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Systems engineering and electronics

Joint decoding algorithm of LDPC codes

Yi Fang, Jianwen Zhang, Lin Wang, 张剑文, Lin Wang

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

Designing a realizable maximum likelihood(ML) decoder for low-density parity-check(LDPC) codes is always a challenging work over an additive white Gaussian noise(AWGN) channel.Although Maxwell decoder is well known for its excellent performance over the binary erasure channel(BEC),it seems that generalizing this algorithm for other channels is difficult.This paper introduces an idea called channel transformation which could realize the conversation between two different channels.A Maxwell decoder is applied to an AWGN channel.In terms of this method,and a joint decoder-BP-Maxwell(BM) decoder is proposed which combines a belief propagation(BP) decoder and a Maxwell decoder,to reduce the gap to the ML decoder in performance.Simulation results show that the BM decoding algorithm could break most small trapping sets to accomplish a lower frame error rate(FER).Moreover it also could eliminate most of the small-scale errors compared with a BP decoder.

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Designing realizable Maximum Likelihood (ML) decoders for low-density parity-check (LDPC) codes has always been a challenging problem. Maxwell decoder is a well-known ML decoder for LDPC codes over a binary erasure channel (BEC). Based on the concept used in the Maxwell decoder, it may be possible to design near-ML decoders for LDPC codes over an additive white Gaussian noise (AWGN) channel. In this paper, we introduce a novel idea called channel transformation, which will be used as a bridge linking a BEC channel and an AWGN channel. With the transformation, we further propose a near-ML decoder for LDPC codes sent over an AWGN channel. The proposed near-ML decoder, combining a Belief Propagation (BP) decoder and a Maxwell decoder, is called the BP-Maxwell (BM) decoder. Simulation results show that compared with a BP decoder, the BM decoding algorithm can accomplish better frame error rates (FERs) by eliminating many of the error frames with a smaller percentage of error bits. Thus, the BM decoder is a step closer to the ML decoder compared with the BP decoder.

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This paper investigates decoding of low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximum-likelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graph-theoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finite-length codes.

IEEE Transactions on Information Theory•Mar 1, 2004•147 citations
Designing a realizable maximum likelihood(ML) decoder for low-density parity-check(LDPC) codes is always a challenging work over an additive white Gaussian noise(AWGN) channel.Although Maxwell decoder is well known for its excellent performance over the binary erasure channel(BEC),it seems that generalizing this algorithm for other channels is difficult.This paper introduces an idea called channel transformation which could realize the conversation between two different channels.A Maxwell decoder is applied to an AWGN channel.In terms of this method,and a joint decoder-BP-Maxwell(BM) decoder is proposed which combines a belief propagation(BP) decoder and a Maxwell decoder,to reduce the gap to the ML decoder in performance.Simulation results show that the BM decoding algorithm could break most small trapping sets to accomplish a lower frame error rate(FER).Moreover it also could eliminate most of the small-scale errors compared with a BP decoder.
Designing realizable Maximum Likelihood (ML) decoders for low-density parity-check (LDPC) codes has always been a challenging problem. Maxwell decoder is a well-known ML decoder for LDPC codes over a binary erasure channel (BEC). Based on the concept used in the Maxwell decoder, it may be possible to design near-ML decoders for LDPC codes over an additive white Gaussian noise (AWGN) channel. In this paper, we introduce a novel idea called channel transformation, which will be used as a bridge linking a BEC channel and an AWGN channel. With the transformation, we further propose a near-ML decoder for LDPC codes sent over an AWGN channel. The proposed near-ML decoder, combining a Belief Propagation (BP) decoder and a Maxwell decoder, is called the BP-Maxwell (BM) decoder. Simulation results show that compared with a BP decoder, the BM decoding algorithm can accomplish better frame error rates (FERs) by eliminating many of the error frames with a smaller percentage of error bits. Thus, the BM decoder is a step closer to the ML decoder compared with the BP decoder.
A new method is presented for low-complexity near-maximum-likelihood (ML) decoding of low-density parity-check (LDPC) codes over the additive white Gaussian noise channel. The proposed method termed belief-propagation-list erasure decoding (BP-LED) is based on erasing carefully chosen unreliable bits performed in case of BP decoding failure. A strategy of introducing erasures into the received vector and a new erasure decoding algorithm are proposed. The new erasure decoding algorithm, called list erasure decoding, combines ML decoding over the BEC with list decoding applied if the ML decoder fails to find a unique solution. The asymptotic exponent of the average list size for random regular LDPC codes from the Gallager ensemble is analyzed. Furthermore, a few examples of irregular quasi-cyclic LDPC as well as randomly constructed regular LDPC codes of short and moderate lengths are studied by simulations and their performance is compared to the tightened upper bound on the LDPC ensemble-average performance and the upper bound on the average performance of random linear codes under ML decoding. A comparison of the BP decoding and BP-LED performance of the WiMAX standard codes and performance of the near-ML BEAST decoding are presented. The new algorithm is applied to decoding a short nonbinary (NB) LDPC code over extensions of the binary Galois field. The obtained simulation results are compared to the tightened upper bound on the ensemble-average performance of the binary image of regular NB LDPC codes.
Recently one decoding algorithm called as beyond belief propagation (BP) algorithm for LDPC Codes is derived using the knowledge of trapping set and different from standard BP algorithm, whose performance outperforms that of the latter over binary symmetric channel (BSC). However, the former is not suitable for additive white Gaussian noise (AWGN) channel. In this paper how to improve bit error rate (BER) performance of Beyond BP Algorithm is investigated over AWGN channel. It is found that this algorithm can be modified through setting appropriate penalty factor and obtains better performance than BP one over AWGN channel. The simulation results show dependently on choosing penalty factor modified Beyond BP Algorithm outperforms BP one over AWGN channel in high signal noise rate (SNR) region or almost whole SNR one and there are some differences for different code rates. Concretely, for medium or low code rate proposed algorithm can obtain more performance gain relative to high one when short frame is used.
The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density parity-check (LDPC) codes from two random ensembles is studied. A tightened union-type upper bound on the ML decoding error probability based on the precise coefficients of the average weight spectrum is presented. For LDPC codes from the Gallager ensemble and the Richardson-Urbanke ensemble, new upper bounds on the ML decoding performance based on computing the rank of submatrices of the code parity-check matrix are derived. A new lower bound on the ML decoding threshold followed from the latter error probability bound is obtained. An improved lower bound on the error probability for codes with a known estimate on the minimum distance is presented as well. A new low-complexity near-ML decoding algorithm for quasi-cyclic LDPC codes is proposed and simulated. Its performance is compared to the simulated belief propagation and ML decoding performance and simulated performance of the best known improved iterative decoding techniques, as well as, with the derived upper bounds on the ML decoding performance and with decoding thresholds obtained by the density evolution technique.
Summary The simplicity of decoding is one of the most important characteristics of the low density parity check (LDPC) codes. Belief propagation (BP) decoding algorithm is a well‐known decoding algorithm for LDPC codes. Most LDPC codes with long lengths have short cycles in their Tanner graphs, which reduce the performance of the BP algorithm. In this paper, we present 2 methods to improve the BP decoding algorithm for LDPC codes. In these methods, the calculation of the variable nodes is controlled by using “multiplicative correction factor” and “additive correction factor.” These factors are obtained for 2 separate channels, namely additive white Gaussian noise (AWGN) and binary symmetric channel (BSC), as 2 functions of code and channel parameters. Moreover, we use the BP‐based method in the calculation of the check nodes, which reduces the required resources. Simulation results show the proposed algorithm has better performance and lower decoding error as compared to BP and similar methods like normalized‐BP and offset‐BP algorithms.
This paper investigates decoding of low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximum-likelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graph-theoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finite-length codes.