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IEEE International Symposium on Information Theory (ISIT)

On the capacity of low-density parity-check codes

Sae-Young Chung, G. David Forney

Published Nov 13, 2002• 5 citations
Publisher

Abstract

We demonstrate that we can translate threshold values of low-density parity-check (LDPC) codes between channels accurately using a simple mapping. We develop some models of density evolution from this observation, namely erasure-channel and reciprocal-channel approximations. The reciprocal-channel approximation, based on dualizing LDPC codes, provides a very accurate model of density evolution for the additive-white Gaussian noise (AWGN) channel.

DOI
10.1109/isit.2001.936183

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We demonstrate that we can translate threshold values of low-density parity-check (LDPC) codes between channels accurately using a simple mapping. We develop some models of density evolution from this observation, namely erasure-channel and reciprocal-channel approximations. The reciprocal-channel approximation, based on dualizing LDPC codes, provides a very accurate model of density evolution for the additive-white Gaussian noise (AWGN) channel.
Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinite-dimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a one-dimensional problem of updating the means of the Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is 10. We show that by using the Gaussian approximation, we can visualize the sum-product decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization.
Focus of this letter is the oldest class of codes that can approach the Shannon limit quite closely, i.e., low-density parity-check (LDPC) codes, and two mathematical tools that can make their design an easier job under appropriate assumptions. In particular, we present a simple algorithmic method to estimate the threshold for regular and irregular LDPC codes on memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels with sum-product decoding, and, to determine how close are the obtained thresholds to the theoretical maximum, i.e., to the Shannon limit, we give a simple and invertible expression of the AWGN channel capacity in the binary input-soft output case. For these codes, the thresholds are defined as the maximum noise level, such that an arbitrarily small bit-error probability can be achieved as the block length tends to infinity. We assume a Gaussian approximation for message densities under density evolution, a widely used simplification of the decoding algorithm.
In this paper, we study the density evolution analysis of spatially coupled low-density parity-check (SC-LDPC) codes over binary input additive white Gaussian noise (BIAWGN) channels under the belief propagation (BP) decoding algorithm. Using reciprocal channel approximation and Gaussian approximation, we propose averaging techniques for the density evolution of SC-LDPC codes over BIAWGN channels. We show that the proposed techniques can closely predict the decoding threshold while offering reduced complexity compared to the existing multi-edge-type density evolution.
Introduction Sequences of irregular low-density parity-check (LDPC) codes that achieve the capacity of the binary erasure channel (BEC) under iterative decoding were first constructed by Luby, et al. in [11]. This was followed by the work of Chung, et al., which provided evidence suggesting that sequences of iteratively decoded LDPC codes can also achieve the channel capacity of the binary-input additive white Gaussian noise (AWGN) channel [3]. Since then, density evolution (DE) [13] has been used to optimize irregular LDPC codes for a variety of memoryless channels (e.g., [6]), and the results suggest, for each channel, that sequences of iteratively decoded LDPC codes can indeed achieve the channel capacity. In fact, the discovery of a channel whose capacity cannot be approached by LDPC codes would be more surprising than a proof that iteratively decoded LDPC codes can achieve the capacity of any binary-input symmetric channel (BISC). The idea of decoding a code transmitted over a c
This paper considers density evolution for low-density parity-check (LDPC) and multi-edge type LDPC (MET-LDPC) codes over the binary input additive white Gaussian noise channel. We first analyze three single-parameter Gaussian approximations for density evolution and discuss their accuracy under several conditions, namely, at low rates, with punctured and degree-one variable nodes. We observe that the assumption of symmetric Gaussian distribution for the density-evolution messages is not accurate in the early decoding iterations, particularly at low rates and with punctured variable nodes. Thus, single-parameter Gaussian approximation methods produce very poor results in these cases. Based on these observations, we then introduce a new density evolution approximation algorithm for LDPC and MET-LDPC codes. Our method is a combination of full density evolution and a single-parameter Gaussian approximation, where we assume a symmetric Gaussian distribution only after density-evolution messages closely follow a symmetric Gaussian distribution. Our method significantly improves the accuracy of the code threshold estimation. Additionally, the proposed method significantly reduces the computational time of evaluating the code threshold compared with full density evolution thereby making it more suitable for code design.