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Asilomar Conference on Signals, Systems and Computers (ACSSC)

Noisy belief propagation decoder

Chu-Hsiang Huang, Yao Li, Lara Dolecek

Published Nov 1, 2014• 3 citations
Publisher

Abstract

This paper analyzes the fundamental performance limits of an LDPC Belief Propagation (BP) decoder implemented on noisy hardware and proposes a robust decoder implementation to improve the resilience to hardware errors. Assuming that the effects of hardware noise in various computational units, i.e., variable nodes and check nodes, can be approximated by Gaussian noise, we develop a Gaussian approximate density evolution for noisy BP decoders. By the Gaussian approximate density evolution, we find that zero residual error rate is achievable for noisy BP decoders as long as the message representations are of arbitrarily high precision. Noisy BP decoding thresholds are then derived for various regular LDPC codes. These decoding thresholds determine maximum allowable communication and computation noise for reliable communication. Next, we propose an averaging BP decoder implementation by averaging over the messages in all up-to-date iterations to cancel out the computation noise. Simulation results demonstrate that on noisy hardware, the averaging BP decoder significantly reduces the residual error rates when compared to the nominal BP decoder.

DOI
10.1109/acssc.2014.7094847

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This paper analyzes the fundamental performance limits of an LDPC Belief Propagation (BP) decoder implemented on noisy hardware and proposes a robust decoder implementation to improve the resilience to hardware errors. Assuming that the effects of hardware noise in various computational units, i.e., variable nodes and check nodes, can be approximated by Gaussian noise, we develop a Gaussian approximate density evolution for noisy BP decoders. By the Gaussian approximate density evolution, we find that zero residual error rate is achievable for noisy BP decoders as long as the message representations are of arbitrarily high precision. Noisy BP decoding thresholds are then derived for various regular LDPC codes. These decoding thresholds determine maximum allowable communication and computation noise for reliable communication. Next, we propose an averaging BP decoder implementation by averaging over the messages in all up-to-date iterations to cancel out the computation noise. Simulation results demonstrate that on noisy hardware, the averaging BP decoder significantly reduces the residual error rates when compared to the nominal BP decoder.
In this paper, we investigate the performance of the belief propagation (BP) algorithm for decoding low-density parity-check codes over the additive white Gaussian noise channel when there is an incorrect estimate of the channel signal-to-noise ratio (SNR) (referred to as "SNR mismatch") at the decoder. At the extremes for over- and underestimation of SNR, the performance of BP tends to that of min-sum algorithm and the channel bit-error rate, respectively. Our results for regular codes indicate that the sensitivity to mismatch increases by increasing the variable-node degree and by decreasing the check-node degree. The effect of variable-node degree, however, appears to be more profound, such that at a given rate, the codes with the smallest variable and check degrees are more robust against SNR mismatch. For irregular codes, by comparing the thresholds of a few ensembles, we demonstrate that the ensemble which performs better in the absence of mismatch can perform worse in the presence of it. To obtain our asymptotic results, we propose a computationally efficient method based on the Gaussian approximation of density evolution in the presence of SNR mismatch. We also show that the asymptotic results are consistent with simulation results for codes with finite block lengths
Departing from traditional communication theory where decoding algorithms are assumed to perform without error, a system where noise perturbs both computational devices and communication channels is considered here. This paper studies limits in processing noisy signals with noisy circuits by investigating the effect of noise on standard iterative decoders for low-density parity-check (LDPC) codes. Concentration of decoding performance around its average is shown to hold when noise is introduced into message-passing and local computation. Density evolution equations for simple faulty iterative decoders are derived. In one model, computing nonlinear estimation thresholds shows that performance degrades smoothly as decoder noise increases, but arbitrarily small probability of error is not achievable. Probability of error may be driven to zero in another system model; the decoding threshold again decreases smoothly with decoder noise. As an application of the methods developed, an achievability result for reliable memory systems constructed from unreliable components is provided.
Departing from traditional communication theory where decoding algorithms are assumed to perform without error, a system where noise perturbs both computational devices and communication channels is considered here. This paper studies limits in processing noisy signals with noisy circuits by investigating the effect of noise on standard iterative decoders for low-density parity-check codes. Concentration of decoding performance around its average is shown to hold when noise is introduced into message-passing and local computation. Density evolution equations for simple faulty iterative decoders are derived. In one model, computing nonlinear estimation thresholds shows that performance degrades smoothly as decoder noise increases, but arbitrarily small probability of error is not achievable. Probability of error may be driven to zero in another system model; the decoding threshold again decreases smoothly with decoder noise. As an application of the methods developed, an achievability result for reliable memory systems constructed from unreliable components is provided.
Summary form only given. The capacity of the message-passing decoder for LDPC codes can be computed by the density evolution algorithm by iteratively computing message densities. The infinite-dimensional problem of iteratively calculating the message densities in the case of the binary input AWGN channel and the belief propagation decoder can be simplified to a one-dimensional problem by using a Gaussian approximation. By assuming Gaussian densities the density evolution simplifies to updating means of Gaussian densities. An improved Gaussian approximation algorithm is suggested for computing the capacity of the BP decoder based on the mutual information measure. An analytical method for computing the mutual information of a transmitted bit with the message corresponding to it is proposed. Using this method we can approximate the non-Gaussian message by a Gaussian message that has the same mutual information with the transmitted bit. Computationally, the algorithm is similar to the Gaussian approximation algorithm proposed in Chung (2001). For various regular LDPC codes that were examined, the algorithm computed threshold values within 0.01dB or less from the exact threshold which is an improvement over the Gaussian approximation. Furthermore, additional insight on the convergence process can be gained from EXIT charts that can be derived from the algorithm. This can assist in designing better irregular LDPC codes.
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.