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【图文】1998 Turbo decoding as an instance of Pearl's “belief propagation” algorithm
140IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.16,NO.2,FEBRUARY1998
TurboDecodingasanInstanceofPearl’s“BeliefPropagation”Algorithm
RobertJ.McEliece,Fellow,IEEE,DavidJ.C.MacKay,andJung-FuCheng
Abstract—Inthispaper,wewilldescribethecloseconnectionbetweenthenowcelebratediterativeturbodecodingalgorithmofBerrouetal.andanalgorithmthathasbeenwellknownintheartificialintelligencecommunityforadecade,butwhichisrelativelyunknowntoinformationtheorists:Pearl’sbeliefpropagationalgorithm.WeshallseethatifPearl’salgorithmisappliedtothe“beliefnetwork”ofaparallelconcatenationoftwoormorecodes,theturbodecodingalgorithmimmediatelyresults.Unfortunately,however,thisbeliefdiagramhasloops,andPearlonlyprovedthathisalgorithmworkswhentherearenoloops,soanexplanationoftheexcellentexperimentalperformanceofturbodecodingisstilllacking.However,weshallalsoshowthatPearl’salgorithmcanbeusedtoroutinelyderivepreviouslyknowniterative,butsuboptimal,decodingalgorithmsforanumberofothererror-controlsystems,includingGallager’slow-densityparity-checkcodes,seriallyconcatenatedcodes,andproductcodes.Thus,beliefpropagationprovidesaveryattrac-tivegeneralmethodologyfordevisinglow-complexityiterativedecodingalgorithmsforhybridcodedsystems.
IndexTerms—Beliefpropagation,error-correctingcodes,iter-ativedecoding,Pearl’sAlgorithm,probabilisticinference,turbocodes.
I.INTRODUCTION
URBOcodes,whichwereintroducedin1993byBerrouetal.[10],arethemostexcitingandpotentiallyimportantdevelopmentincodingtheoryinmanyyears.Manyofthestructuralpropertiesofturbocodeshavenowbeenputonafirmtheoreticalfooting[7],[18],[20],[21],[27],[45],andseveralinnovativevariationsontheturbothemehaveappeared[5],[8],[9],[12],[27],[48].
Whatisstilllacking,however,isasatisfactorytheoreticalexplanationofwhytheturbodecodingalgorithmperformsaswellasitdoes.Whilewecannotyetannounceasolutiontothisproblem,webelievethattheanswermaycomefromaclosestudyofPearl’sbeliefpropagationalgorithm,whichislargelyunknowntoinformationtheorists,butwellknownintheartificialintelligencecommunity.(Thefirstmentionofbeliefpropagationinacommunicationspaper,andindeedthe
ManuscriptreceivedSeptember27,1996;revisedMay3,1997.ThisworkwassupportedbyNSFGrantNCR-9505975,AFOSRGrant5F-0313,andagrantfromQualcomm,Inc.AportionofR.J.McEliece’scontributionwasdonewhilehewasvisitingtheSonyCorporationinTokyo.ThecollaborationbetweenD.J.C.MacKayandR.J.McEliecewasbegunat,andpartiallysupportedby,theNewtonInstituteforMathematicalSciences,Cambridge,U.K.
R.J.McElieceiswiththeDepartmentofElectricalEngineering,CaliforniaInstituteofTechnology,Pasadena,CA91125USA.
D.J.C.MacKayiswiththeCavendishLaboratory,DepartmentofPhysics,DarwinCollege,CambridgeUniversity,CambridgeCB3OHEU.K.
J.-F.ChengiswithSalomonBrothersInc.,NewYork,NY10048USA.PublisherItemIdentifierS)00170-X.
paperthatmotivatedthisone,isthatofMacKayandNeal[37].Seealso[38]and[39].)
Inthispaper,wewillreviewtheturbodecodingalgorithmasoriginallyexpoundedbyBerrouetal.[10],butwhichwasperhapsexplainedmorelucidlyin[3],[18],or[50].WewillthendescribePearl’salgorithm,firstinitsnatural“AI”setting,andthenshowthatifitisappliedtothe“beliefnetwork”ofaturbocode,theturbodecodingalgorithmim-mediatelyresults.Unfortunately,however,thisbeliefnetworkhasloops,andPearl’salgorithmonlygivesexactanswerswhentherearenoloops,sotheexistingbodyofknowledgeaboutPearl’salgorithmdoesnotsolvethecentralproblemofturbodecoding.Still,itisinterestingandsuggestivethatPearl’salgorithmyieldstheturbodecodingalgorithmsoeasily.Furthermore,weshallshowthatPearl’salgorithmcanalsobeusedtoderiveeffectiveiterativedecodingalgorithmsforanumberofothererror-controlsystems,includingGallager’slow-densityparity-checkcodes,therecentlyintroducedlow-densitygeneratormatrixcodes,seriallyconcatenatedcodes,andproductcodes.Someofthese“BP”decodingalgorithmsagreewiththeonespreviouslyderivedbyadhocmethods,andsomearenew,butallprovetoberemarkablyeffective.Inshort,beliefpropagationprovidesanattractivegeneralmethodfordevisinglow-complexityiterativedecodingalgorithmsforhybridcodedsystems.Thisisthemessageofthepaper.(AsimilarmessageisgiveninthepaperbyKschischangandFrey[33]inthisissue.)
Hereisanoutlineofthepaper.InSectionII,wederivesomesimplebutimportantresultsabout,andintroducesomecompactnotationfor,“optimalsymboldecision”decodingalgorithms.InSectionIII,wedefinewhatwemeanbyaturbocode,andreviewtheturbodecodingalgorithm.Ourdefinitionsaredeliberatelymoregeneralthanwhathasprevi-ouslyappearedintheliterature.Inparticular,ourtransmittedinformationisnotbinary,butrathercomesfrom
-aryprobability
distributionsinsteadofthetraditional“log-likelihoodratios.”Furthermore,thereadermaybesurprisedtofindnodiscussionof“interleavers,”whichareanessentialcomponentofallturbo-codingsystems.Thisisbecause,aswewillarticulatefullyinourconcludingremarks,webelievethattheinter-leaver’scontributionistomaketheturbocodea“good”code,butithasnothingdirectlytodowiththefactthattheturbodecodingalgorithmisagoodapproximationtoanoptimaldecoder.InSectionIV,wechangegears,andgiveatutorialoverviewofthegeneralprobabilisticinferenceproblem,withspecialreferencetoBayesianbeliefnetworks.InSectionV,
/98$10.00(C)1998IEEE
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JOINT SOURCE-CHANNEL LIST SEQUENCE DECODING
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