinonlyonemultiplexingoperationattheinputofACS,whereasforthedetectorin[27],multiplexersarerequiredattheinputandattheoutputofACSunitsaswellasinBMadditions.Moreover,themultiplexingoperationsattheinputofACSarerelativelycomplex.ThereducedcomplexitysystemwassimulatedtoverifytheproperoperationandtoevaluatethepossiblelossinBERcomparedwithfullimplementation.Theresultingcomparisonoftrellis-codedE PR4channels,atauserdensityof3.0,implementingan8/9codeandafulltrellisfrom[11]withstationaryTMTRcodewithreducedtrellis,isshowninFig.7.AbaselineperformanceisshownfortheRLL-coded,rateTABLEIVEQUIREDIZESANDPERATIONSINACSU 16/17EPR4detector.Simulationsconfirmthat,indeed,thereisasmallloss—below0.3dB—atlowerSNRs,andvirtuallynolossathigherSNRwithBERlowerthan .Additivewhite 656IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.19,NO.4,APRIL2001 Fig.7.ComparisonofBERversusSNRfortrellis-codedE PR4channels(SNRlevelsareshownbeforeequalization).Gaussiannoise(AWGN)isaddedtotheinputoftheanalogfrontendofthechannel.SNRisdefinedasthezero-to-peakamplitudeoftheinputsignaloverthenoisepowerwithintwotimestheNyquistbandwidth.BERcurvesaregeneratedwith100errorsforeachdatapointinthegraph(i.e., bitsareexercisedforBERat ).Fig.8demonstratestheperfor-manceofthereducedcomplexitydetectorwhenusedwithanerrorpostprocessor.Fourperformancecurvesareshown;theycorrespondtothefollowingdetectors:1)E PR4withstationaryTMTRcode,fulltrellis;2)E PR4withstationaryTMTRcode,reducedtrellis;3)E PR4withstationaryTMTRcode,fulltrelliswithideal4)E PR4withstationaryTMTRcode,reducedtrelliswithidealpostprocessing.Allgraphsareshownwithrate8/9(16:18)codes;i.e.,therewasnocode-rateadjustmentforeventualparitybitsusedtoen-hancethepostprocessing.Postprocessingsystemsaredenoted inFig.8.Theidealpostprocessingisimplementedinasimulatorbynotcountingerrorsoftype[ ]and[ ],thus,assumingidealpostprocessorperformance.Thepurposeofthissetupistoverifytheperformanceofthereducedcomplexitydetectorinthepresenceoftheerrorpostprocessor.AsseenfromFig.8,thereisnoperformancedegradationbetweenfullandreducedtrellisimplementationswithorwithoutapostprocessor.VI.APPLICATIONOFTHEETHODTOARTIALTodemonstratethewideapplicationrangeoftheproposedmethodforreductionofdetectorcomplexity,itwasappliedtoseveralhigherorderPRchannelssuitableforapplicationathigheruserdensities.Athigheruserdensities,channelBERcanbeimprovedbyemployinga32-stateViterbidetectorthatcorrespondstotargetsthatbettermatchthechannelresponse.ThePRoftype matchestheLorentzianchannelatauserdensityof3.0.Theresultingtargethas11discretesignallevels[ Fig.8.ComparisonofBERversusSNRforfullandreducedimplementationsofatrellis-codedE PR4channelwithandwithoutpostprocessing. Fig.9.Performancecomparisonofthenew[121][211]targetversusE withtrellisandRLLcodes. ]andrequiresa32-stateViterbide-tector.Thedominanterroreventsforthischannelare[ withasquareddistanceof10,[ ]withsquareddistanceof20,andasinglebiterroreventwithasquareddistanceof22.Thus,thecodinggainobtainedbyusingtheTMTRcodethateliminatesallshort-distanceerroreventsis3.4dB.Withcode-ratelossaccountedfor,thenewtargetdemonstratesabout1.7dBimprovementovertheE PR4channelwitha16/17RLLcode,orabout1dBimprovementwiththeuseoftheTMTRcode,asshowninFig.9.ApplicationoftheQMTRcodetothe32-statetrellisperma-nentlyeliminatestwostates(01010and10101)ineverystep,asshowninFig.10(a).ApplicationoftheTMTRcodetothe32-statetrelliseliminatesthestates01010and10101aswellastwomorestates,butresultsinastep-by-steptimevaryingstruc-ture,asshowninFig.10(b)and(c).Usingatwo-stepdetectorstructureforbothQMTRandTMTRcodesresultsinstationarytrellises,asshowninFig.11(a).TwostatesareremovedinthecaseoftheQMTRcode,andfourstatesareremovedwhenusingaTMTRcodere-sultingintwopossibletrellisstructures,asshowninFig.11(b)and(c). etal.:REDUCEDCOMPLEXITYFORHIGH-ORDERPRCHANNELS (a)(b)(c)Fig.10.(a)Single-steptrellisdiagramwithQMTRcode,(b)single-steptrellisdiagramwithTMTRcode(versionA),and(c)single-steptrellisdiagramwithTMTRcode(versionB).ThereductionmethoddescribedinSectionIVwasappliedtothenewtargetwitha16/18TMTRcodeanda16/17QMTRcode.Whentheincomingsampleispositive,anyBMthatcor-respondstoatargetvalueof orbelowisremovedfromthetrellis.Whenthesampleisnegative,BMsthatcorrespondtotargets 2oraboveareeliminated.TodemonstratetheeffectoftrellisreductioninBER,thefollowingtestcaseswereselected:1)newtargetwithQMTRcode(rate16/17);2)newtargetwithTMTRcode(rate16/18);3)newtargetwithQMTRcodewithidealpostprocessing(rate8/9,assumesadding3bitsfor6bytesforrequiredparityforpostprocessing);4)newtargetwithTMTRcodewithidealpostprocessing(rate48/58,assumesadding4bitsfor6bytesforrequiredparityforpostprocessing).Allfourofthesecasesweresimulatedtwicewithfullandwithreducedcomplexitydetectors.TheresultssummarizedinFig.12shownoperformancedegradationinanyofthecases.Thepostprocessingoperationsareassumedidealinthesimulations.Threeerroreventtypeshadsimplybeenomittedinerrorcounting.Theremovederrorpatternswereasfollows:•inQMTR: •inTMTR: Toeachdatablockof6userbytes,evenandoddparitybitsareaddedwithadditionalbitsneededtoavoidcodeconstraintVII.CSTIMATIONTheimplementationofasingle-stepE PR4Viterbidetectorrequiressevenbranchmetricsunits(BMU),16ACSunits,and16survivalregisters.Eachsingle-stepACSunitconsistsoftwoaddersandonecomparator,usuallyimplementedasasubtractor.Asthesubtractorsrequireonlycarrychainforimplementationofthecomparison,theyarelesscomplexthanadders.Forfulltwo-stepimplementation,BMsareaddedtogethertoform39differenttwo-stepBMs.Afour-wayACSunitconsistsoffouraddersandsixparallelsubtractors. 658IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.19,NO.4,APRIL2001 (a)(b)(c)Fig.11.(a)Two-steptrellisdiagramwithQMTRcode,(b)two-steptrellisdiagramwithTMTRcode(versionA),and(c)two-steptrellisdiagramwithTMcode(versionB). Fig.12.PerformancecomparisonoffullversusreducedcomplexityimplementationforTMTRandQMTRcodeswithandwithoutpostprocessing.TableIshowsthenumberofaddersandcomparatorsrequiredandtheequivalentsizesofone-,two-,three-,andfour-wayACSunits,assumingthesizeofanadderis1.5timeslargerthanthatofacomparator.TableVshowsthatreducedcomplexitydetectorimplementstheViterbialgorithmatlessthana30%ofhardwareincreasefromsingle-stepwhileachievingtwo-stepthroughput.Fulltwo-stepimplementationisapproximately2.7timesbiggerthanissingle-stepimplementation.ToestimatethesavingsinhardwarerequirementswhenthereductionmethodisappliedtothePR(8),thefollowingtestcasesareselected:1)stationaryTMTRcodewithtwo-stepdetector;2)QMTRcodewithtwo-stepdetector;3)QMTRcodewithsingle-stepdetector.TableVIshowsthenumberofACSunitsusedwithQMTRandTMTRcodesforsingle-stepViterbiimplementation,where etal.:REDUCEDCOMPLEXITYFORHIGH-ORDERPRCHANNELSTABLEVOMPARISONBETWEENMPLEMENTATIONSOF PR4DETECTORS TABLEVIOMPARISONOFETECTORSFORTHEARGETWITH TABLEVIIOMPARISONOFETECTORSFORTHEARGETWITH TMTR_AdenotesthetrellisusedintheoddcycleandTMTR_Bdenotesthetrellisusedintheevencycle.Italsocomparesthehardwarecomplexityoftheregular,single-step32-stateViterbiusingaQMTRcodewiththereducedsingle-stepstructureandshowsthatthereducedtrellismethodresultsina45%reductioninsize. 660IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.19,NO.4,APRIL2001ThenumberofACSunitsrequiredandtheestimatedsizeofdifferenttwo-stepdetectorsareshowninTableVII.Thetablesummarizesthecomplexitiesofthefulltwo-stepimplementa-tionsandthesavingsthatcorrespondtodifferentcodes.There-duced-complexitydetectorshowsabouta45%reductioninsizeforbothQMTRandTMTRcodes,comparedwiththefullim-VIII.CTheproposedmethodsignificantlyreducestheimplementa-tioncomplexityoftheViterbidetectorforhigh-orderPRchan-nels.Thisisachievedbyeliminatinglesslikelytakenbranchesfromthetrellis,inconjunctionwithappliedtrelliscoding.Theeliminationusesasimplemethodbasedonthesignoftheinputsample.Thesmallerareaandpowerofthedetector,achievedbyusingasimpleassignmentofACSunits,couldbetradedforspeedenhancement.SystemlevelsimulationshaveshownthatthereisnosignificantlossinBERwhencomparedwithfulltrellisimplementation.Themethodisdemonstratedonatrellis-codedE PR4channelresultingina50%complexityreduction.Similarcomplexityandperformanceresultsareachievedfora32-statedetectorthatmatchesadifferentequalizationtarget,withdifferentrecordingcodesanddetectorimplementations.CKNOWLEDGMENTThisworkwasinitiatedbythelateR.Yamasaki,forwhoseguidancetheauthorsaredeeplygrateful.[1]G.Forney,Jr.,“Maximum-likelihoodsequenceestimationofdigitalse-quencesinthepresenceofintersymbolinterference,”IEEETrans.In-form.Theory,vol.IT-18,pp.363–378,May1972.[2]A.J.Viterbi,“Errorboundsforconvolutionalcodesandanasymptoti-callyoptimumdecodingalgorithm,”IEEETrans.Inform.Theory,vol.IT-13,pp.260–269,Apr.1967.[3]G.D.Forney,Jr.,“TheViterbialgorithm,”Proc.IEEE,vol.61,pp.268–278,Mar.1973.[4]P.KabalandS.Pasupathy,“Partial-responsesignaling,”IEEETrans.,vol.COM-23,pp.921–934,Sept.1975.[5]H.KobayashiandD.T.Tang,“Applicationofpartial-responsechannelcodingtomagneticrecordingsystems,”IBMJ.Res.Devel.,vol.14,pp.368–375,July1970.[6]H.ThaparandA.Patel,“Aclassofpartialresponsesystemsforin-creasingstoragedensityinmagneticrecording,”IEEETrans.Magn.vol.MAG-23,pp.3666–3668,Sept.1987.[7]P.BlackandT.Meng,“A140MB/s32-stateradix-4Viterbidecoder,”IEEEJ.Solid-StateCircuits,vol.27,pp.1877–1885,Dec.1992.[8]R.KarabedandP.Siegel,“Codingforhigherorderpartialresponsechannels,”inProc.1995SPIEInt.Symp.onVoice,VideoandData,vol.2605,Philadelphia,PA,Oct.1995,pp.115–126.[9]J.MoonandB.Brickner,“Maximumtransitionruncodesfordatastoragesystems,”IEEETrans.Magn.,vol.32,pp.3992–3994,Sept.[10]B.E.Moision,P.H.Siegel,andE.Soljanin,“Distance-enhancingcodesfordigitalrecording,”IEEETrans.Magn.,vol.34,pp.69–74,Jan.1998.[11]R.Karabed,P.H.Siegel,andE.Soljanin,“Constrainedcodingforbi-narychannelswithhighintersymbolinterference,”IEEETrans.Inform.,vol.45,pp.1777–1797,Sept.1999.[12]W.G.Bliss,“An8/9ratetime-varyingtrelliscodeforhighdensitymag-neticrecording,”inProc.Dig.IEEEInt.Magn.Conf.,InterMag’97NewOrleans,p.BS-09. 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,“Trelliscodingsystemfordiscstoragesystems,”U.S.Patent6.032.284,Feb.29,2000.[15]B.Nikolic,M.Leung,andL.Fu,“Arate8/9slidingblockcodewithstationarydetectorformagneticrecording,”IEEETrans.Magn.,vol.37,pp.1653–1657,May2001,tobepublished.[16]B.NikolicandM.Leung,“Slidingblock(rate8/9)trelliscodeformag-neticrecording,”U.S.Patent6,081,210,June27,2000.[17]K.Knudson-FitzpatrickandC.S.Modlin,“Time-varyingMTRcodesforhighdensitymagneticrecording,”inProc.1997GlobalTelecomm.Conf.(GLOBECOM’97),Conf.Record,Phoenix,AZ,pp.1250–1253.[18]K.A.SchouhamerImmink,P.H.Siegel,andJ.K.Wolf,“Codesfordigitalrecorders,”IEEETrans.Inform.Theory,vol.44,pp.2260–2299,Oct.1998.[19]T.Nishiya,K.Tsukano,T.Hirai,S.Mita,andT.Nara,“Rate16/17max-imumtransitionrun(3;11)codeonanEEPRMLchannelwithanerror-correctingpostprocessor,”IEEETrans.Magn.,vol.35,pp.4378–4386,Sept.1999.[20]R.Wood,“Turbo-PRML:AcompromiseEPRMLdetector,”Trans.Magn.,vol.29,pp.4018–4020,Nov.1993.[21]T.Conway,“Anewtargetresponsewithparitycodingforhighdensitymagneticrecordingchannels,”IEEETrans.Magn.,vol.34,pp.2382–2386,July1998.[22]M.V.EyubogluandS.U.H.Qureshi,“Reduced-statesequenceestima-tionforcodedmodulationonintersymbolinterferencechannels,”J.Select.AreasCommun.,vol.7,pp.989–995,Aug.1989.[23]H.ShafieeandJ.Moon,“Areduced-complexitytrellissearchdecodingalgorithmforextendedclassIVpartialresponsesystems,”inCOMM/ICC’92,vol.1,Chicago,IL,pp.120–124.[24]R.Behrensetal.,“Methodandapparatusforreduced-complexityViterbi-typesequencedetectors,”U.S.Patent5.291.499,Mar.1994.[25]H.KobayashiandD.T.Tang,“Ondecodingofcorrelativelevelcodingsystemswithambiguityzonedetection,”IEEETrans.Commun.,vol.COM-19,pp.467–477,Aug.1971.[26]R.He,J.R.Cruz,andH.Song,“Areduced-statesequenceestimationtechniqueanditsapplications,”IEEETrans.Magn.,vol.35,pp.2331–2333,Sept.1999.[27]K.Tsukano,T.Nishiya,T.Hirai,andT.Nara,“SimplifiedEEPRViterbidetectorbasedonatransformedradix-4trellisforadiskdrive,”Trans.Magn.,vol.35,pp.4387–4401,Sept.1999.[28]J.Bergmans,DigitalBasebandTransmissionandRecording.Dordrecht:KluwerAcademic,1996.[29]G.Fettweis,R.Karabed,P.H.Siegel,andH.K.Thapar,“Reduced-com-plexityViterbidetectorarchitecturesforpartialresponsesignaling,”inProc.1995GlobalTelecommun.Conf.(Globecom’95),Singapore,pp.[30]R.T.BehrensandA.J.Armstrong,“Anadvancedread/writechannelformagneticdiskstorage,”inProc.Conf.Record26thAsilomarConf.Sig-nals,Syst.Comput.,PacificGrove,CA,Oct.26–28,1992,pp.956–960.[31]B.Nikolic,M.Leung,L.Fu,V.G.Oklobdzija,andR.Yamasaki,“Re-ducedcomplexitysequencedetectionforE PR4magneticrecordingchannel,”inProc.1999IEEEGlobalConf.Commun.,GLOBECOM’99,Conf.Record,vol.1,partB,RiodeJaneiro,Brazil,pp.960–964. MichaelLeungreceivedtheB.Eng.degreeinelec-tronicandcommunicationengineeringfromtheUni-versityofBirmingham,Birmingham,U.K.,in1987,andtheM.S.andPh.D.degreesinelectricalengi-neeringfromStanfordUniversity,Stanford,CA,in1988and1993,respectively.HewastheManageroftheAdvancedChannelAr-chitectureGroupinTexasInstrument’sStorageProd-uctsGroup,SanJose,CA,wherehehadworkedonthedefinitionofsignalprocessingarchitecturesandimplementationofadvancedHDDstoragechannelICs.HeiscurrentlytheManageroftheDSPArchitectureGroupwithTripathTechnology,SantaClara,CA,andisworkingonarchitectureofADSLandopticaltransceiverICproducts.Histechnicalinterestsincludecommunicationchanneltheory,coding,DSParchitectureandimplementation,andhigh-speedCMOSdigitaldesign. etal.:REDUCEDCOMPLEXITYFORHIGH-ORDERPRCHANNELS BorivojeNikolic(S’93–M’99)receivedtheDipl.IngandM.Sc.degreesinelectricalengineeringfromUniversityofBelgrade,Yugoslavia,andthePh.D.degreefromtheUniversityofCalifornia,Davis,in1992,1994,and1999,respectively.HeisanAssistantProfessorintheDepartmentofElectricalEngineeringandComputerSciencesoftheUniversityofCaliforniaatBerkeley.HewasonthefacultyoftheUniversityofBelgradefrom1992to1996.Duringhisgraduatestudies,hespenttwoyearswithSiliconSystems,Inc.,TexasInstrumentsStorageProductsGroup,SanJose,CA,workingondiskdrivesignalprocessingelectronics.Hisresearchinterestsincludehigh-speedandlow-powerdigitalintegratedcircuitsandVLSIimplementationofcommunicationsandsignalprocessingalgorithms.HereceivedtheCollegeofEngineeringbestdoctoraldissertationawardandtheAnilK.JainprizeforthebestdoctoraldissertationinelectricalandcomputerengineeringattheUniversityofCaliforniaatDavisin1999,aswellastheCityofBelgradeawardforthebestDiplomathesisin1992. LeoKi-ChunFureceivedtheB.SdegreeinelectricalengineeringandcomputersciencefromtheUniversityofCaliforniaatBerkeleyin1993.From1992to1993,heworkedasaResearchAssistantintheCenterofX-rayOpticsatLawrenceBerkeleyLaboratory,whereheconductedresearchondiffractionpropertiesofX-raythroughvariesmetal.In1993,hejoinedtheAdvancedChannelArchitectureGroupinSiliconSystems,Inc.,lateracquiredbyTexasInstrumentsIncorporated(TI)asTIsStorageProductsGroup,SanJose,CA,whereheworkedasaSystemEngineeronsignalprocessingarchitectureandmodelingDSPalgorithmonadvancedHDDstoragechannelICs.Histechnicalinterestsincludecoding,DSParchitecture,andimplementation.In2000,hejoinedtheDSPArchitectureGroupatTripathTechnology,SantaClara,CA,andiscurrentlyworkingonthearchitectureofADSLandopticaltransceiverIC TaehyunJeonreceivedtheB.S.degreefromYonseiUniversity,Seoul,Korea,in1989,andtheM.S.andPh.D.degreesfromtheUniversityofMinnesota,Minneapolis,in1993and1997,respectively,allinelectricalengineering.From1994to1997,hewasaResearchAssistantwiththeUniversityofMinnesota,Minneapolis.Uponcompletionofhisgraduatestudies,hejoinedMotorola,Inc.,whereheworkedontheadvancedreadchannelarchitecturedevelopmentforhigh-den-sitydatastoragesystems.Since1998,hehasbeenwithTexasInstruments,Inc.,SanJose,CA,whereheiscurrentlyaMemberoftheTechnicalStaff.Hiscurrentresearchinterestsincludecommunicationtheoryandsignalprocessingfordigitaldatatransmissionsystemswithemphasisonchannelcharacterizationandmodeling,modulationcoding,equalization,anddetection.