Performance Analysis of ML-Based Feedback Carrier Phase Synchronizers for Coded Signals

PerformanceAnalysisofML-BasedFeedbackCarrier

PhaseSynchronizersforCodedSignals

NeleNoels,HeidiSteendam,Member,IEEE,andMarcMoeneclaey,Fellow,IEEE

Abstract—Thispaperconsiderscarrierphaserecoveryintrans-missionsystemswithaniterativelydecodableerror-controlcode[turbocodes,low-densityparitycheck(LDPC)codes],whoselargecodinggainsenablereliablecommunicationatverylowsignal-to-noiseratio(SNR).Wecomparethreetypesoffeedbackphasesynchronizers,whichareallbaseduponthemaximum-like-lihood(ML)estimationprinciple:adata-aided(DA)synchronizer,anon-code-aided(NCA)synchronizer,andaniterativecode-aided(CA)synchronizer.Weintroduceablockwiseforward–backwardrecursivephaseestimator,andweshowthatthemean-squarephaseerror(MSPE)oftheNCAsynchronizerequalsthatoftheDAsynchronizerwhenthecarrierphaseisconstantandtheloopfiltergainisthesameforbothsynchronizers.Whenthesignalisaffectedbyphasenoise,theNCAsynchronizer(ascomparedwiththeDAsynchronizer)yieldsalargerMSPEduetophasefluctuations.Wealsoshowthat,atthenormaloperatingSNRoftheconsideredcode,theperformanceoftheCAsynchronizerisveryclosetothatofaDAsynchronizerthatknowsalltransmittedsymbolsinadvance.

IndexTerms—Carriersynchronization,errorcontrolcoding,feedbackphaseestimation.

I.INTRODUCTION

HElastdecadehasseenthedevelopmentofpowerfulerrorcorrectingcodessuchasturbocodesandlow-densityparitycheck(LDPC)codes.Theimpressivebiterrorrate(BER)performanceoftheassociatediterativedecodingprocessesim-plicitlyassumescoherentdetection,meaningthatthecarrierphasemustberecoveredaccuratelybeforethedataisdecoded.However,sincethedecoderusuallyoperatesatextremelylowsignal-to-noiseratio(SNR)values,accuratecarrierrecoveryisachallengingtask.Numerouseffortstotacklethisproblemhaveresultedinamyriadofdifferentreceivers[1]–[10].

In[1]and[2],thephaseestimatorignoreserror-controlcodingandassumesthatthetransmittedsymbolsaremutuallyindependent[non-code-aided(NCA)operation],whereasin[3]–[10],thecodepropertiesareexploitedinthephaseesti-mationprocess(CAoperation).In[11],itwasshownthatthesecondapproachispotentiallymoreaccurate.

Theiterativeschemein[5],whichisbasedontheexpecta-tion-maximizationalgorithm,isoptimalinthesensethatitcon-vergestothetruemaximum-likelihood(ML)carrierphaseesti-mate[12],[13].Thealgorithmdoesnotrequiremodificationof

ManuscriptreceivedSeptember2,2005;revisedApril11,2006.Theasso-ciateeditorcoordinatingthereviewofthismanuscriptandapprovingitforpub-licationwasDr.MounirGhogho.

TheauthorsarewiththeTelecommunicationsandInformationProcessingDepartment,GhentUniversity,B-9000Ghent,Belgium(e-mail:nnoels@telin.ugent.be;hs@telin.ugent.be;mm@telin.ugent.be).DigitalObjectIdentifier10.1109/TSP.2006.887108

T

thedecoderoperation,andtheresultingreceiverisonlymargin-allymorecomplexthantheconventionalreceiverthataprioriknowstheexactvalueofthephase.Unfortunately,itsperfor-mancerapidlydegradesinthepresenceofatime-varyingcar-rierphase.

In[2],[6]–[8],and[10],feedbackphaseestimationhasbeenadoptedtocopewithcarrierphasevariations.TheML-basedre-ceiverproposedin[10]combinesthelowcomplexityfromtheapproachin[5]withtheabilitytoautomaticallytrackaslowlyvaryingcarrierphase.Simulationresultsin[10]showtheinter-estingpotentialofthisapproach.Asopposedtothealgorithmsin[2],[6],and[7],thederivationofthephaseestimationalgo-rithmstemsdirectlyfromtheMLcriterionandcanthereforebeseenasthefeedbackcounterpartofthereceiverpresentedin[5].Moreover,itscomputationalcomplexityislowerthanthatofthealgorithmsin[8]and[9],whichmodifythedecoderoper-ationbyeithertakingintoaccountthephasestatisticsorusingper-survivorphaseestimatesinsidethedecoder.

Thiscontributionzoomsinontheapproachthatwasadoptedin[10].Bymeansoftheoreticalanalysisandcomputersimu-lations,wecomparethetrackingperformancesresultingfromtheiterativecode-aided(CA)synchronizerfrom[10],thedata-aided(DA)synchronizer,whichknowsalltransmittedsymbolsinadvance,andtheNCAsynchronizer,whichneglectstheun-derlyingencodingrule.ItisshownthatCAfeedbackphasees-timationoutperformsNCAfeedbackestimationwhenthephasetobeestimatedvarieswithtime;whenthecarrierphaseiscon-stantovertheobservationinterval,bothsynchronizersyieldes-sentiallythesamemean-squarephaseerror(MSPE).Wealsoshowthat,atthenormaloperatingSNRoftheconsideredcode,theperformanceoftheCAsynchronizerisveryclosetothatofaDAsynchronizerthatknowsalltransmittedsymbolsinad-vance.ThisillustratestheoptimalityoftheCAsynchronizer.

II.MAXIMUM-LIKELIHOODCRITERION

Weconsiderthetransmissionofanarbitrarysequenceof

overancomplex-valuedsymbols

additivewhiteGaussiannoise(AWGN)channel.Thejoint

isdenotedasprobabilitymassfunctionofthesymbols

.Assuminglinearmodulationusingsquare-rootNyquisttransmitpulses,andmatchedfilteringatthecorrectdecisioninstants,thediscrete-timebasebandobservationisgivenby

(1)

where

1The

denotestheunknowncarrierphase,1andthesequenceconsistsofindependentzero-meancomplex-valued

carrierphaseisinitiallyassumedtobeconstantovertheobservation

interval.Later,theobservationmodelwillbeextendedtoallowatime-varyingcarrierphase.

1053-587X/$25.00©2007IEEE

1130Gaussiannoiseterms;andarestatistically

independentandhaveavarianceequalto

.Letusdenotebyatrialvalueofthetruecarrierphasethathastobeestimatedbythesynchronizer.Then,theMLestimateofthecarrierphaseisthevalueofthatmakeszerothederiva-tiveofthelog-likelihoodfunctionwithre-spectto[14].Theprobabilitydensityofresultingfrom(1),giventhedatasequenceandatrialvalueofthecarrierphase,is(withinafactornotdependingon)givenby

(2)

Thelikelihoodfunctionofthecarrierphaseisobtained

byaveraging

overthesymbolvector,i.e.,.Fromasimilarreasoningasin[5]and[11],thederivative

ofthelog-likelihoodfunctionwithrespecttocanbemanipulatedintothefollowingform:

(3)

where

(4)

istheaposterioriexpectationofthesymbolcondi-tionedonand,withdenotingthemarginalaposterioriprobability(APP)ofthesymbol,and

thesetofconstellationpointswithsymbol

energy

.Whenthedatasymbolvectorconsistsofknownpilotsym-bols,weobtainequal

to1for

andzerootherwise,yieldingin(4).Thelog-likelihoodfunctionthatcorrespondstothetrans-missionofpilotsymbolsisdenoted.

Inthecaseofuncodedtransmission,thesymbolsare

statisticallyindependent,sotheAPPsof

reduceto(5)

where

(6)

As(5)dependsonlyon,wewilldenotethecorrespondingaposterioriaverageofthesymbolas.Thelog-likelihoodfunctionthatcorrespondstothetransmissionofsta-tisticallyindependentsymbolsisdenoted.

Thispaperconsiderssystemswithaniterativelydecodableerror-controlcode(turbo,LDPCcodes).Thedatasymbolvector

isobtainedfromtheencodingofase-quenceofinformationbitsandapropermappingofthecoded

IEEETRANSACTIONSONSIGNALPROCESSING,VOL.55,NO.3,MARCH2007

Fig.1.Generalstructureofadiscrete-timefeedbackcarriersynchronizer.

bitsonthesignalconstellation.Inthiscase,theAPPsin(4)are

afunctionofallcomponentsofthevector.Toavoidthecom-putationalcomplexityassociatedwiththeirexactevaluation,2themarginalAPPsareapproximatelycomputedbymeansoftheiterativeapplicationofthesum–product(SP)algorithmonafactorgraphwithcycles[15].Ifthecyclesinthegrapharelarge(whichisreasonableforwell-designedturboandLDPCcodes),thisiterativeprocedure(afterconvergence)yieldsmar-ginalAPPsthatareveryclosetothecorrectmarginalAPPs.The

correspondinglog-likelihoodfunctionisdenoted

.III.ML-BASEDPHASETRACKINGFORCODEDSIGNALSThegeneralstructureofafirstorderdiscrete-timefeedback

carriersynchronizerorphase-lockedloop(PLL)isshowninFig.1[16].Thephaseestimateisupdatedoncepersymbolin-terval,accordingtothefollowingforward3recursion

(7)

In(7),istheloopfiltergain,and

denotesthephaseerrordetector(PED)output.Therecursionstartswithaninitialphaseestimate,thatcanbeobtainedfromafeedforwardsynchro-nizeroperatingonashortpilotsequence[16].

Inthefollowing,weconsiderthreetypesofML-basedPEDs.

TheDAPED(basedon

)assumesthatalldatasymbolsareknown.TheNCAPED(basedon)assumesthatthedatasymbolsareindependent,whereastheCAPED(basedon

)takesthecodepropertiesintoaccount.Weobtain

from(3)that

DAoperationNCAoperationCAoperation

(8)

ComparisonofthePEDoutputsforNCAandCAoperationwiththatforDAoperationindicatesthattheaposteriorimeancanbeconsideredasasoftdecision(SD)regarding,based

uponthereceivedsample

orthereceivedsamplesequenceandthephaseestimate.Notefrom(8)thattheDAand

theNCAPEDoutputdependonlyon

;thisisincon-trastwiththeCAPEDoutputwhosecomputationdependson

theentirevector:allsamples

havetobe2Inprinciple,the

exactmarginalAPPsPr[ajr;󰀒]canbeobtainedasasum-mationofjointAPPsPr[ajr;󰀒],which,inturn,canbecomputedfrom(2)and

Bayes’rule.However,thecomputationalcomplexityofthisprocedureincreasesexponentiallywiththesequencelengthK.

3Wespeakofaforwardrecursionwhenthephaseupdatingisperformedfromthefirstsymbolintervaltothelast.

NOELSetal.:PERFORMANCEANALYSISOFML-BASEDFEEDBACKCARRIERPHASESYNCHRONIZERSFORCODEDSIGNALS1131

rotatedoveranangleandfedtotheSPalgorithmforpro-ducingtheSD

.Hence,inthecaseofCAoperation,theentirereceivedblockmustbeprocessed

times,whereasthereceivedblockisprocessedonlyonceinthecaseofDAorNCAoperation.

InordertoavoidthehighcomputationalcomplexityresultingfromtheCAPED,thefollowingiterativeCAPLLhasbeenproposedin[10].Duringthethiteration,thefeedbacksyn-chronizergeneratesestimatesessentiallyaccordingto(7),butwiththePEDoutputgivenby

iterativeCAoperation

(9)

where

,andistheaposterioriexpectationofthesymbolconditionedonand.Hence,fromthephasevector,there-ceivedvectorisprocessedtocompute

for,afterwhichthePLLgeneratesthephasevector

.Theiterativeprocessisinitializedbymeansofaphasevector,whichcanbeobtainedfromaPLLwithNCAop-eration.Whenconvergenceisachievedafteriterations,thevectorhasbeenprocessedtimes.When,considerablesavingsincomputationtimehavebeenobtainedascomparedwiththenoniterativePLLthatusestheCAPEDoutputfrom(8).Moreover,whenappliedtoaturboorLDPCreceiverwithiterativeMAPdetection/decoding,theproposedphaseestimation/compensationschemeyieldsverylowadditionalcomplexitywhenthesynchronizeriterationsaremergedwiththedecoderiterations[4],[5],[10],i.e.,aftereachsynchronizeriterationonlyonedecoderiterationisperformedwithoutresettingextrinsicprobabilities.

IV.TRACKINGPERFORMANCEANALYSIS

A.AnalyticalResults

ComputingtheexacttrackingperformanceoftheiterativeCA

feedbackphaseestimatorismuchmoredifficultthanfortheNCAandDAsynchronizers,becauseoftheiterationsinvolvedandthedependenceofthesoftdecisionsontheentirephasevector.Instead,wewillproceedassumingthat,atthenormalop-eratingSNRoftheconsiderederror-correctingcode,theMSPEresultingfromtheiterativeCAphaseestimatorconvergestotheMSPEresultingfromafictitiousDAphaseestimatorthatknowsalldatasymbolsinadvance.

Amotivationforthisassumptionreadsasfollows.Notethatin(8)theCAPEDoutputreducestotheDAPEDoutputwhentheAPPisonefor

andzerootherwise.ThisindicatesthattheCAPLLessen-tiallybehavesliketheDAPLL,providedthattheratios

arelikelytobe

muchsmallerthan1forallandall.Letusintroducetheindicatorfunction,whichequalsone

when

foratleastone,andequals

zerootherwise.Then,weobtain

(10)

wheredenotesthesetoflegitimatecodedsymbolsequences

oflength.Weassumethat

forandotherwise,wherethequantitiesanddenotethe

rateofthecodeandthenumberofconstellationpoints,respec-tively.Withandforall,(10)isnothingbutthe(verysmall)symbolerrorrateresultingfromanoptimalmax-imumaposterioriprobabilitysymboldecoder[17].Hence,forsmallphaseerrors,thefractionofsymbolintervalsforwhich

isverysmall,sothatwecansafelyassumethatthe

CAPLLoperationcloselyresemblestheDAPLLoperation,atthenormaloperatingSNRofthecode.

WewillnowcomputetheperformanceoftheDAandtheNCAphaseestimator.AssumingthatatthelowSNRsupportedbycapacity-approachingcodes,itisnotpossibletocomputere-liabledatadecisionswithouttakingintoaccountthecodestruc-ture,weexpecttheNCAPLLtoperformsignificantlyworsethanaDAPLLwithperfectknowledgeonthedatasymbols.Inordertoallowatime-varyingcarrierphase,theobservationmodel(1)ismodifiedinto

(11)

whereisthephaseduringthethsymbolinterval.Anoftenusedphasenoise(PN)modelisbasedonadiscreteWienerprocess(randomwalk)

(12)

characterizedbyindependentandidenticallydistributed(i.i.d.)

Gaussianincrements

withzeromeanandstandarddeviation,descriptiveofthephasenoiseintensity.Itisassumedthatandarestatisticallyindependent,andthatisuni-formlydistributedin.Wedefinethephaseestimationerrorduringthethsymbolperiodas.4TheDAandNCAPEDoutputsfrom(8)thatdependonlyoncanbedecomposedasthesumoftheiraverageandtheirzero-meanstatisticalfluctuation

(13)

with

(14)

denotingthePEDcharacteristicandtheloopnoiseofthesyn-chronizer,respectively.WeshowintheAppendixthat.

4This

definitionoftheestimationerroragreeswiththeframeworkin[16].Itshouldbenoted,however,thatinverseof󰀞,i.e.,accordingto^theestimationerrorisusuallydefinedasthe

󰀒

0󰀒.1132Assumingsmallphaseerrors,thefollowinglinearizationapplies[16]:

(15)

whereistheslopeofthePEDcharacteristicandistheloopnoiseat.Substituting(15)into(7),weobtain

(16)

where

.WeshowintheAppendixthat

(17)

where

DAoperation

NCAoperation

(18)

and

.Solving(16)yields

(19)

Thephaseerror(19)attheoutputofthePLLconsistsoftwocontributions,whicharecausedbythenoise(AWGN,PN)af-fectingtheobservationandbytheinitialphaseerror,respec-tively.Inallpracticalcases,thequantityissmallerthan1sothatthephaseerror(19)exhibitsadecayingacquisi-tiontransientnearthestartoftheobservationinterval.Assumingauniformlydistributedinitialphaseerror,themeanacquisitiontimeintheabsenceofnoiseiswellapproximatedby[16]

(20)

where

istheone-sidebandwidth(normalizedtothesymbolrate)oftheclosed-loopfilterwithimpulse

response

and-transform.Wehave

(21)

Theapproximationin(20)and(21)isvalidforsmall.Itfol-lowsfrom(20)thatalargerresultsinafasteracquisition.Thesamegoesforalarger.Attheendoftheacquisitionperiodthephaseerrorentersthetrackingmode,duringwhich(19)canbesafelyapproximatedby

(22)

Itiseasilyseenfrom(22)thatthesteady-statephaseerrorhaszeromeanandthatthesteady-stateMSPEisgivenby

MSPE

MSPE

(23)

IEEETRANSACTIONSONSIGNALPROCESSING,VOL.55,NO.3,MARCH2007

with

MSPEMSPE

(24)

Thelinearizedsteady-stateMSPEfrom(24)consistoftwocon-tributions:anadditivenoisecontributionandaphasenoisecon-tribution.ThePNcontributionisinverselyproportionaltotheslopeofthePEDcharacteristic,whereastheAWGNcontribu-tiondoesnotdependontheslopeofPEDcharacteristic.This

isbecause,forgivenvaluesofand

,thereductionofthePEDslopeoftheNCAPLL(ascomparedwiththeDAPLL)ispreciselycompensatedbythereductionoftheloop

bandwidth

andofthephasenoisevariance.AlargervalueofyieldsalargerAWGNcontributionbutasmallerPNcontribution,andviceversa.Whenthecarrierphase

istime-invariant

,theMSPEcanbemadearbitrarilysmallbyreducingthevalueof.Asmall,however,impliesalargeacquisitiontime.Whenthecarrierphaseistime-varying

,thereexistsanoptimalvalueforthatminimizesthe

steady-stateMSPE.Solvingthisoptimizationproblemyields

MSPE

(25)

whereandMSPEdenotetheoptimalvalueforandthecorrespondingminimumvalueforthelinearizedsteady-stateMSPE,respectively.

B.NumericalResultsandDiscussion

Bymeansofexample,weconsiderabinaryphase-shiftkeying(BPSK)signalconstellation,anobservationintervalof

symbolperiodsandarate1/3turbocode.Theturbo

encoderconsistsoftheparallelconcatenationoftwoidenticalnonrecursivesystematicconvolutionalencoderswithgenerator

polynomials

andinoctalnotation,separatedbyapseudorandominterleaverofsize333bits.Fig.2showstheBERafterone,two,andteniterationsofthecoherentturbodecoder/detector.

InFig.3,theslope

oftheNCAPEDisplottedagainsttheSNR.MonteCarlosimulationtechniqueswereusedtoevaluatethestatisticalexpectationinvolvedintheexpressionof(see(A10)oftheAppendix).Weobservethat,atthenormaloperatingSNRoftheturbocode(say,BER),theslopeoftheNCAPEDcharacteristicisstrictlysmallerthan1,i.e.,thanistheslopeoftheDAPED.ThetradeoffonincaseisillustratedinFig.4showingthenumericalevaluationof(24)for

2.77dB,whichisintheoperatingrangeof

theturbocodefromFig.2,and2.AstheslopeoftheDAPEDislargerthanthatoftheNCAPED,theoptimizedDAPLLyieldsasmalleracquisitiontimeandasmallerMSPEthantheoptimizedNCAPLL.TheoptimalloopfiltergainisaroundwiththeNCAPED,andreducesto

withtheDAPED.TheminimumMSPEforthe

NOELSetal.:PERFORMANCEANALYSISOFML-BASEDFEEDBACKCARRIERPHASESYNCHRONIZERSFORCODEDSIGNALS1133

Fig.2.BERofcoherentturboreceiverforarate1/3turbocodedBPSKsignalinAWGN.

Fig.3.SlopeoftheNCAPEDcharacteristicat󰀞=0,inthecaseofBPSK.

NCAPLLisafactorof0.57largerthantheminimumMSPEfortheDAPLL.Thisisconsistentwith(25)andwiththebehaviorofdepictedinFig.3.

InFig.5,wehaveplottedthesimulatedMSPEattheoutputoftheDA,theNCA,andtheiterativeCAPLLasafunctionofthesymbolindex.Wehavetaken2.77dBand

.Thecarrierphaseisassumedtobeeitherconstant

overtheobservationinterval(CCP)ortoperformarandomwalk

with

2(WPN).TheiterationoftheCAPLLisanFig.4.Linearizedsteady-stateMSPEasafunctionoftheloopparameter󰀕.

Fig.5.MSPEofafirstorderPLLwith󰀕=0:04,trackingaconstantcarrierphase(CCP)orWienerphasenoisewith󰀛=2(WPN).

NCArecursion.Inbothcases,wefindthattheMSPEoftheit-erativeCAPLLbecomesessentiallyequaltotheMSPEofthe

DAPLLafteronlytwoiterations(i.e.,foriterations).ThisconfirmsthevalidityoftheassumptionmadeatthebeginningofSectionIV-A.AstheslopeoftheNCAPEDissmallerthanthatoftheDAPED,theacquisitiontimeandthePNcontribu-tiontothetrackingMSPEarelargerforNCAoperationthanfor

DAoperation(CAoperation,

).Conversely,theAWGNcontributiontothetrackingMSPE,isthesameforNCAopera-tionandforDAoperation(CAoperation,).Thisimplies

1134Fig.6.MSPEofafirst-orderDAPLLwith󰀕=0:04,trackingaconstantcarrierphaseandusingforward–backwardphaseupdating.

that,whenthecarrierphaseistime-invariant(CCP),thetrackingMSPEcannotbereducedbyperforming,aftertheinitialNCArecursion,iterationsintheCAmode.

V.FORWARD–BACKWARDPHASEUPDATING

Duringtheacquisitionperiodatthebeginningoftheobser-vationinterval(seeFig.5),theMSPEmayassumelargevalues.

Assumingthattheacquisitiontransientisnolongerpresentattheendoftheobservationinterval,accuratephaseestimatesatthebeginningoftheobservationintervalcanbeobtainedbycar-ryingoutanadditionalrecursion,usingasinitialphaseestimatetheestimateobtainedattheendofthefirstrecursionandup-datingthephaseestimatesfromthelastsymboltothefirstac-cordingtothefollowingbackwardrecursion:

backwardrecursion

(26)

TheresultofthisprocedureisshowninFig.6forafirst-order

DAPLLwith

,andfor2.77dBand.ForaDAPLLoranNCAPLL,theedgeeffectthatarisesneartheendoftheobservationintervalcanbeexplainedasfollows.Substituting(15)into(26),weobtain

(27)

with

,andgivenby(18).Takingintoaccountthat

isgiven

by(22)with

,solving(27)yields(28)

IEEETRANSACTIONSONSIGNALPROCESSING,VOL.55,NO.3,MARCH2007

Fig.7.f=2󰀕g(0)n(10󰀕g(0))

asafunctionofn.

where.Approximatingby0

for

andtakingintoaccountthatthenoisesamplesaremutuallyindependent,itcanbeverifiedthat

MSPE

MSPE

(29)

where

(30)

andMSPEandMSPEaregivenby(24).Theterms

proportionalto

donotoccurin(23)andarearesultofthereusingofthesamples.Thefunction

isplotinFig.7,for

.For,itequalszero.Astheproductofalinearlyincreasingfunctionandanexponentiallydecreasing

function,

firstincreasesandthendecreaseswithincreasing.Forinfinite,convergestozero.ThisexplainstheshapeoftheedgeeffectintheMSPEneartheendoftheobserva-tioninterval(where

issmall).Nearthestartoftheobser-vationinterval(where

islarge),isnegligiblysmalland(29)reducesto(23).Assumingthat

ises-sentiallyzerofor

,theedgeeffectcanbecircum-ventedbytakingasfinalphaseestimatestheestimatesatin-stants

fromtheforwardrecursionandthosewithindexes

fromthebackwardrecursion.

Intheabove,wehaveconsideredonlytwosuccessiverecur-sions,oneintheforwardandoneinthebackwarddirection,buttheforward–backwardphaseupdatingprincipleiseasilyex-tendedtoahighernumberofrecursions.Inordertoreducetheeffectofacquisitiontransients,theiterativeCAPLLismodifiedasfollows.Insteadofperformingonlyforwardrecursions,an

alternationofforward(foriterations

andback-ward(foriterations

)recursionsiscarriedout,witheachrecursionusingasinitialphaseestimatetheestimateobtainedattheendofthepreviousiteration.

Letusreconsidertheturbo-codedBPSKsignalfromSectionIV.Fig.8showstheMSPEperformanceoftheNCAandCAfeedbackphaseestimatorswithforward–backward

phaseupdatingand

,forseveralvaluesofthephasenoisevariance

asafunctionof.ThesimulatedNOELSetal.:PERFORMANCEANALYSISOFML-BASEDFEEDBACKCARRIERPHASESYNCHRONIZERSFORCODEDSIGNALS1135

Fig.8.MSPEofafirstorderML-basedPLLwith󰀕=0:04.CAoperationversusNCAoperation,simulationversusanalyticalcomputation.

MSPE(continuouslines)iscomparedwiththelinearizedana-lyticalresultfromSectionIV(dashedlines).Thediscrepancy

(whichisquitesmallfor

largerthanabout4dB)be-tweenthesimulationsandthecomputationsistheresultoftherandomoccurrenceofnonlinearphenomenasuchascycleslipsandhang-ups[16].Intheabsenceofphasenoise,theCAal-gorithmyieldsessentiallythesameMSPEastheconventionalNCAalgorithmsincethelatterisgiven‘moretime’bythefor-ward-backwardrecursiontoletitstransientfadeout.Inthepres-enceofphasenoise,theMSPEoftheCAsynchronizerislowerthanthatoftheconventionalNCAalgorithm.Therelativead-vantageoftheCAsynchronizerovertheNCAsynchronizerin-creaseswiththevalueofthephasenoisevariance.Itspeaksfor

itselfthattheestimationaccuracyincreaseswith

.VI.CONCLUSION

ThiscontributionhasstudiedtheeffectivenessofCAandNCAML-basedfeedbackphasesynchronizersatthelowSNRssupportedbypowerfuliterativelydecodablecodessuchasturbocodesorLDPCcodes.Undernormaloperatingconditions,theMSPEresultingfromtheiterativeCAsynchronizerconvergestotheMSPEresultingfromaDAsynchronizerthatknowsalldatasymbolsinadvance.ThisillustratestheoptimalityoftheCAML-basedfeedbackphasesynchronizer.Byvirtueofaforward-

backwardmultiple-recursionestimator,thelinearizedMSPEoftheNCAfeedbacksynchronizerequalsthatoftheCAfeedbacksynchronizer,whenthecarrierphaseisessentiallyconstantovertheobservationintervalandtheloopfiltergainisthesameforbothsynchronizers.Conversely,thepresenceofWienerphasenoiseresultsinaNCAfeedbacksynchronizerMSPEthatislargerthantheCAfeedbacksynchronizerMSPE.TheNCAsyn-chronizeralsoyieldsalargeracquisitiontimethantheCAsyn-chronizer.However,assumingthattheacquisitiontransientisshorterthantheobservationinterval,theeffectoftheacquisitiontransientonthephaseerrorcanbecircumventedbycarryingoutanalternationofforwardandbackwardphaseupdatingre-cursions,witheachrecursionusingasinitialphaseestimatetheestimateobtainedattheendofthepreviousrecursion.

APPENDIX

WefirstconsiderNCAfeedbackphasesynchronization.

Takingin(3)

and,wefindthattheNCAPEDoutputfrom(8)canberewrittenasfollows:

(A1)

wheretheprobabilitydensityis(withinanirrel-evantfactor)givenby

with

asin(6).

Thedecomposition(13),(14)of(A1)asthesumofthePED

characteristic

andtheloopnoiseyields(A2)

(A3)

Takingthefirstandthesecondderivative(withrespect

tothetruecarrierphase

)ofbothsidesofthenor-malizationconstraint

,andusing,itcanbeverifiedthat

(A4)

and

(A5)

Itfollowsdirectlyfromand(A4)that.Takingintoaccountthat

and(11),thePEDslopeandtheloopnoise

at

aregivenby

(A6)(A7)

1136Becauseofthestatisticalpropertiesof,theloopnoiseat

iswhite5anditspowerspectraldensityisgivenby

(A8)

Takingintoaccount(A5)and(3),weobtain

(A9)

with

(A10)

Itcanbeverifiedthat

from(A10)equalstheratioofthemodifiedCramér–Raobound(MCRB)tothetrueCramér–Raobound(CRB)relatedtotheestimationofanunknownbutdeter-ministiccarrierphasefromthenoisyobservationofuncoded

datasymbols[14].AsCRB

MCRB,andCRBconvergestoMCRBforlargeSNR,itisfoundthat

,and

.TheaboveanalysisremainsvalidforDAoperationpro-videdthatwereplace

with,andwith,whichis(withinanirrelevantfactor)givenby.Inthiscase,weobtain

(A11)

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NeleNoelsreceivedtheDiplomadegreeinelectricalengineeringfromGhentUniversity,Gent,Belgium,in2001.SheiscurrentlyworkingtowardsthePh.D.degreeattheDepartmentofTelecommunicationsandInformationProcessing,GhentUniversity.

Hermainresearchinterestsareincarrierandsymbolsynchronization.Sheistheauthorofseveralpapersininternationaljournalsandconferenceproceedings.

HeidiSteendam(M’00)receivedtheDiplomaandPh.D.degrees,bothinelectricalengineering,fromGhentUniversity,Gent,Belgium,in1995and2000,respectively.

SheisProfessorattheDepartmentofTelecom-municationsandInformationProcessing,GhentUni-versity.Hermainresearchinterestsareinstatisticalcommunicationtheory,carrierandsymbolsynchro-nization,bandwidth-efficientmodulationandcoding,spread-spectrum(multicarrierspread-spectrum),andsatelliteandmobilecommunication.Sheistheau-thorofmorethan50scientificpapersininternationaljournalsandconferenceproceedings.

MarcMoeneclaey(M’93–SM’99–F’02)receivedtheDiplomaandPh.D.degrees,bothinelectricalengineering,fromtheUniversityofGhent,Gent,Belgium,in1978and1983,respectively.

From1978to1999,heheldvariouspositionsfortheBelgianNationalFundforScientificResearch(NFWO),fromResearchAssistanttoResearchDirector,atGhentUniversity.HeispresentlyaPro-fessorintheDepartmentofTelecommunicationsandInformationProcessing(TELIN),GhentUniversity.Hisresearchinterestsincludestatisticalcommu-nicationtheory,carrierandsymbolsynchronization,bandwidth-efficientmodulationandcoding,spread-spectrum,andsatelliteandmobilecommu-nication.Heistheauthorofmorethan200scientificpapersininternationaljournalsandconferenceproceedings.TogetherwithProf.H.Meyr(RWTHAachen)andDr.S.Fechtel(SiemensAG),hehascoauthoredthebookDigitalCommunicationReceivers—Synchronization,ChannelEstimation,andSignalProcessing(NewYork:Wiley,1998).

Dr.MoeneclaeyhasbeenactiveinvariousinternationalconferencesasTechnicalProgramCommitteememberandSessionchairman.From1992to1994,heservedasEditorforSynchronizationfortheIEEETRANSACTIONSONCOMMUNICATIONS.Hewasco-GuestEditorfortheDecember2001IEEEJOURNALOFSELECTEDAREASINCOMMUNICATIONS(JSAC)specialissueonSignalSynchronizationinDigitalTransmissionSystems.From1993to2002,hewasanexecutiveCommitteeMemberoftheIEEECommunicationsandVehicularTechnologySocietyJointChapter,BeneluxSection.