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IEEE Transactions on Communications

Dynamics and performance analysis of analog iterative decoding for low-density parity-check (LDPC) codes

Saied Hemati, Amir H. Banihashemi

Published Jan 1, 2006• 71 citations
Publisher

Abstract

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.

DOI
10.1109/tcomm.2005.861668

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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.
Conventional iterative decoding with flooding or parallel schedule can be formulated as a fixed-point problem solved iteratively by the successive substitution (SS) method. In this work, we investigate the dynamics of continuous-time asynchronous analog implementation of iterative decoding, and show that it can be approximated as the application of the well-known successive overrelaxation (SOR) method for solving the fixed-point problem. We observe that SOR with the optimal relaxation factor can considerably improve the performance of iterative decoding for short low-density parity-check (LDPC) codes compared to SS. Our simulation results for the application of SOR 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 infinite maximum number of iterations and infinitesimal relaxation factor represents the performance of analog continuous-time asynchronous 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. The proposed 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-
Iterative decoding with flooding schedule can be formulated as a fixed-point problem solved iteratively by successive substitution (88) method. In this work, we model continuous-time analog (asynchronous) iterative decoding by a first-order differential equation, and show that it can be approximated as the application of the well-known successive over relaxation (SOR) method for solving the fixed-point problem. Simulation results for belief propagation (sum-product) and min-sum algorithms confirm that SOR, which is in general superior to the simpler 88 method, can considerably improve the performance of iterative decoding for short codes. The improvement in performance increases with the maximum number of iterations and by reducing the step size in SOR, and the asymptotic result, corresponding to infinite maximum number of iterations and infinitesimal step size represents the performance of continuous-time analog iterative decoding. This means that under ideal circumstances continuous-time analog decoders can outperform their discrete-time digital counterparts by a large margin. Moreover, the results obtained by the proposed model are surprisingly close to the results of circuit simulation of a min-sum analog decoder presented in [S. Hemati et al., 2003]. Our work also suggests a general framework for improving iterative decoding algorithms on graphs with cycles, even for synchronous digital implementations.
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.
Density evolution is often used to determine the performance of an ensemble of low-density parity-check (LDPC) codes under iterative message-passing algorithms. Conventional density evolution techniques over memoryless channels are based on the independence assumption amongst all the processed messages in variable and check nodes. This assumption is valid for many algorithms such as standard belief propagation (BP) and min-sum (MS) algorithms. However, there are other important iterative algorithms such as successive relaxation (SR) versions of BP and MS, and differential decoding with binary message passing (DD-BMP) algorithm of Mobini et. al., for which this assumption is not valid. The dependence created among messages for these algorithms is due to the introduction of memory in the iterative algorithm. In this work, we propose a model for iterative decoding algorithms with memory which covers SR and DD-BMP algorithms as special cases. Based on this model, we derive a Bayesian network for iterative algorithms with memory over memoryless channels and use this representation to analyze the algorithms using density evolution. The density evolution technique is developed based on truncating the memory of the decoding process and approximating it with a finite order Markov process, and can be implemented efficiently. As an example, we apply our technique to analyze the performance of DD-BMP on regular LDPC code ensembles, and make a number of interesting observations with regard to the performance/complexity trade off of DD-BMP in comparison with BP and MS algorithms.