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NOVEMBER2004 VOLUME52 NUMBER11 IECMBT (ISSN0090-6778) TRANSACTIONSLETTERS Coding AnAlgorithmforDetectingUnreliableCodeSequenceSegmentsandItsApplications ....... ......J.FreudenbergerandB.Stender 1833 DecodingofLow-DensityParity-CheckCodesOverFinite-StateBinaryMarkovChannels .. ...... ......... .....J.Garcia-Frias 1840 DigitalCommunications AccurateComputationofthePerformanceofM-aryOrthogonalSignalingonaDiscreteMemorylessChannel. ......... . J.Hamkins 1844 High-RateRecursiveConvolutionalCodesforConcatenatedChannelCodes . ....... F.Daneshgaran,M.Laddomada,andM.Mondin 1846 Fading/Equalization ANewBase-StationReceiverforIncreasingDiversityOrderinaCDMACellularSystem...... W.Choi,C.Yi,J.Y.Kim,andD.I.Kim 1851 PowerControlandDiversityinFeedbackCommunicationsOveraFadingChannel... ....... ....... . I.SaarinenandA.Mämmelä 1857 SpreadSpectrum LargeSetofCISpreadingCodesforHigh-CapacityMC-CDMA... ......... .....B.Natarajan,Z.Wu,C.R.Nassar,andS.Shattil 1862 Synchronization OntheMiller–ChangLowerBoundforNDACarrierPhaseEstimation...... .......G.N.Tavares,L.M.Tavares,andM.S.Piedade 1867 TRANSACTIONSPAPERS Coding PerformanceAnalysisandDesignCriteriaforFinite-AlphabetSource-ChannelCodes. ...... ........ .A.HedayatandA.Nosratinia 1872 Near-CapacityCodinginMulticarrierModulationSystems. ...... ......... ....M.Ardakani,T.Esmailian,andF.R.Kschischang 1880 PerformanceofCodedOQPSKandMIL-STDSOQPSKWithIterativeDecoding.... ..... ......... ......L.LiandM.K.Simon 1890 DesignandPerformanceofTurboGallagerCodes . ...... ...... ...... ......... ..... ...... ...... ......G.Colavolpe 1901 ReducedComplexityMAP-BasedIterativeMultiuserDetectionforCodedMulticarrierCDMASystems ..... ...... ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ......J.Li,K.B.Letaief,andZ.Cao 1909 DigitalCommunications ANovelTrellis-ShapingDesignWithBothPeakandAveragePowerReductionforOFDMSystems.. ....... ........ .. H.Ochiai 1916 Reduced-DelayProtectionofDSLSystemsAgainstNonstationaryDisturbances ....... . D.Toumpakaris,J.M.Cioffi,andD.Gardan 1927 Fading/Equalization TrainingSequenceOptimizationinMIMOSystemsWithColoredInterference...... ...... ........ ..... T.F.WongandB.Park 1939 DistributionFunctionsofSelectionCombinerOutputinEquallyCorrelatedRayleigh,Rician,andNakagami-mFadingChannels. ... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... ...Y.ChenandC.Tellambura 1948 Networks Delay-LimitedThroughputofAdHocNetworks .. ...... ...... ........ ....... ..... ...... ..E.PerevalovandR.S.Blum 1957 OpticalCommunication PerformanceAnalysisandTradeoffsforDual-PulsePPMonOpticalCommunicationChannelsWithDirectDetection.. ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... .....M.K.SimonandV.A.Vilnrotter 1969 (ContentsContinuedonBackCover) TLFeBOOK (ContentsContinuedfromFrontCover) PersonalCommunicationSystems AccurateSimulationofMultipleCross-CorrelatedRicianFadingChannels .. ....... ....... ....K.E.BaddourandN.C.Beaulieu 1980 SignalProcessing OFDMSystemsinthePresenceofPhaseNoise:ConsequencesandSolutions ...... ...... ........ ......S.WuandY.Bar-Ness 1988 Synchronization Pilot-AssistedMaximum-LikelihoodFrequency-OffsetEstimationforOFDMSystems..... ......... ...... . J.H.YuandY.T.Su 1997 TransmissionSystems AClassofNonlinearSignal-ProcessingSchemesforBandwidth-EfficientOFDMTransmissionWithLowEnvelopeFluctuation.... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... .... R.DinisandA.Gusmão 2009 Transmission/Reception Pre-DFTProcessingUsingEigenanalysisforCodedOFDMWithMultipleReceiveAntennas ......... .. D.HuangandK.B.Letaief 2019 AbstractsofForthcomingManuscripts.... ..... ...... ...... ...... ........ ...... ...... ...... ...... ..... .... 2028 InformationforAuthors.. ...... ...... ..... ...... ...... ....... ........ ..... ...... ...... ...... ..... .... 2031 PAPERS SCHEDULED TO BE PUBLISHED IN THE NEXT ISSUE DECEMBER2004 TransactionsLetters OnSoft-InputSoft-OutputDecodingUsing“BoxandMatch”Techniques..... ..P.A.Martin,A.Valembois,M.P.C.Fossorier,andD.P.Taylor AnEfficientAlgorithmtoComputetheEuclideanDistanceSpectrumofaGeneralIntersymbolInterferenceChannelandItsApplications ... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... . J.Li,K.R.Narayanan,andC.N.Georghiades Reduced-ComplexityErrorStateDiagramsinTCMandISIChannelPerformanceEvaluation ...... ......... ..... W.E.RyanandZ.Tang AModifiedBlahutAlgorithmforDecodingReed-SolomonCodesBeyondHalftheMinimumDistance. ..... ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...S.Egorov,G.Markarian,andK.Pickavance Lattice-Reduction-AidedBroadcastPrecoding.... ...... ........ ....... ...... ... C.Windpassinger,R.F.H.Fischer,andJ.B.Huber Bit-InterleavedTurbo-EqualizationOverStaticFrequency-SelectiveChannels:ConstellationMappingImpact . ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... .....A.DejongheandL.Vandendorpe SharingofARQSlotsinGilbert-ElliotChannels .. ...... ...... ......... ...... ..... ......C.-H.Hou,J.-F.Chang,andD.-Y.Chen Doppler-ChannelBlindIdentificationforNon-CircularTransmissionsinMultiple-AccessSystems... ......... ..A.NapolitanoandM.Tanda AnAlternativeExpressionfortheSymbolErrorProbabilityofMPSKinthePresenceofI-QUnbalance ..... ......... ..S.ParkandD.Yoon MatchedFilterBoundforBinarySignalingOverDispersiveFadingChannelsWithReceiveDiversity. ...... ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... . H.Hadinejad-Mahram,D.Dahlhaus,andD.Blömker TransactionsPapers ThresholdValuesandConvergencePropertiesofMajority-BasedAlgorithmsforDecodingRegularLow-DensityParity-CheckCodes.. .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... ..P.ZarrinkhatandA.H.Banihashemi Graph-BasedMessage-PassingSchedulesforDecodingLDPCCodes...... ...... ........ ...... ......H.XiaoandA.H.Banihashemi AMoreAccurateOne-DimensionalAnalysisandDesignofIrregularLDPCCodes .. ...... ........ ...M.ArdakaniandF.R.Kschischang SubspaceAlgorithmsforErrorLocalizationWithQuantizedDFTCodes.... ...... ...... ........ ...... ....G.RathandC.Guillemot OnFinite-StateVectorQuantizationforNoisyChannels... ...... ...... ......... ..... ...... ...... . P.YahampathandM.Pawlak ParametricConstructionofNyquist-IPulses ..... ...... ...... ........ ....... ..... ...... .....N.C.BeaulieuandM.O.Damen SERandOutageofThreshold-BasedHybridSelection/Maximal-RatioCombiningOverGeneralizedFadingChannels . ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... ...... ..X.ZhangandN.C.Beaulieu ANSFrequencyDomainApproachforContinuousTimeDesignofCAP/ICOMWaveform.. ...... ......... .... X.TangandI.L.-J.Thng OCDMACodedFreeSpaceOpticalLinksforWirelessOpticalMeshNetworks..... ..... ......... ...... . B.HamzehandM.Kavehrad PerformanceModelingofOpticalBurstSwitchingWithFiberDelayLines .. ...... ..... ......... ...... ...... .X.LuandB.L.Mark BandwidthEfficientWDMChannelAllocationforFour-WaveMixingEffectMinimization.. ......... .V.L.L.Thing,P.Shum,andM.K.Rao CharacterizingOutageRatesforSpace-TimeCommunicationOverWidebandChannels .... ........ ....... .. G.BarriacandU.Madhow ParallelInterferenceCancellationforUplinkMultirateOverlayCDMAChannels.... ..... ....... ........ ...... ...H.YanandS.Roy AbstractsofForthcomingManuscripts.... ..... ...... ...... ...... ...... ........ ...... ...... ...... ..... ...... .... TLFeBOOK IEEE COMMUNICATIONS SOCIETY ThefieldofinterestoftheCommunicationsSocietyconsistsofalltelecommunicationsincludingtelephone,telegraphy,facsimile,andpoint-to-pointtelevision,byelectromagneticpropagationincludingradio;wire;aerial; underground,coaxial,andsubmarinecables;waveguides,communicationsatellites,andlasers;inmarine,aeronautical,spaceandfixedstationservices;repeaters,radiorelaying,signalstorage,andregeneration;telecommuni- cationerrordetectionandcorrection;multiplexingandcarriertechniques;communicationswitchingsystems;datacommunications;andcommunicationtheory. 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DigitalObjectIdentifier10.1109/TCOMM.2004.839885 TLFeBOOK IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 1833 Transactions Letters ________________________________________________________________ An Algorithm for Detecting Unreliable Code Sequence Segments and Its Applications Jürgen Freudenberger and Boris Stender Abstract—Let the Viterbi algorithm be applied for max- The contribution of this letter is a modified Viterbi decoder imum-likelihood decoding of a terminated convolutional code forterminatedconvolutionalcodes.Thenewalgorithmdoesnot using a trellis. We propose an additional procedure that permits only output a single estimate, but marks code sequence seg- areceivertolocateunreliablesegmentswithinanestimatedcode mentsasunreliableifanerroneousdecisionforthesesegments sequence. This reliability output may be used, for example, to requestretransmissions,insystemswitherrorconcealment,orin islikely.Wedonotconsidersymbol-by-symbolreliabilities,but channel-codingsystemswithunequalerrorprotection. possibleerroreventsforViterbidecoding,i.e.,pathsinthede- codertrelliswhichdivergefromthemostlikelycodesequence Index Terms—Convolutional codes, list decoding, reliability output,repeatrequest,Viterbidecoding. andlateronre-merge.Inadditiontothiserasure-declaringde- coding algorithm, we present a soft-output version. Here the decoder puts out not an erasure or estimate alone, but an esti- I. INTRODUCTION matewithareliabilityindicator.Thisreliabilityindicatoristhe DECODING with erasures is a form of decoding where thresholdforwhichthecorrespondingcodesegmentwouldhave the decoder rejects received data if a reliable decision is beenerased,hadanerasureoptionbeenused. notapparent.Therefore,thereliabilityoftheestimateistested Thepaperisstructuredasfollows.InSectionII,weintroduce and the decoded message is only accepted if the decision is notations,andinSectionIII,wedescribethetwoversionsofour sufficientlyreliable,whereasanerasureisdeclaredifareliable reliabilityoutputdecodingalgorithmindetail.Inthesubsequent decisionisnotevident.Thereareseveraldecodingschemeswith section,weconsidertheirapplicationinrepeat-requestschemes. erasures which have been applied to coded automatic repeat First,wecomparetheperformanceofthereliabilityoutputalgo- request (ARQ). Those schemes may be classified according rithmwiththatofYamamotoandItoh’sscheme[4].Second,we to their erasure-decision rules [1], [2]. An important class of introduce an adaptive-feedback error-correction scheme. Sim- erasure-decisionrulesarelikelihood-ratiotestswheretheratio ilar to standard hybrid ARQ, the new approach is packet ori- oflikelihood(metric)valuesaretested[2]–[5].Frequently,the ented.However,weproposetotransmitincrementalredundancy erasure decision is based on explicit error detection. For this only for particular unreliable code sequence segments which purpose, usually a high-rate cyclic code is used exclusively can be determined using the new algorithm. Finally, we give for error detection. This concept was generalized to error- someconcludingremarksinSectionV. locating codes by Wolf and Elspas [6]. With error-locating codes, a block of received digits is regarded as subdivided II. NOTATIONS into mutually exclusive subblocks. Errors occurring within TraditionalViterbidecodingisexplainedbymeansofatrellis particular subblocks are detected at the receiver. In addition, diagram. A trellis is a directed graph. The set of nodes of the the receiver is able to determine which particular subblocks graph is decomposed into a union of disjoint containerrors.Similarly,itisdesirabletogeneralizetheconcept subsets that are called levels of the of likelihood-ratio decision. In addition to the likelihood-ratio trellis. A node of the level may be connected with a test, the receiver should be able to determine which parts node of the level by one or several branches. of a codeword (or which segments of a convolutional code Eachbranch isdirectedfromanode oflevel toanode sequence) are unreliable. of the nextlevel . We assume that the end levelshaveonly onenode,namely, and . Inthefollowing,weconsiderrate binaryterminated PaperapprovedbyR.D.Wesel,theEditorforCodingandCommunication convolutionalcodes[7]usedforcommunicationoveramemo- TheoryoftheIEEECommunicationsSociety.ManuscriptreceivedNovember 13,2003;revisedMay7,2004.ThisworkwassupportedbytheDFG(Deutsche rylesschannel.Assumeaterminatedcodesequence consistsof Forschungsgemeinschaft)underGrantBo867/9-1.Thispaperwaspresentedin -tuples , i.e., . partattheInternationalSymposiumonInformationTheory,Lausanne,Switzer- The encoder of the convolutional code is forced to return land,June2002. TheauthorsarewiththeDepartmentofTelecommunicationsandApplied to its initial state by appending dummy -tuples to the Information Theory, University of Ulm, D-89081 Ulm, Germany (e-mail: information sequence, where denotes the memory of the [email protected]; [email protected] convolutional code. The corresponding information sequence ulm.de). DigitalObjectIdentifier10.1109/TCOMM.2004.836556 is , with . For 0090-6778/04$20.00©2004IEEE TLFeBOOK 1834 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 Fig.1. Divergingpathslabelingsegmentsasunreliable. the decoder trellis of a convolutional code, a node Definition 1: Let be a given threshold. Con- corresponds to a possible encoder state. Each branch of the sider the ordered list of the most likely paths decoder trellis is labeled by a -tuple . There is suchthat a one-to-one correspondence between each codeword in with the code and a path in the trellis, such that . We denote code sequence segments and and path segments by and , for all paths respectively. We call the segments merged if we Forlevels whereallpathsin aremerged,i.e., have . Let be the received sequence. ,wedefinethedecoderoutput asreliable,oth- A Viterbi decoder selects the code sequence erwiseasunreliable. whichmaximizestheconditionalprobability ,orequiv- Example 1: Fig. 1 shows the conceptual situation with this alently, which minimizes the Viterbi metric reliabilitycriterion,wherethefigureillustratesonlypaths .Thestraightlineindicatesthemostlikelycodesequence.Fur- (1) thermore,therearethreeunreliablesegmentsindicated,where with (2) foreachsegment,thereareseveralcandidatepathswithmetric (3) differenceslessorequal .Thevalues denotethe metricdifferencesofmergingpaths. In order to evaluate (1) efficiently, we assign additional node NotethatthisreliabilitycriterionissimilartotheoneSchaub and partial path metrics and , respectively. and Modestino [11] introduced for a symbol-by-symbol era- is the metric of the local survivor entering node from level sure-declaringVA.Itisalsorelatedtothevariouslist-typegen- . The complexity of the Viterbi algorithm (VA) for a eralizationsoftheVAwhichhaveappearedintheliterature[8], terminated convolutional code is , where [12]–[15].Thatis,astraightforwardmethodforlabelingthede- denotes the overall constraint length of the code, which is the coderoutputwouldbeasfollows.First,determinetheordered minimum number of memory elements required to realize the list andmark initiallyasreliable.Considerallpathsinthe correspondingconvolutionalencoder.Foradetaileddiscussion orderedlist startingwith andendingwith .Trace of the trellis representation of convolutional codes or Viterbi thecorrespondingpathsfromnode tonode andmarkthe decoding, see, for example, [8] and [9] or the textbooks [7] decoder output as unreliable if a path is not merged with and [10]. attime .Thecomplexityofthismethodincreaseslinearly withincreasinglistsize.However,itisnotnecessarytogenerate the list of paths explicitly in order to mark the decoder output accordingtoDefinition1.Thefollowinglemmaprovidessome III. RELIABILITYOUTPUTALGORITHM insightconcerningthisissue. An error event with Viterbi decoding occurs whenever the Compare any path from with the path . estimatedpathdivergesfromthecorrectpath.Inmanypractical The two paths may diverge and re-merge many times, but the situations, it is useful to decide whether such an error event end states of the paths are the same. We say that a path is is likely or not. For this purpose, Yamamoto and Itoh [4] a one-loop path if it diverges from the path only once. introduced the following reliability test. Let be a given Furthermore, let the symbol denote path concatenation, i.e., threshold. Consider the first and second most likely paths with and . , such that . The estimated Lemma 1: In order to mark the decoder output as re- path is considered reliable if , liableorunreliableaccordingtoDefinition1,itis sufficientto otherwise unreliable. The underlying idea is that the most considerallone-looppathsintheorderedlist . unreliable decision during the VA occurred at the node , Proof: Assumethatweinitiallymark asreliable.By wherethebestandthesecond-bestpathsmerged.However,for back-tracingapath fromnode tonode ,wechange moderateorhighsignal-to-noiseratios(SNRs),anerroneously thelabelofthedecoderoutput fromreliabletounreliable estimated path will remerge with the correct path after a few ifthepathisnotmergedwith attime .Letapath , transitions.Itisthereforereasonabletodeclareonlypartsofthe havetwoormoreloops estimated sequence as unreliable if . , withrespecttothepath We therefore consider the following criterion. . Regarding ,the code-sequence TLFeBOOK IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 1835 Fig.2. Exampleofalabeledtrellis(ROAI). segmentscorrespondingto shouldbemarkedunre- tween the list entry and the node metric. Trace back liable.Weshowthatthefollowingone-looppaths allbrancheswithmetricdifferencelessthanorequal , tothenodethreshold.Labelthecorrespondingnodes all have a metric less than or equal to at level with new thresholds, which are the , from which we conclude that . In fact, differences of the node threshold at level and the , , metric differences of the particular branches. If dif- because hastheleastmetricamongallpaths.Forthepath ferent paths merge in a node at level , find the , we have , and maximum of all according thresholds at level and we obtain , ,andso on.From assignittothenodeatlevel . follows .Thus,bytracingbackthe Step5) If there existsa branch different from between one-looppaths ,weobtainthesamelabelingaswith level and whichconnectsnodeslabeledwith .Similarly,bytracingbackallone-looppathsin ,weobtain thresholds not equal to zero, mark the estimate thedesiredlabeling,accordingtoDefinition1. asunreliable,otherwise,markitreliable. It follows from Lemma 1 that we may use an ordered list Step6) If ,thencontinuewiththenextlevel.Decrement of all one-loop paths contained in instead of the byoneandgotoStep4). complete list . A recursive algorithm to construct the list of Theorem1: Algorithm1labelsthemaximum-likelihoodes- the most likely one-loop paths is given in [13]. However, the timateaccordingtoDefinition1. following nonrecursive algorithm determines these unreliable AproofofTheorem1isgivenintheAppendix.Notethatour positionswithoutexplicitlygeneratingalist. reliabilitycriterionisthesameasthatofYamamotoandItoh[4] ifwedeclarethecompletecodesequenceasunreliableoncean unreliable segment has occurred. In particular, the bounds de- Algorithm1ReliabilityOutputAlgorithm(ROA)I: rivedin[4]onthefirstevent-errorprobabilityandontheevent ofthefirsterasurearealsovalidforourscheme.Thesebounds Step1) Assignmetriczerototheinitialnode and dependonthepathenumeratorfunctionofthecodeandonthe set . metricthreshold.Forlowchannel-errorrates,theseboundscan Step2) Each node oflevel isprocessedasfollows(for- be used to determine the appropriate threshold values. How- ward pass). Calculate the partial path metrics corre- ever,forpoorchannelconditions,theseunionboundsdiverge. sponding to the branches that enter the node Inthiscase,thresholdvalueshavetobedeterminedbycomputer by ,where isthe simulations. node metric ofthisbranch’s predecessornode . Example2: Fig.2depictsalabeledtrellisfortherate Moreover, assign the node metric convolutional code with memory and generator to the node , where denotes the branch with matrix (encoder in con- the smallest partial path metric among all branches trollercanonicalform).Fourinformationsymbolsfollowedby merging in node . Store a list with all partial path twodummyzerosareencoded.Weconsidertransmissionover metrics. thebinarysymmetricchannelwithacrossoverprobability Step3) If ,thencontinuewiththenextlevel.Increment .Therefore,wehavebranchmetricvalues by one and go to Step 2). Otherwise, label the ter- and . Let the received sequence be minatingnode withathresholdvalue ,allothers .Thelistundereachnodecomprisesthemet- withthresholdvalue0,andgotoStep4). ricscorrespondingtoincomingbranches.Thebackwardpassof Step4) Eachnode oflevel withthresholdgreaterthan0 the ROA I starts at the terminating node (node in Fig. 2). isprocessedasfollows(backwardpass).Foreachel- Above each node which is reprocessed during the backward ementinthemetriclist,determinethedifferencebe- pass,thenodethresholdisdepicted.Additionally,allbranches TLFeBOOK 1836 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 Fig.3. Exampleofalabeledtrellis(ROAII). aremarkedwhichwillbetracedback.Inthisexample,thetermi- isnowindependentofthereceivedsequenceandisstillofthe natingnode isinitializedwiththreshold .Thenode sameorderastheVA. metric for node is 5.8, and the differences of the incoming Example 3: Fig. 3 depicts a labeled trellis for the ROA II, pathmetricstothenodemetricare0.0and6.9.Consequently, whereweusedthesamesettingsasinExample2.Aboveeach onlythelocalsurvivoristracedback.Notethatasaresult,all node, the node’s reliability value is shown. With ROA II, all nodescorrespondingtotheglobalsurvivorpath arelabeled branches in the trellis are traced back. The terminating node withthreshold ,because isalwaysgreaterthanorequalto is initialized with value 0.0. The differences of the incoming anyotherthreshold.Atnode ,thedifferencesare0.0and2.3. branch metrics to the terminating node’s metric are 0.0 and Asthesevaluesaresmallerthan2.5,bothincomingbranchesare 6.9.Bothincomingbranchesaretracedback,andthenodesat tracedback.Moreover,node islabeledwiththenewthreshold level are labeled with these values. As 6.9 is the .Now,twopathsaretracedbackuntiltheymerge smallest value that can be found at a node connected by the innode .Ifdifferentpathsmergeinanodeatlevel ,we last branches, which is greater than 0, this value is given to assign the maximum of all thresholds, e.g., node is marked theoutput.Ifwecontinuethebacktracing,wehavetoaccumu- with threshold 2.5. Finally, we notice that the estimated code late all the metric differences until some paths merge. In this segment corresponding to the path from node to node is case,wecontinuewiththeminimumvalue.Thefinalreliability markedunreliable,becausetherearenodeswhichdonotcorre- output is . Finally, marking the spondtothesurvivorpath thatarelabeledwiththresholds estimatedcodesegmentcorrespondingtoreliabilityvaluesless greaterthan0. than as unreliable, we obtain the same labelingas in Intheremainingpartofthissection,weconsideramodifica- Example2. tionofAlgorithm1suchthatthenewalgorithm(ROAII)deter- mines a vector of reliability values. Steps 1–3 IV. APPLICATIONSANDSIMULATIONRESULTS of ROA II are the same as with ROA I, except that we do not assumeapredeterminedthreshold .Therefore,inStep3,we The above-presented algorithm is applicable for repeat- label only the terminating node with value 0 and continue request strategies. In this case, the complete received code withStep4,whereeachnode oflevel isprocessedasfol- sequenceshouldbestored,andthereceivermaysimplyrequest lows.Foreachelementinthemetriclist,determinethediffer- retransmission of code sequence segments that are marked encebetweenthelistentryandthenodemetric.Tracebackall unreliable. The marked symbols are then substituted by (or branchesandlabelthecorrespondingnodesatlevel with combined with) the newly received symbols and the decoding the new values. The new values are the node values at level procedure restarts. The transmission terminates when finally plusthemetricdifferencesoftheparticularbranch.Ifdifferent no unreliable symbols are determined. The performance of paths merge in a node at level , find the minimum of all such repeat-request schemes may be measured in terms of thesevalues.Thenodeslabeledwithvalue0correspondtothe errorprobabilities,e.g.,thebit-errorprobability,afterthefinal survivorpath .Thereliabilityvalue isthesmallestvalue, decoding step. Furthermore, the effective transmission rate is exceptthevaluezerofromthesurvivor,whichcanbefoundata ofgreatimportance,whichisdefinedastheratioofthenumber labelfromnodes or .Thelabelingofthedecoderoutput of decoded data symbols to the number of transmitted code canbeperformedbasedonthesereliabilityvalues.Thus,Step symbols.Inthefollowing,weinvestigatesucharepeat-request 5 in Algorithm 1 can be omitted. In order to obtain an erasure scheme by means of computer simulations. We compare our labelingofthedecoderoutput,wedeclareallpositions ,where schemewiththestrategyproposedbyYamamotoandItoh[4]. thereliabilityvalue islessthanorequaltoachosenthreshold YamamotoandItoh’salgorithm(YIA)isalsobasedonViterbi , as unreliable. This second version of the reliability output decoding,hasthesameorderofdecodingcomplexity,anduses algorithmrequiresmorecalculations,becauseeachnodeisre- the same likelihood test. However, YIA does not determine processedduringthebackwardpass.However,thecomplexity the exact lengths of unreliable segments. With Yamamoto and TLFeBOOK IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 1837 Fig.5. ARQschemewithROAII. noiselessfeedbacklink.Ifactualfeedbacklinkswithnoiseare treated,thenthelisttobetransmittedoverthereturnchannelhas tobeprotectedbyforward-errorcorrection.Ifaretransmission Fig. 4. Simulation results for a memory m = 3 optimum free distance is requested, then the transmitter calculates the code bit posi- convolutionalcode(G(D)=(1+D+D +D ;1+D+D ),M =32). tionstobesentnext.Thisisdonesimultaneouslyatthereceiver, AWGNchannelwithSNRsE =N 2 f(cid:0)2;(cid:0)1;0;1dBg.Thevaluesbeside asthisinformationisneededintheMRCunit.Theschemeused each simulation point are the respective metric thresholds. For the ROA algorithm,thecodesequencesareterminatedafter1000informationbits,while for simulationcan be described as follows.First, calculatethe withYIA,thesequencesarenotterminated. numberofcodebits thataretobetransmitted.Foreachunre- liablesegment,wechooseavaluewhichisthesumofasmall Itoh’s scheme, a retransmission of the latest code tuples is constant andafraction ofthesegmentlength .As should requested if the survivor is declared unreliable, where is a beaninteger,wecalculate .Second,thepartic- preselectedconstant.Thisresultsinarateloss,ascanbeseen ular code bit positions have to be found. For this purpose, the from the following simulation results. For these simulations, receiverregistersthenumberoftransmissionsforeachcodebit, we consideredtransmission overabinary-inputadditivewhite andselects positionswhichincrementtheredundancy. Gaussian noise (BIAWGN) channel, an ideal return channel, Example4: Forsimulations,weassumeperfecterrordetec- andmaximumratiocombiningofretransmittedsymbols.Fig.4 tion. We encode each data block with a memory presents the corresponding results.1 Note that both schemes terminatedconvolutionalcode(133145175).Thenwe achievethesameBERsforagivenmetricthresholdand perform an equidistant puncturing, such that every third code value. This is due to the fact that both schemes are based on bitistransmitted(givinganinitialcoderate ). the same likelihood test. However, with the scheme based on Theparameters and for retransmissionare chosentobe 2 ROA I, the effective transmission rate, necessary to achieve and1/24,respectively.Theincrement isfixedtobe1/10.In a certain BER, is significantly improved for all considered Fig.6,weseethethroughputreachedwiththisschemeaswell SNRs. asthecapacitycurvefortheBIAWGNchannel.Theasteriskat Inthefollowing,wedescribeadifferentsystemwhereonlya position( 1.62;0.331)indicatestheSNRwhereaBERof fewbitsareretransmittedwitheachrepeatrequest.Fig.5shows isobtainedwiththeterminatedrate mothercode.The theflowchartforatransmissionofonedatablockwiththisARQ achieved performance gain of about 2.9 dB is a result of the scheme. For the initial transmission, the data block is first en- feedbackscheme. coded with a convolutional encoder. Then the code sequence ispuncturedtogetaninitialcoderate.Aftertransmissionover V. CONCLUSIONS the BIAWGN channel, all code bits of a block are maximum ratiocombined(MRC).Thenthe aggregatedcodesequenceis We have devised a new algorithm that permits a receiver to locate unreliable segments within an estimated convolutional decoded using the ROA II. Afterwards, error detection should beapplied.Theerror-detectionpartispicturedbythetermtest code sequence. The reliability criterion is based on the list of the most likely codewords which satisfy a likelihood-ratio inFig.5.Ifanerrorisdetected,alistwithunreliablesegments mustbesentback.Theinterceptionpointlabeled indicatesthe test.Inprinciple,thesamereliabilityoutputdecodingcouldbe performedusingoneofthevariouslist-typegeneralizationsof 1For YIA, we used M =32 as in the original paper [4]. Small gains the VA. However, the complexity of list-output algorithms, in can be obtained by using optimized values of M for different thresholds general,dependsonthelistsize .Itmayincreaselinearlywith and SNR values. However, the differences are moderate, compared with a fixed value of M. , where the list size itself depends on the received sequence TLFeBOOK 1838 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 deriveanequivalentexpressionintermsoftheset .Consider thenode .Apath passingthrough canberepresentedas concatenationofaheadandatailpath: .If ,thenatleastthepathwithminimum metricamongallpathspassingthrough isanelementof . Thispathhasalsominimumheadandminimumtailpathmetric. Thus,wehave (4) wheretheminimumistakenoverallheadandtailpaths,ending orstartinginnode . With the help of the list of all branch metrics that were stored in Step 2, we can calculate the accumulated metric ofanytail path Fig.6. BIAWGNcapacityandthroughputofROA-ARQ. andincreaseswithincreasingchannelerrorrate.Theworst-case complexityofthenewalgorithmisindependentofthelistsize, andisofthesamecomplexityorderastheVA. wherewehaveusedtheabbreviation ,i.e., The reliability output may be used to request retransmis- denotes the difference between the list entry corresponding to sions. The performance of such a repeat-request scheme has the particular branch and the node metric at node . Substi- been compared with that of Yamamoto and Itoh [4] and is tutingthisinto(4),weget shown to provide a significant improvement in throughput. In addition, an adaptive-feedback error-correction scheme has been presented. The corresponding simulation results indicate that near-optimum performance for the BIAWGN channel is attainable with moderate complexity and for very short block sizes. This is true for a wide range of SNRs. However, while Notice that only the sum depends on the particular tail standard hybrid ARQ schemes require only a single bit of path. In particular, we have and information per packet over the feedback path, we have to . Therefore, canceling these send much more information in order to obtain this level of termsweobtain performance. APPENDIX PROOFOFTHEOREM1 Proof: Steps1–3oftheROAIareessentiallytheforward passofaVA.WiththeROAI,weadditionallystorealistofall Let denotethenodethresholdatnode with .Note partialpathmetricsforeachnodeinthetrellis.Thus,following thatthebackwardpassoftheROAI(Steps4–6)isessentially the branches with the smallest partial path metric in the list, aVAwhichcalculatesthenodethresholdsforeachnode starting in the terminating node , and ending in the initial . To see this, consider Step 4, where the node threshold is node ,wecanfindthebestpath .Inthefollowingproof, calculated as . Branches which weconsidertheadditionallabelingoftheoutputsequence . stem from a state need not to be considered, because AccordingtoDefinition1,wehavetoshowthatouralgorithm thecorrespondingmetricdifferenceswouldresultinathreshold labelsthedecoderoutput reliableifallpathsinthelist value less than zero. Thus, by finite induction, we have are merged.That is,during thebackward pass(Steps4–6) the .Finally,weobservethat algorithmshouldmarktheestimate atlevel reliable,iffall .ItfollowsfromStep4thatwetracebackallbranchesto paths aremergedatthislevel.Otherwise, hastobe nodesforwhichthisinequalityisfulfilled.Therefore,allnodes labeledunreliable.Insteadofdirectlyconsideringpathsthrough arefound.Moreover,iffthereexistmorethanonestate thetrellis,weconsidercertaintrellisnodes.Wethereforedefine at time or more than one state at time ,thenthereisabranch correspondingtoapath there exists a path passing through .Consequently, hastobemarkedunreliable. Thatis, isthesetofallnodesinthetrellisrepresentation Remarks: Theaboveproofisonlyvalidfortrellisrepresen- whichareconnectedbyanypath .Forconvenience,we tations which have no parallel branches from some state to resume . We will now some state . In particular, it does not hold for partial unit TLFeBOOK IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 1839 memorycodes.However,thevalidityofthealgorithmcanalso [5] T.Hashimoto,“OntheerrorexponentofconvolutionallycodedARQ,” beshownforthiscase.Moreover,wewouldliketomentionthat IEEETrans.Inform.Theory,vol.40,pp.567–575,Mar.1994. [6] J.K.WolfandB.Elspas,“Error-locatingcodes—anewconceptinerror the expression in the proof corresponds to the control,”IEEETrans.Inform.Theory,vol.IT-9,pp.113–117,Apr.1963. labelingofROA-II.Inordertoobtainanerasurelabelingofthe [7] R.JohannessonandK.S.Zigangirov,FundamentalsofConvolutional decoderoutputofROA-II,wedeclareallpositions ,wherethe Coding. Piscataway,NJ:IEEEPress,1999. [8] G.D.Forney,Jr.,“Convolutionalcodes{II}:Maximumlikelihoodde- reliability value is less than or equal to a chosen threshold coding,”Inform.Control,vol.25,pp.222–266,July1974. as unreliable. Arguments similar to the proof of Theorem 1 [9] ,“TheViterbialgorithm,”Proc.IEEE,vol.61,pp.268–278,Mar. showthattheresultinglabelingsatisfiesDefinition1. 1973. [10] M. Bossert, Channel Coding for Telecommunications. New York: Wiley,1999. [11] T.SchaubandJ.W.Modestino,“AnerasuredeclaringViterbidecoder REFERENCES anditsapplicationstoconcatenatedcodingsystems,”inProc.IEEEInt. Conf.Communications,1986,pp.1612–1616. [1] T.HashimotoandM.Taguchi,“Performanceofexpliciterrordetection [12] T. Hashimoto, “A list-type reduced-constraint generalization of the andthresholddecisionindecodingwitherasures,”IEEETrans.Inform. Viterbi algorithm,” IEEE Trans. Inform. Theory, vol. IT-33, pp. Theory,vol.43,pp.1650–1655,Sept.1997. 866–876,Nov.1987. [2] T.Hashimoto,“CompositeschemeLR+Thfordecodingwitherasures [13] V.V.Zyablov,V.G.Potapov,andV.R.Sidorenko,“Maximum-likeli- and its effective equivalence to Forney’s rule,” IEEE Trans. Inform. hoodlistdecodingusingtrellises,”ProblemyPeredachiInformatsii,vol. Theory,vol.45,pp.78–93,Jan.1999. 29,pp.3–10,1993. [3] G.D.Forney,Jr.,“Exponentialerrorboundsforerasure,list,andde- [14] N.SeshadriandC.E.W.Sundberg,“ListViterbidecodingalgorithm cisionfeedbackschemes,”IEEETrans.Inform.Theory,vol.IT-14,pp. withapplications,”IEEETrans.Commun.,vol.42,pp.313–323,Feb. 206–220,Mar.1968. 1994. [4] H.YamamotoandK.Itoh,“Viterbidecodingalgorithmforconvolutional [15] C.NillandC.E.W.Sundberg,“ListandsoftsymboloutputViterbi codeswithrepeatrequest,”IEEETrans.Inform.Theory,vol.IT-26,pp. algorithms:Extensionsandcomparisions,”IEEETrans.Commun.,vol. 540–547,Sept.1980. 43,pp.277–287,Feb.1995. TLFeBOOK

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NOVEMBER2004 VOLUME52 NUMBER11 IECMBT (ISSN0090-6778) TRANSACTIONSLETTERS Coding AnAlgorithmforDetectingUnreliableCodeSequenceSegmentsandItsApplications ....... ......J.FreudenbergerandB.Stender 1833 DecodingofLow-DensityParity-CheckCodesOverFinite-StateBinaryMarkovChannels .. ...... ......... .....J.Garcia-Frias 1840 DigitalCommunications AccurateComputationofthePerformanceofM-aryOrthogonalSignalingonaDiscreteMemorylessChannel. ......... . J.Hamkins 1844 High-RateRecursiveConvolutionalCodesforConcatenatedChannelCodes . ....... F.Daneshgaran,M.Laddomada,andM.Mondin 1846 Fading/Equalization ANewBase-StationReceiverforIncreasingDiversityOrderinaCDMACellularSystem...... W.Choi,C.Yi,J.Y.Kim,andD.I.Kim 1851 PowerControlandDiversityinFeedbackCommunicationsOveraFadingChannel... ....... ....... . I.SaarinenandA.Mämmelä 1857 SpreadSpectrum LargeSetofCISpreadingCodesforHigh-CapacityMC-CDMA... ......... .....B.Natarajan,Z.Wu,C.R.Nassar,andS.Shattil 1862 Synchronization OntheMiller–ChangLowerBoundforNDACarrierPhaseEstimation...... .......G.N.Tavares,L.M.Tavares,andM.S.Piedade 1867 TRANSACTIONSPAPERS Coding PerformanceAnalysisandDesignCriteriaforFinite-AlphabetSource-ChannelCodes. ...... ........ .A.HedayatandA.Nosratinia 1872 Near-CapacityCodinginMulticarrierModulationSystems. ...... ......... ....M.Ardakani,T.Esmailian,andF.R.Kschischang 1880 PerformanceofCodedOQPSKandMIL-STDSOQPSKWithIterativeDecoding.... ..... ......... ......L.LiandM.K.Simon 1890 DesignandPerformanceofTurboGallagerCodes . ...... ...... ...... ......... ..... ...... ...... ......G.Colavolpe 1901 ReducedComplexityMAP-BasedIterativeMultiuserDetectionforCodedMulticarrierCDMASystems ..... ...... ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ......J.Li,K.B.Letaief,andZ.Cao 1909 DigitalCommunications ANovelTrellis-ShapingDesignWithBothPeakandAveragePowerReductionforOFDMSystems.. ....... ........ .. H.Ochiai 1916 Reduced-DelayProtectionofDSLSystemsAgainstNonstationaryDisturbances ....... . D.Toumpakaris,J.M.Cioffi,andD.Gardan 1927 Fading/Equalization TrainingSequenceOptimizationinMIMOSystemsWithColoredInterference...... ...... ........ ..... T.F.WongandB.Park 1939 DistributionFunctionsofSelectionCombinerOutputinEquallyCorrelatedRayleigh,Rician,andNakagami-mFadingChannels. ... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... ...Y.ChenandC.Tellambura 1948 Networks Delay-LimitedThroughputofAdHocNetworks .. ...... ...... ........ ....... ..... ...... ..E.PerevalovandR.S.Blum 1957 OpticalCommunication PerformanceAnalysisandTradeoffsforDual-PulsePPMonOpticalCommunicationChannelsWithDirectDetection.. ..... .... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... .....M.K.SimonandV.A.Vilnrotter 1969 (ContentsContinuedonBackCover) TLFeBOOK (ContentsContinuedfromFrontCover) PersonalCommunicationSystems AccurateSimulationofMultipleCross-CorrelatedRicianFadingChannels .. ....... ....... ....K.E.BaddourandN.C.Beaulieu 1980 SignalProcessing OFDMSystemsinthePresenceofPhaseNoise:ConsequencesandSolutions ...... ...... ........ ......S.WuandY.Bar-Ness 1988 Synchronization Pilot-AssistedMaximum-LikelihoodFrequency-OffsetEstimationforOFDMSystems..... ......... ...... . J.H.YuandY.T.Su 1997 TransmissionSystems AClassofNonlinearSignal-ProcessingSchemesforBandwidth-EfficientOFDMTransmissionWithLowEnvelopeFluctuation.... ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... .... R.DinisandA.Gusmão 2009 Transmission/Reception Pre-DFTProcessingUsingEigenanalysisforCodedOFDMWithMultipleReceiveAntennas ......... .. D.HuangandK.B.Letaief 2019 AbstractsofForthcomingManuscripts.... ..... ...... ...... ...... ........ ...... ...... ...... ...... ..... .... 2028 InformationforAuthors.. ...... ...... ..... ...... ...... ....... ........ ..... ...... ...... ...... ..... .... 2031 PAPERS SCHEDULED TO BE PUBLISHED IN THE NEXT ISSUE DECEMBER2004 TransactionsLetters OnSoft-InputSoft-OutputDecodingUsing“BoxandMatch”Techniques..... ..P.A.Martin,A.Valembois,M.P.C.Fossorier,andD.P.Taylor AnEfficientAlgorithmtoComputetheEuclideanDistanceSpectrumofaGeneralIntersymbolInterferenceChannelandItsApplications ... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... . J.Li,K.R.Narayanan,andC.N.Georghiades Reduced-ComplexityErrorStateDiagramsinTCMandISIChannelPerformanceEvaluation ...... ......... ..... W.E.RyanandZ.Tang AModifiedBlahutAlgorithmforDecodingReed-SolomonCodesBeyondHalftheMinimumDistance. ..... ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...S.Egorov,G.Markarian,andK.Pickavance Lattice-Reduction-AidedBroadcastPrecoding.... ...... ........ ....... ...... ... C.Windpassinger,R.F.H.Fischer,andJ.B.Huber Bit-InterleavedTurbo-EqualizationOverStaticFrequency-SelectiveChannels:ConstellationMappingImpact . ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... .....A.DejongheandL.Vandendorpe SharingofARQSlotsinGilbert-ElliotChannels .. ...... ...... ......... ...... ..... ......C.-H.Hou,J.-F.Chang,andD.-Y.Chen Doppler-ChannelBlindIdentificationforNon-CircularTransmissionsinMultiple-AccessSystems... ......... ..A.NapolitanoandM.Tanda AnAlternativeExpressionfortheSymbolErrorProbabilityofMPSKinthePresenceofI-QUnbalance ..... ......... ..S.ParkandD.Yoon MatchedFilterBoundforBinarySignalingOverDispersiveFadingChannelsWithReceiveDiversity. ...... ...... ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... . H.Hadinejad-Mahram,D.Dahlhaus,andD.Blömker TransactionsPapers ThresholdValuesandConvergencePropertiesofMajority-BasedAlgorithmsforDecodingRegularLow-DensityParity-CheckCodes.. .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... ..P.ZarrinkhatandA.H.Banihashemi Graph-BasedMessage-PassingSchedulesforDecodingLDPCCodes...... ...... ........ ...... ......H.XiaoandA.H.Banihashemi AMoreAccurateOne-DimensionalAnalysisandDesignofIrregularLDPCCodes .. ...... ........ ...M.ArdakaniandF.R.Kschischang SubspaceAlgorithmsforErrorLocalizationWithQuantizedDFTCodes.... ...... ...... ........ ...... ....G.RathandC.Guillemot OnFinite-StateVectorQuantizationforNoisyChannels... ...... ...... ......... ..... ...... ...... . P.YahampathandM.Pawlak ParametricConstructionofNyquist-IPulses ..... ...... ...... ........ ....... ..... ...... .....N.C.BeaulieuandM.O.Damen SERandOutageofThreshold-BasedHybridSelection/Maximal-RatioCombiningOverGeneralizedFadingChannels . ..... ...... .... .. ...... ..... ...... ...... ...... ..... ...... ...... ...... ...... ..... ...... ...... ..X.ZhangandN.C.Beaulieu ANSFrequencyDomainApproachforContinuousTimeDesignofCAP/ICOMWaveform.. ...... ......... .... X.TangandI.L.-J.Thng OCDMACodedFreeSpaceOpticalLinksforWirelessOpticalMeshNetworks..... ..... ......... ...... . B.HamzehandM.Kavehrad PerformanceModelingofOpticalBurstSwitchingWithFiberDelayLines .. ...... ..... ......... ...... ...... .X.LuandB.L.Mark BandwidthEfficientWDMChannelAllocationforFour-WaveMixingEffectMinimization.. ......... .V.L.L.Thing,P.Shum,andM.K.Rao CharacterizingOutageRatesforSpace-TimeCommunicationOverWidebandChannels .... ........ ....... .. G.BarriacandU.Madhow ParallelInterferenceCancellationforUplinkMultirateOverlayCDMAChannels.... ..... ....... ........ ...... ...H.YanandS.Roy AbstractsofForthcomingManuscripts.... ..... ...... ...... ...... ...... ........ ...... ...... ...... ..... ...... .... TLFeBOOK IEEE COMMUNICATIONS SOCIETY ThefieldofinterestoftheCommunicationsSocietyconsistsofalltelecommunicationsincludingtelephone,telegraphy,facsimile,andpoint-to-pointtelevision,byelectromagneticpropagationincludingradio;wire;aerial; underground,coaxial,andsubmarinecables;waveguides,communicationsatellites,andlasers;inmarine,aeronautical,spaceandfixedstationservices;repeaters,radiorelaying,signalstorage,andregeneration;telecommuni- cationerrordetectionandcorrection;multiplexingandcarriertechniques;communicationswitchingsystems;datacommunications;andcommunicationtheory. 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DigitalObjectIdentifier10.1109/TCOMM.2004.839885 TLFeBOOK IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 1833 Transactions Letters ________________________________________________________________ An Algorithm for Detecting Unreliable Code Sequence Segments and Its Applications Jürgen Freudenberger and Boris Stender Abstract—Let the Viterbi algorithm be applied for max- The contribution of this letter is a modified Viterbi decoder imum-likelihood decoding of a terminated convolutional code forterminatedconvolutionalcodes.Thenewalgorithmdoesnot using a trellis. We propose an additional procedure that permits only output a single estimate, but marks code sequence seg- areceivertolocateunreliablesegmentswithinanestimatedcode mentsasunreliableifanerroneousdecisionforthesesegments sequence. This reliability output may be used, for example, to requestretransmissions,insystemswitherrorconcealment,orin islikely.Wedonotconsidersymbol-by-symbolreliabilities,but channel-codingsystemswithunequalerrorprotection. possibleerroreventsforViterbidecoding,i.e.,pathsinthede- codertrelliswhichdivergefromthemostlikelycodesequence Index Terms—Convolutional codes, list decoding, reliability output,repeatrequest,Viterbidecoding. andlateronre-merge.Inadditiontothiserasure-declaringde- coding algorithm, we present a soft-output version. Here the decoder puts out not an erasure or estimate alone, but an esti- I. INTRODUCTION matewithareliabilityindicator.Thisreliabilityindicatoristhe DECODING with erasures is a form of decoding where thresholdforwhichthecorrespondingcodesegmentwouldhave the decoder rejects received data if a reliable decision is beenerased,hadanerasureoptionbeenused. notapparent.Therefore,thereliabilityoftheestimateistested Thepaperisstructuredasfollows.InSectionII,weintroduce and the decoded message is only accepted if the decision is notations,andinSectionIII,wedescribethetwoversionsofour sufficientlyreliable,whereasanerasureisdeclaredifareliable reliabilityoutputdecodingalgorithmindetail.Inthesubsequent decisionisnotevident.Thereareseveraldecodingschemeswith section,weconsidertheirapplicationinrepeat-requestschemes. erasures which have been applied to coded automatic repeat First,wecomparetheperformanceofthereliabilityoutputalgo- request (ARQ). Those schemes may be classified according rithmwiththatofYamamotoandItoh’sscheme[4].Second,we to their erasure-decision rules [1], [2]. An important class of introduce an adaptive-feedback error-correction scheme. Sim- erasure-decisionrulesarelikelihood-ratiotestswheretheratio ilar to standard hybrid ARQ, the new approach is packet ori- oflikelihood(metric)valuesaretested[2]–[5].Frequently,the ented.However,weproposetotransmitincrementalredundancy erasure decision is based on explicit error detection. For this only for particular unreliable code sequence segments which purpose, usually a high-rate cyclic code is used exclusively can be determined using the new algorithm. Finally, we give for error detection. This concept was generalized to error- someconcludingremarksinSectionV. locating codes by Wolf and Elspas [6]. With error-locating codes, a block of received digits is regarded as subdivided II. NOTATIONS into mutually exclusive subblocks. Errors occurring within TraditionalViterbidecodingisexplainedbymeansofatrellis particular subblocks are detected at the receiver. In addition, diagram. A trellis is a directed graph. The set of nodes of the the receiver is able to determine which particular subblocks graph is decomposed into a union of disjoint containerrors.Similarly,itisdesirabletogeneralizetheconcept subsets that are called levels of the of likelihood-ratio decision. In addition to the likelihood-ratio trellis. A node of the level may be connected with a test, the receiver should be able to determine which parts node of the level by one or several branches. of a codeword (or which segments of a convolutional code Eachbranch isdirectedfromanode oflevel toanode sequence) are unreliable. of the nextlevel . We assume that the end levelshaveonly onenode,namely, and . Inthefollowing,weconsiderrate binaryterminated PaperapprovedbyR.D.Wesel,theEditorforCodingandCommunication convolutionalcodes[7]usedforcommunicationoveramemo- TheoryoftheIEEECommunicationsSociety.ManuscriptreceivedNovember 13,2003;revisedMay7,2004.ThisworkwassupportedbytheDFG(Deutsche rylesschannel.Assumeaterminatedcodesequence consistsof Forschungsgemeinschaft)underGrantBo867/9-1.Thispaperwaspresentedin -tuples , i.e., . partattheInternationalSymposiumonInformationTheory,Lausanne,Switzer- The encoder of the convolutional code is forced to return land,June2002. TheauthorsarewiththeDepartmentofTelecommunicationsandApplied to its initial state by appending dummy -tuples to the Information Theory, University of Ulm, D-89081 Ulm, Germany (e-mail: information sequence, where denotes the memory of the [email protected]; [email protected] convolutional code. The corresponding information sequence ulm.de). DigitalObjectIdentifier10.1109/TCOMM.2004.836556 is , with . For 0090-6778/04$20.00©2004IEEE TLFeBOOK 1834 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 Fig.1. Divergingpathslabelingsegmentsasunreliable. the decoder trellis of a convolutional code, a node Definition 1: Let be a given threshold. Con- corresponds to a possible encoder state. Each branch of the sider the ordered list of the most likely paths decoder trellis is labeled by a -tuple . There is suchthat a one-to-one correspondence between each codeword in with the code and a path in the trellis, such that . We denote code sequence segments and and path segments by and , for all paths respectively. We call the segments merged if we Forlevels whereallpathsin aremerged,i.e., have . Let be the received sequence. ,wedefinethedecoderoutput asreliable,oth- A Viterbi decoder selects the code sequence erwiseasunreliable. whichmaximizestheconditionalprobability ,orequiv- Example 1: Fig. 1 shows the conceptual situation with this alently, which minimizes the Viterbi metric reliabilitycriterion,wherethefigureillustratesonlypaths .Thestraightlineindicatesthemostlikelycodesequence.Fur- (1) thermore,therearethreeunreliablesegmentsindicated,where with (2) foreachsegment,thereareseveralcandidatepathswithmetric (3) differenceslessorequal .Thevalues denotethe metricdifferencesofmergingpaths. In order to evaluate (1) efficiently, we assign additional node NotethatthisreliabilitycriterionissimilartotheoneSchaub and partial path metrics and , respectively. and Modestino [11] introduced for a symbol-by-symbol era- is the metric of the local survivor entering node from level sure-declaringVA.Itisalsorelatedtothevariouslist-typegen- . The complexity of the Viterbi algorithm (VA) for a eralizationsoftheVAwhichhaveappearedintheliterature[8], terminated convolutional code is , where [12]–[15].Thatis,astraightforwardmethodforlabelingthede- denotes the overall constraint length of the code, which is the coderoutputwouldbeasfollows.First,determinetheordered minimum number of memory elements required to realize the list andmark initiallyasreliable.Considerallpathsinthe correspondingconvolutionalencoder.Foradetaileddiscussion orderedlist startingwith andendingwith .Trace of the trellis representation of convolutional codes or Viterbi thecorrespondingpathsfromnode tonode andmarkthe decoding, see, for example, [8] and [9] or the textbooks [7] decoder output as unreliable if a path is not merged with and [10]. attime .Thecomplexityofthismethodincreaseslinearly withincreasinglistsize.However,itisnotnecessarytogenerate the list of paths explicitly in order to mark the decoder output accordingtoDefinition1.Thefollowinglemmaprovidessome III. RELIABILITYOUTPUTALGORITHM insightconcerningthisissue. An error event with Viterbi decoding occurs whenever the Compare any path from with the path . estimatedpathdivergesfromthecorrectpath.Inmanypractical The two paths may diverge and re-merge many times, but the situations, it is useful to decide whether such an error event end states of the paths are the same. We say that a path is is likely or not. For this purpose, Yamamoto and Itoh [4] a one-loop path if it diverges from the path only once. introduced the following reliability test. Let be a given Furthermore, let the symbol denote path concatenation, i.e., threshold. Consider the first and second most likely paths with and . , such that . The estimated Lemma 1: In order to mark the decoder output as re- path is considered reliable if , liableorunreliableaccordingtoDefinition1,itis sufficientto otherwise unreliable. The underlying idea is that the most considerallone-looppathsintheorderedlist . unreliable decision during the VA occurred at the node , Proof: Assumethatweinitiallymark asreliable.By wherethebestandthesecond-bestpathsmerged.However,for back-tracingapath fromnode tonode ,wechange moderateorhighsignal-to-noiseratios(SNRs),anerroneously thelabelofthedecoderoutput fromreliabletounreliable estimated path will remerge with the correct path after a few ifthepathisnotmergedwith attime .Letapath , transitions.Itisthereforereasonabletodeclareonlypartsofthe havetwoormoreloops estimated sequence as unreliable if . , withrespecttothepath We therefore consider the following criterion. . Regarding ,the code-sequence TLFeBOOK IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 1835 Fig.2. Exampleofalabeledtrellis(ROAI). segmentscorrespondingto shouldbemarkedunre- tween the list entry and the node metric. Trace back liable.Weshowthatthefollowingone-looppaths allbrancheswithmetricdifferencelessthanorequal , tothenodethreshold.Labelthecorrespondingnodes all have a metric less than or equal to at level with new thresholds, which are the , from which we conclude that . In fact, differences of the node threshold at level and the , , metric differences of the particular branches. If dif- because hastheleastmetricamongallpaths.Forthepath ferent paths merge in a node at level , find the , we have , and maximum of all according thresholds at level and we obtain , ,andso on.From assignittothenodeatlevel . follows .Thus,bytracingbackthe Step5) If there existsa branch different from between one-looppaths ,weobtainthesamelabelingaswith level and whichconnectsnodeslabeledwith .Similarly,bytracingbackallone-looppathsin ,weobtain thresholds not equal to zero, mark the estimate thedesiredlabeling,accordingtoDefinition1. asunreliable,otherwise,markitreliable. It follows from Lemma 1 that we may use an ordered list Step6) If ,thencontinuewiththenextlevel.Decrement of all one-loop paths contained in instead of the byoneandgotoStep4). complete list . A recursive algorithm to construct the list of Theorem1: Algorithm1labelsthemaximum-likelihoodes- the most likely one-loop paths is given in [13]. However, the timateaccordingtoDefinition1. following nonrecursive algorithm determines these unreliable AproofofTheorem1isgivenintheAppendix.Notethatour positionswithoutexplicitlygeneratingalist. reliabilitycriterionisthesameasthatofYamamotoandItoh[4] ifwedeclarethecompletecodesequenceasunreliableoncean unreliable segment has occurred. In particular, the bounds de- Algorithm1ReliabilityOutputAlgorithm(ROA)I: rivedin[4]onthefirstevent-errorprobabilityandontheevent ofthefirsterasurearealsovalidforourscheme.Thesebounds Step1) Assignmetriczerototheinitialnode and dependonthepathenumeratorfunctionofthecodeandonthe set . metricthreshold.Forlowchannel-errorrates,theseboundscan Step2) Each node oflevel isprocessedasfollows(for- be used to determine the appropriate threshold values. How- ward pass). Calculate the partial path metrics corre- ever,forpoorchannelconditions,theseunionboundsdiverge. sponding to the branches that enter the node Inthiscase,thresholdvalueshavetobedeterminedbycomputer by ,where isthe simulations. node metric ofthisbranch’s predecessornode . Example2: Fig.2depictsalabeledtrellisfortherate Moreover, assign the node metric convolutional code with memory and generator to the node , where denotes the branch with matrix (encoder in con- the smallest partial path metric among all branches trollercanonicalform).Fourinformationsymbolsfollowedby merging in node . Store a list with all partial path twodummyzerosareencoded.Weconsidertransmissionover metrics. thebinarysymmetricchannelwithacrossoverprobability Step3) If ,thencontinuewiththenextlevel.Increment .Therefore,wehavebranchmetricvalues by one and go to Step 2). Otherwise, label the ter- and . Let the received sequence be minatingnode withathresholdvalue ,allothers .Thelistundereachnodecomprisesthemet- withthresholdvalue0,andgotoStep4). ricscorrespondingtoincomingbranches.Thebackwardpassof Step4) Eachnode oflevel withthresholdgreaterthan0 the ROA I starts at the terminating node (node in Fig. 2). isprocessedasfollows(backwardpass).Foreachel- Above each node which is reprocessed during the backward ementinthemetriclist,determinethedifferencebe- pass,thenodethresholdisdepicted.Additionally,allbranches TLFeBOOK 1836 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 Fig.3. Exampleofalabeledtrellis(ROAII). aremarkedwhichwillbetracedback.Inthisexample,thetermi- isnowindependentofthereceivedsequenceandisstillofthe natingnode isinitializedwiththreshold .Thenode sameorderastheVA. metric for node is 5.8, and the differences of the incoming Example 3: Fig. 3 depicts a labeled trellis for the ROA II, pathmetricstothenodemetricare0.0and6.9.Consequently, whereweusedthesamesettingsasinExample2.Aboveeach onlythelocalsurvivoristracedback.Notethatasaresult,all node, the node’s reliability value is shown. With ROA II, all nodescorrespondingtotheglobalsurvivorpath arelabeled branches in the trellis are traced back. The terminating node withthreshold ,because isalwaysgreaterthanorequalto is initialized with value 0.0. The differences of the incoming anyotherthreshold.Atnode ,thedifferencesare0.0and2.3. branch metrics to the terminating node’s metric are 0.0 and Asthesevaluesaresmallerthan2.5,bothincomingbranchesare 6.9.Bothincomingbranchesaretracedback,andthenodesat tracedback.Moreover,node islabeledwiththenewthreshold level are labeled with these values. As 6.9 is the .Now,twopathsaretracedbackuntiltheymerge smallest value that can be found at a node connected by the innode .Ifdifferentpathsmergeinanodeatlevel ,we last branches, which is greater than 0, this value is given to assign the maximum of all thresholds, e.g., node is marked theoutput.Ifwecontinuethebacktracing,wehavetoaccumu- with threshold 2.5. Finally, we notice that the estimated code late all the metric differences until some paths merge. In this segment corresponding to the path from node to node is case,wecontinuewiththeminimumvalue.Thefinalreliability markedunreliable,becausetherearenodeswhichdonotcorre- output is . Finally, marking the spondtothesurvivorpath thatarelabeledwiththresholds estimatedcodesegmentcorrespondingtoreliabilityvaluesless greaterthan0. than as unreliable, we obtain the same labelingas in Intheremainingpartofthissection,weconsideramodifica- Example2. tionofAlgorithm1suchthatthenewalgorithm(ROAII)deter- mines a vector of reliability values. Steps 1–3 IV. APPLICATIONSANDSIMULATIONRESULTS of ROA II are the same as with ROA I, except that we do not assumeapredeterminedthreshold .Therefore,inStep3,we The above-presented algorithm is applicable for repeat- label only the terminating node with value 0 and continue request strategies. In this case, the complete received code withStep4,whereeachnode oflevel isprocessedasfol- sequenceshouldbestored,andthereceivermaysimplyrequest lows.Foreachelementinthemetriclist,determinethediffer- retransmission of code sequence segments that are marked encebetweenthelistentryandthenodemetric.Tracebackall unreliable. The marked symbols are then substituted by (or branchesandlabelthecorrespondingnodesatlevel with combined with) the newly received symbols and the decoding the new values. The new values are the node values at level procedure restarts. The transmission terminates when finally plusthemetricdifferencesoftheparticularbranch.Ifdifferent no unreliable symbols are determined. The performance of paths merge in a node at level , find the minimum of all such repeat-request schemes may be measured in terms of thesevalues.Thenodeslabeledwithvalue0correspondtothe errorprobabilities,e.g.,thebit-errorprobability,afterthefinal survivorpath .Thereliabilityvalue isthesmallestvalue, decoding step. Furthermore, the effective transmission rate is exceptthevaluezerofromthesurvivor,whichcanbefoundata ofgreatimportance,whichisdefinedastheratioofthenumber labelfromnodes or .Thelabelingofthedecoderoutput of decoded data symbols to the number of transmitted code canbeperformedbasedonthesereliabilityvalues.Thus,Step symbols.Inthefollowing,weinvestigatesucharepeat-request 5 in Algorithm 1 can be omitted. In order to obtain an erasure scheme by means of computer simulations. We compare our labelingofthedecoderoutput,wedeclareallpositions ,where schemewiththestrategyproposedbyYamamotoandItoh[4]. thereliabilityvalue islessthanorequaltoachosenthreshold YamamotoandItoh’salgorithm(YIA)isalsobasedonViterbi , as unreliable. This second version of the reliability output decoding,hasthesameorderofdecodingcomplexity,anduses algorithmrequiresmorecalculations,becauseeachnodeisre- the same likelihood test. However, YIA does not determine processedduringthebackwardpass.However,thecomplexity the exact lengths of unreliable segments. With Yamamoto and TLFeBOOK IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 1837 Fig.5. ARQschemewithROAII. noiselessfeedbacklink.Ifactualfeedbacklinkswithnoiseare treated,thenthelisttobetransmittedoverthereturnchannelhas tobeprotectedbyforward-errorcorrection.Ifaretransmission Fig. 4. Simulation results for a memory m = 3 optimum free distance is requested, then the transmitter calculates the code bit posi- convolutionalcode(G(D)=(1+D+D +D ;1+D+D ),M =32). tionstobesentnext.Thisisdonesimultaneouslyatthereceiver, AWGNchannelwithSNRsE =N 2 f(cid:0)2;(cid:0)1;0;1dBg.Thevaluesbeside asthisinformationisneededintheMRCunit.Theschemeused each simulation point are the respective metric thresholds. For the ROA algorithm,thecodesequencesareterminatedafter1000informationbits,while for simulationcan be described as follows.First, calculatethe withYIA,thesequencesarenotterminated. numberofcodebits thataretobetransmitted.Foreachunre- liablesegment,wechooseavaluewhichisthesumofasmall Itoh’s scheme, a retransmission of the latest code tuples is constant andafraction ofthesegmentlength .As should requested if the survivor is declared unreliable, where is a beaninteger,wecalculate .Second,thepartic- preselectedconstant.Thisresultsinarateloss,ascanbeseen ular code bit positions have to be found. For this purpose, the from the following simulation results. For these simulations, receiverregistersthenumberoftransmissionsforeachcodebit, we consideredtransmission overabinary-inputadditivewhite andselects positionswhichincrementtheredundancy. Gaussian noise (BIAWGN) channel, an ideal return channel, Example4: Forsimulations,weassumeperfecterrordetec- andmaximumratiocombiningofretransmittedsymbols.Fig.4 tion. We encode each data block with a memory presents the corresponding results.1 Note that both schemes terminatedconvolutionalcode(133145175).Thenwe achievethesameBERsforagivenmetricthresholdand perform an equidistant puncturing, such that every third code value. This is due to the fact that both schemes are based on bitistransmitted(givinganinitialcoderate ). the same likelihood test. However, with the scheme based on Theparameters and for retransmissionare chosentobe 2 ROA I, the effective transmission rate, necessary to achieve and1/24,respectively.Theincrement isfixedtobe1/10.In a certain BER, is significantly improved for all considered Fig.6,weseethethroughputreachedwiththisschemeaswell SNRs. asthecapacitycurvefortheBIAWGNchannel.Theasteriskat Inthefollowing,wedescribeadifferentsystemwhereonlya position( 1.62;0.331)indicatestheSNRwhereaBERof fewbitsareretransmittedwitheachrepeatrequest.Fig.5shows isobtainedwiththeterminatedrate mothercode.The theflowchartforatransmissionofonedatablockwiththisARQ achieved performance gain of about 2.9 dB is a result of the scheme. For the initial transmission, the data block is first en- feedbackscheme. coded with a convolutional encoder. Then the code sequence ispuncturedtogetaninitialcoderate.Aftertransmissionover V. CONCLUSIONS the BIAWGN channel, all code bits of a block are maximum ratiocombined(MRC).Thenthe aggregatedcodesequenceis We have devised a new algorithm that permits a receiver to locate unreliable segments within an estimated convolutional decoded using the ROA II. Afterwards, error detection should beapplied.Theerror-detectionpartispicturedbythetermtest code sequence. The reliability criterion is based on the list of the most likely codewords which satisfy a likelihood-ratio inFig.5.Ifanerrorisdetected,alistwithunreliablesegments mustbesentback.Theinterceptionpointlabeled indicatesthe test.Inprinciple,thesamereliabilityoutputdecodingcouldbe performedusingoneofthevariouslist-typegeneralizationsof 1For YIA, we used M =32 as in the original paper [4]. Small gains the VA. However, the complexity of list-output algorithms, in can be obtained by using optimized values of M for different thresholds general,dependsonthelistsize .Itmayincreaselinearlywith and SNR values. However, the differences are moderate, compared with a fixed value of M. , where the list size itself depends on the received sequence TLFeBOOK 1838 IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 deriveanequivalentexpressionintermsoftheset .Consider thenode .Apath passingthrough canberepresentedas concatenationofaheadandatailpath: .If ,thenatleastthepathwithminimum metricamongallpathspassingthrough isanelementof . Thispathhasalsominimumheadandminimumtailpathmetric. Thus,wehave (4) wheretheminimumistakenoverallheadandtailpaths,ending orstartinginnode . With the help of the list of all branch metrics that were stored in Step 2, we can calculate the accumulated metric ofanytail path Fig.6. BIAWGNcapacityandthroughputofROA-ARQ. andincreaseswithincreasingchannelerrorrate.Theworst-case complexityofthenewalgorithmisindependentofthelistsize, andisofthesamecomplexityorderastheVA. wherewehaveusedtheabbreviation ,i.e., The reliability output may be used to request retransmis- denotes the difference between the list entry corresponding to sions. The performance of such a repeat-request scheme has the particular branch and the node metric at node . Substi- been compared with that of Yamamoto and Itoh [4] and is tutingthisinto(4),weget shown to provide a significant improvement in throughput. In addition, an adaptive-feedback error-correction scheme has been presented. The corresponding simulation results indicate that near-optimum performance for the BIAWGN channel is attainable with moderate complexity and for very short block sizes. This is true for a wide range of SNRs. However, while Notice that only the sum depends on the particular tail standard hybrid ARQ schemes require only a single bit of path. In particular, we have and information per packet over the feedback path, we have to . Therefore, canceling these send much more information in order to obtain this level of termsweobtain performance. APPENDIX PROOFOFTHEOREM1 Proof: Steps1–3oftheROAIareessentiallytheforward passofaVA.WiththeROAI,weadditionallystorealistofall Let denotethenodethresholdatnode with .Note partialpathmetricsforeachnodeinthetrellis.Thus,following thatthebackwardpassoftheROAI(Steps4–6)isessentially the branches with the smallest partial path metric in the list, aVAwhichcalculatesthenodethresholdsforeachnode starting in the terminating node , and ending in the initial . To see this, consider Step 4, where the node threshold is node ,wecanfindthebestpath .Inthefollowingproof, calculated as . Branches which weconsidertheadditionallabelingoftheoutputsequence . stem from a state need not to be considered, because AccordingtoDefinition1,wehavetoshowthatouralgorithm thecorrespondingmetricdifferenceswouldresultinathreshold labelsthedecoderoutput reliableifallpathsinthelist value less than zero. Thus, by finite induction, we have are merged.That is,during thebackward pass(Steps4–6) the .Finally,weobservethat algorithmshouldmarktheestimate atlevel reliable,iffall .ItfollowsfromStep4thatwetracebackallbranchesto paths aremergedatthislevel.Otherwise, hastobe nodesforwhichthisinequalityisfulfilled.Therefore,allnodes labeledunreliable.Insteadofdirectlyconsideringpathsthrough arefound.Moreover,iffthereexistmorethanonestate thetrellis,weconsidercertaintrellisnodes.Wethereforedefine at time or more than one state at time ,thenthereisabranch correspondingtoapath there exists a path passing through .Consequently, hastobemarkedunreliable. Thatis, isthesetofallnodesinthetrellisrepresentation Remarks: Theaboveproofisonlyvalidfortrellisrepresen- whichareconnectedbyanypath .Forconvenience,we tations which have no parallel branches from some state to resume . We will now some state . In particular, it does not hold for partial unit TLFeBOOK IEEETRANSACTIONSONCOMMUNICATIONS,VOL.52,NO.11,NOVEMBER2004 1839 memorycodes.However,thevalidityofthealgorithmcanalso [5] T.Hashimoto,“OntheerrorexponentofconvolutionallycodedARQ,” beshownforthiscase.Moreover,wewouldliketomentionthat IEEETrans.Inform.Theory,vol.40,pp.567–575,Mar.1994. [6] J.K.WolfandB.Elspas,“Error-locatingcodes—anewconceptinerror the expression in the proof corresponds to the control,”IEEETrans.Inform.Theory,vol.IT-9,pp.113–117,Apr.1963. labelingofROA-II.Inordertoobtainanerasurelabelingofthe [7] R.JohannessonandK.S.Zigangirov,FundamentalsofConvolutional decoderoutputofROA-II,wedeclareallpositions ,wherethe Coding. Piscataway,NJ:IEEEPress,1999. [8] G.D.Forney,Jr.,“Convolutionalcodes{II}:Maximumlikelihoodde- reliability value is less than or equal to a chosen threshold coding,”Inform.Control,vol.25,pp.222–266,July1974. as unreliable. Arguments similar to the proof of Theorem 1 [9] ,“TheViterbialgorithm,”Proc.IEEE,vol.61,pp.268–278,Mar. showthattheresultinglabelingsatisfiesDefinition1. 1973. [10] M. Bossert, Channel Coding for Telecommunications. New York: Wiley,1999. [11] T.SchaubandJ.W.Modestino,“AnerasuredeclaringViterbidecoder REFERENCES anditsapplicationstoconcatenatedcodingsystems,”inProc.IEEEInt. Conf.Communications,1986,pp.1612–1616. [1] T.HashimotoandM.Taguchi,“Performanceofexpliciterrordetection [12] T. Hashimoto, “A list-type reduced-constraint generalization of the andthresholddecisionindecodingwitherasures,”IEEETrans.Inform. Viterbi algorithm,” IEEE Trans. Inform. Theory, vol. IT-33, pp. Theory,vol.43,pp.1650–1655,Sept.1997. 866–876,Nov.1987. [2] T.Hashimoto,“CompositeschemeLR+Thfordecodingwitherasures [13] V.V.Zyablov,V.G.Potapov,andV.R.Sidorenko,“Maximum-likeli- and its effective equivalence to Forney’s rule,” IEEE Trans. Inform. hoodlistdecodingusingtrellises,”ProblemyPeredachiInformatsii,vol. Theory,vol.45,pp.78–93,Jan.1999. 29,pp.3–10,1993. [3] G.D.Forney,Jr.,“Exponentialerrorboundsforerasure,list,andde- [14] N.SeshadriandC.E.W.Sundberg,“ListViterbidecodingalgorithm cisionfeedbackschemes,”IEEETrans.Inform.Theory,vol.IT-14,pp. withapplications,”IEEETrans.Commun.,vol.42,pp.313–323,Feb. 206–220,Mar.1968. 1994. [4] H.YamamotoandK.Itoh,“Viterbidecodingalgorithmforconvolutional [15] C.NillandC.E.W.Sundberg,“ListandsoftsymboloutputViterbi codeswithrepeatrequest,”IEEETrans.Inform.Theory,vol.IT-26,pp. algorithms:Extensionsandcomparisions,”IEEETrans.Commun.,vol. 540–547,Sept.1980. 43,pp.277–287,Feb.1995. TLFeBOOK

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