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Bounds on the error probability ofml decoding for block and turbo-block codes

Bornes de la probabilité d’erreur pour des codes en blocs et des turbo codes en blocs avec décodage à maximum de vraisemblance

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Abstract

The performance of either structured or random turbo-block codes and binary, systematic block codes operating over the additive white Gaussian noise (Awgn) channel, is assessed by upper bounds on the error probalities of maximum likelihood (Ml) decoding. These bounds on the block and bit error probability which depend respectively on the distance spectrum and the input-output weight enumeration function (Iowef) of these codes, are compared, for a variety of cases, to simulated performance of iterative decoding and also to some reported simulated lower bounds on the performance ofMl decoders. The comparisons facilitate to assess the efficiency of iterative decoding (as compared to the optimalMl decoding rule) on one hand and the tightness of the examined upper bounds on the other. We focus here on uniformly interleaved and parallel concatenated turbo-Hamming codes, and to that end theIowefs of Hamming and turbo-Hamming codes are calculated by an efficient algorithm. The usefulness of the bounds is demonstrated for uniformly interleaved turbo-Hamming codes at rates exceeding the cut-off rate, where the results are compared to the simulated performance of iteratively decoded turbo-Hamming codes with structured and statistical interleavers. We consider also the ensemble performance of ‘repeat and accumulate’ (Ka) codes, a family of serially concatenated turbo-block codes, introduced by Divsalar, Jin and McEliece. Although, the outer and inner codes possess a very simple structure: a repetitive and a differential encoder respectively, our upper bounds indicate impressive performance at rates considerably beyond the cut-off rate. This is also evidenced in literature by computer simulations of the performance of iteratively decodedRa codes with a particular structured interleaver.

Résumé

Les performances sur canal gaussien de turbo codes en bloc binaires systématiques, avec entrelacement structuré ou aléatoire sont estimées, à l’aide de bornes supérieures sur les probabilités d’erreurs, pour un décodage à maximum de vraisemblance. Ces bornes relatives à des blocs ou à des bits dépendent respectivemet du spectre des distances et de la fonction d’énumération des poids d’entrée et de sortie des codes. Ces bornes sont comparées, pour différents cas de figure, aux performances du décodage itératif obtenues par simulation et à certaines bornes inférieures présentées dans la lit-térature. Ces comparaisons permettent de juger de l’efficacité du décodage itératif vis-à-vis du décodage à maximum de vraisemblance, et d’évaluer la pertinence des bornes supérieures d’erreurs proposées dans cet article. Une attention particulière est portée aux codes turbo-Hamming avec concaténation parallèle et entrelacement uniforme (ou statistique). Les fonctions d’énumeration de poids pour les codes de Hamming et les codes turbo-Hamming sont calculées à l’aide d’algorithmes originaux. L’utilité des bornes d’erreurs proposées pour les codes turbo-Hamming est démontrée, pour des débits qui vont bien au delà de la coupure, par une bonne concordance avec les résultats de simulation relatifs au décodage itératif, et cela pour différents types d’entrelacement. Nous considérons également le cas particulier des codesRa (répétition et accumulation) et d’une famille de codes en bloc à concaténation série et entrelacement uniforme introduite par Divsalar, Jin et McEliece. Les codes extérieur et intérieur y ont une structure très simple: un codeur à répétition et un codeur différentiel respectivement. Malgré cela les performances que l’on peut observer à partir des bornes supérieures de probabilités d’erreurs ou des simulations relatives au décodage itératif sont impressionnantes, y compris au delà du débit de coupure.

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  1. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Igal Sason & Shlomo Shamai (Shitz)

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  1. Igal Sason
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  2. Shlomo Shamai (Shitz)
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The research was supported by the Neaman Institute with the consortium of «software radio»

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Sason, I., Shamai (Shitz), S. Bounds on the error probability ofml decoding for block and turbo-block codes. Ann. Télécommun. 54, 183–200 (1999). https://doi.org/10.1007/BF02998579

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