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Convergence Analysis

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Convergence Analysis of a New Classof Flexible Hybrid Concatenated CodesBin Zhao and Matthew C. ValentiLane Dept. of Comp. Sci. & Elect. Eng.West Virginia UniversityMorgantown, WV 26506-6109email: [email protected], [email protected]– This paper focuses on a new class of flexiblehybrid concatenated codes which are able to achieve anydesired bit error rate performance between that of a par-allel and serial concatenated convolutional code. The maincontribution of this paper is a convergence analysis of thedecoder, which is facilitated by an alternative representationofthecodestructureandanextrinsicinformation(EXIT)chart. In addition to giving some insight into the behavioranddesignofthisparticulartypeofhybridcode,thispa-p er also provides n ew insight into the relationship betweenparallel and serial concatenated codes as well as the hybridconcatenated code prop osed by Divsalar and Pollara.Keywor ds– Turbo codes, hybrid codes, serial concatenatedcodes, parallel concatenated codes, EXIT chart, iterativedecoding convergence analysis.I. IntroductionIn [1], [2], a new class of hybrid concatenated codes wereproposed and shown to split the bit error performance ofparallel concatenated convolutional codes (PCCCs) and se-rial concatenated convolutional codes (SCCCs) in both thewaterfall and error floor regions. The hybrid code was gen-erated by puncturing the output of a slightly m odified par-ent SCCC encoder comprised of a pair of recursive system-atic convolutional (RSC) encoders. The main modificationto the SCCC encoder involved an interleaver design whichessentially interleaves the systematic and parity bits of theouter code independently. This constraint on the inter-leaver ensures that the output of the inner encoder canbe split into four distinct fields, the global systematic bits(systematic outer / systematic inner (So/Si)), the “single”parity due to only the outer encoder (parity outer / system-atic inner (Po/Si)), the single parity due to only the innerencoder ( systema tic outer / parity inner (So/Pi)), and the“double” parity bits (parity outer/parity inner (Po/Pi)).It is the presence of the double parity bits that differen-tiates this SCCC from a PCCC. In particular, puncturingall the double parity bits generates a PCCC code whilekeeping all of them maintains a SCCC code. Puncturinga fraction of the double parity bits creates a hybrid codewhose performance is between that of a PCCC and SCCCcode. When half of the bits in both Po/Piand So/Pifieldsare deleted, the code operates as a punctured (rate 1/3)SCCC, but as more Po/Pibits are punctured (and moreThis work w as supported by the Office of Naval Research undergrant N00014-00-0655.0-7803-7339-1/02/$17.00c°2002 IEEESo/Pibits maintained) the code behaves more like a PCCC.More specifically, as more double parity bits are punctured,the “waterfall” or “turbo cliff” occurs at a lower SNR whilethe error floor is raised. When all the double parity bitsare punctured (and all single parity bits maintained) thecode performance is identical to that of an unpuncturedrate 1/3 PCCC.The benefit of this approach is that it gives codes witherror performance between the two extremes of PCCC andSCCC codes. Since the hybrid code is created by simplypuncturing the output of a SCCC encoder, no additionalhardware is required to implement it. In fact, because aPCCC code can be generated by appropriately puncturinga standard SCCC code (with appropriate interleaver), itmight be wise for IC designers and manufacturers to focuson SCCC products rather than on PCCCs. However, itshould be noted that although a SCCC decoder can beused to decode a PCCC code, th e complexity of the SCCCdecoder is 1.5 times tha t of the PCCC decoder since theinner code is clocked at twice the rate of the outer code.Furthermore, the proposed code structure lends itself toapplicationinhybridFEC/ARQsytemes[1].A hybrid concatenated coding scheme was previouslyproposed in [3]. This technique, termed hybrid concate-nated convolutional codes (HCCCs), involves a convolu-tional code concatenat ed in parallel with a SCCC. A unionbounding analysis showed that HCCCs perform better thanboth SCCCs and PCCCs in both the waterfall region anderror floor. While our proposed code does not simultane-ously perform better than both PCCCs and SCCCs, itscomplexity is much lower. In fact, we show later t hat theproposed code is a subclass of the HCCC of [3].II. A Class of Flexible Hybrid CodesTheproposedhybridencoderisshowninFig. 1andiscomposed of two major parts: a rate 1/4 parent SCCC anda puncturing unit. The parent SCCC has identical innerand outer encoders separated by a structured interleaver.Both constituent codes are recursive systematic convolu-tional (RSC) codes with the same generator (35,23). Thestructured interleaver is designed such that it will map thesystematic bits into the first half of the interleaved frameand parity bits into the second half. This is equivalent tointerleaving the systematic and parity bits independentlyand then putting t hem in cascade so that all the interleavedsystematic bits are introduced to the input of the innerOuter codeINT(P)INT(S)Inner codeinputoutputstructured interleaverPunctureFig. 1. Proposed hybrid concatenated encoder.encoder before any of the interleaved parity bits. Sincethe systematic and parity bits are interleaved separately,there are actually two sub interleaving units (INT(S) andINT(P)) within this structured interleaver, each of whichis implemented as an S-random interleaver.Because of the interleaver design, the output codewordcan be divided into t he f our fields shown in Fig. 7. Theglobal systematic output (So/Si)iscomprisedofthesys-tematic bits at the output of the inner encoder that cor-respond to the interleaved systematic output of the outerencoder (i.e. the first half of the inner encoder’s system-atic output). There are two single parity fields, So/Pi andPo/Si. The So/Pi field is composed of the parity bits at theoutput of the inner encoder that are created using the inter-leaved systematic output of the outer encoder (i.e. the firsthalf of the inner encoder’s parity output), while the Po/Sifield is composed of the systematic bits at the output of theinner encoder due to the interleaved parity output of theouter encoder (i.e. the second half of the inner encoder’ssystematic output). The double parity field Po/Pi is theparity output of the inner encoder which is generated usingthe interleaved


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