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Biennial Symposium on Communications

Successive relaxation for decoding of LDPC codes

Hua Xiao, Sina Tolouei, Amir H. Banihashemi

Published Jun 1, 2008• 13 citations
Publisher

Abstract

The application of successive relaxation (SR) to the fixed-point problem associated with the iterative decoding of low-density parity-check (LDPC) codes is studied. We consider finite-length codes decoded by belief propagation (BP) and a well-known approximation of it, referred to as min-sum (MS), over a binary input additive white Gaussian noise (BIAWGN) channel. For both algorithms, we show that the application of SR in different domains results in different error correcting performance. In particular, the performances of SR in log-likelihood ratio (LLR) and LR domains for BP and MS are compared, and it is shown that SR-MS-LLR has the best performance. For the tested codes, SR-MS-LLR outperforms standard BP by up to about 0.5dB, offering an attractive solution in terms of performance/complexity tradeoff. We also investigate the effects of quantization on SR-MS-LLR and demonstrate that at least 6–7 bits of quantization is required to capture close to floating-point performance.

DOI
10.1109/bsc.2008.4563216

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The application of successive relaxation (SR) to the fixed-point problem associated with the iterative decoding of low-density parity-check (LDPC) codes is studied. We consider finite-length codes decoded by belief propagation (BP) and a well-known approximation of it, referred to as min-sum (MS), over a binary input additive white Gaussian noise (BIAWGN) channel. For both algorithms, we show that the application of SR in different domains results in different error correcting performance. In particular, the performances of SR in log-likelihood ratio (LLR) and LR domains for BP and MS are compared, and it is shown that SR-MS-LLR has the best performance. For the tested codes, SR-MS-LLR outperforms standard BP by up to about 0.5dB, offering an attractive solution in terms of performance/complexity tradeoff. We also investigate the effects of quantization on SR-MS-LLR and demonstrate that at least 6–7 bits of quantization is required to capture close to floating-point performance.
The application of successive relaxation (SR) to the fixed-point problem associated with the iterative decoding of low-density parity-check (LDPC) codes was proposed by Hemati et al.. The simulation results presented by Hemati et al. for the SR version of belief propagation (BP) in the likelihood ratio (LR) domain and that of min-sum (MS) in the log-likelihood ratio (LLR) domain are based on the assumption of all-zero codeword transmission. This assumption however results in erroneous error rates when SR is applied in the LR domain. Here, we correct the simulation results reported by Hemati et al. for SR-BP in the LR domain. Furthermore, we investigate the performance of SR-BP and SR-MS in the LLR and LR domains, respectively. The results for a binary input additive white Gaussian noise (BIAWGN) channel show that for both BP and MS, the application of SR in the two domains of LR and LLR results in different error correcting performance. In particular, for the tested codes, it is shown that among the four algorithms, SR-MS-LLR has the best performance. It outperforms standard MS and BP by up to about 0.6 dB and 0.3 dB, respectively, offering an attractive solution in terms of performance/complexity tradeoff.
Conventional iterative decoding with flooding or parallel schedule can be formulated as a fixed-point problem solved iteratively by a successive substitution (SS) method. In this paper, we investigate the dynamics of a continuous-time (asynchronous) analog implementation of iterative decoding, and show that it can be approximated as the application of the well-known successive relaxation (SR) method for solving the fixed-point problem. We observe that SR with the optimal relaxation factor can considerably improve the error-rate performance of iterative decoding for short low-density parity-check (LDPC) codes, compared with SS. Our simulation results for the application of SR to belief propagation (sum-product) and min-sum algorithms demonstrate improvements of up to about 0.7 dB over the standard SS for randomly constructed LDPC codes. The improvement in performance increases with the maximum number of iterations, and by accordingly reducing the relaxation factor. The asymptotic result, corresponding to an infinite maximum number of iterations and infinitesimal relaxation factor, represents the steady-state performance of analog iterative decoding. This means that under ideal circumstances, continuous-time (asynchronous) analog decoders can outperform their discrete-time (synchronous) digital counterparts by a large margin. Our results also indicate that with the assumption of a truncated Gaussian distribution for the random delays among computational modules, the error-rate performance of the analog decoder, particularly in steady state, is rather independent of the variance of the distribution. The proposed simple model for analog decoding, and the associated performance curves, can be used as an "ideal analog decoder" benchmark for performance evaluation of analog decoding circuits.
Abstract—Conventional iterative decoding with flooding or par-allel schedule can be formulated as a fixed-point problem solved iteratively by a successive substitution (SS) method. In this paper, we investigate the dynamics of a continuous-time (asynchronous) analog implementation of iterative decoding, and show that it can be approximated as the application of the well-known successive relaxation (SR) method for solving the fixed-point problem. We observe that SR with the optimal relaxation factor can consider-ably improve the error-rate performance of iterative decoding for short low-density parity-check (LDPC) codes, compared with SS. Our simulation results for the application of SR to belief prop-agation (sum-product) and min-sum algorithms demonstrate im-provements of up to about 0.7 dB over the standard SS for ran-domly constructed LDPC codes. The improvement in performance increases with the maximum number of iterations, and by accord-
Various log-likelihood-ratio-based belief-propagation (LLR-BP) decoding algorithms and their reduced-complexity derivatives for low-density parity-check (LDPC) codes are presented. Numerically accurate representations of the check-node update computation used in LLR-BP decoding are described. Furthermore, approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check-node update, or symbol-node update, or both. In particular, two main approaches for simplified check-node updates are presented that are based on the so-called min-sum approximation coupled with either a normalization term or an additive offset term. Density evolution is used to analyze the performance of these decoding algorithms, to determine the optimum values of the key parameters, and to evaluate finite quantization effects. Simulation results show that these reduced-complexity decoding algorithms for LDPC codes achieve a performance very close to that of the BP algorithm. The unified treatment of decoding techniques for LDPC codes presented here provides flexibility in selecting the appropriate scheme from performance, latency, computational-complexity, and memory-requirement perspectives.
Reduced-Complexity Decoding of LDPC Codes Various log-likelihood-ratio-based belief-propagation (LLR- BP) decoding algorithms and their reduced-complexity derivatives for low-density parity-check (LDPC) codes are presented. Numerically accurate representations of the check-node update computation used in LLR-BP decoding are described. Furthermore, approximate representation of the decoding computations are shown to achieve a reduction in complexity, by simplifying the check-node update or symbol-node update, or both. In particular, two main approaches for simplified check-node updates are presented that are based on the so-called min-sum approximation coupled with either a normalization term or an additive offset term. Density evolution is used to analyze the performance of these decoding algorithms, to determine the optimum values of the key parameters, and to evaluate finite quantization effects. Simulation results show that these reduced-complexity decoding algorithms for LDPC codes achieve a performance very close to that of the BP algorithm. The unified treatment of decoding techniques for LDPC codes presented here provides flexibility in selecting the appropriate scheme from a performance, latency, computational complexity, and memory-requirement perspective.
This paper presents a Relaxed Half-Stochastic (RHS) low-density parity-check (LDPC) decoding algorithm that uses some elements of the sum-product algorithm (SPA) in its variable nodes, but maintains the low-complexity interleaver and check node structures characteristic of stochastic decoders. The algorithm relies on the principle of successive relaxation to convert binary stochastic streams to a log-likelihood ratio (LLR) representation. Simulations of a (2048, 1723) RS-LDPC code show that the RHS algorithm can outperform 100-iterations floating-point SPA decoding. We describe approaches for low-complexity implementation of the RHS algorithm. Furthermore, we show how the stochastic nature of the belief representation can be exploited to lower the error floor.