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Wave Electronics and its Application in Information and Telecommunication Systems (WECONF)

About Some Irregular Degree Distributions of LDPC Codes in Two-State Channels

Andrei A. Ovchinnikov, Anna A. Fominykh

Published May 31, 2021• 2 citations
Publisher

Abstract

The paper examines the decoding of low-density parity-check codes in channels with memory. Exact degree distributions optimized for a set of parameters of Gilbert and Gilbert-Elliott channels are given. The decoding performance of low-density parity-check codes constructed using obtained degree distributions are presented under belief propagation decoding over Gilbert-Elliott channel.

DOI
10.1109/weconf51603.2021.9470627

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The paper examines the decoding of low-density parity-check codes in channels with memory. Exact degree distributions optimized for a set of parameters of Gilbert and Gilbert-Elliott channels are given. The decoding performance of low-density parity-check codes constructed using obtained degree distributions are presented under belief propagation decoding over Gilbert-Elliott channel.
The problem of low-density parity-check (LDPC) decoding in channels with memory has been attracting increasing attention in the literature. In this thesis, we use LDPC codes in an estimation-decoding scheme for the Gilbert-Elliott (GE) channel and more general Markov-modulated channels. The major accomplishments of this thesis include both analysis and design components. To analyze our estimation-decoding scheme, we derive density evolution for the GE channel, which has previously been used largely in memoryless channels. Furthermore, we develop techniques that use density evolution results to more efficiently characterize the space of Markov-modulated parameters. We begin by applying a characterization to the GE channel, following which we generalize this characterization into a partial ordering of Markov-modulated channels in terms of probability of symbol error. We also consider the design problem of developing LDPC degree sequences that are optimized for the GE channel. We obtain a novel design tool that approximates density evolution for our estimation-decoding algorithm, and present degree sequences that represent the best known codes in the GE channel. We also present a method of generalizing this tool to Markov-modulated channels, and give some of the first optimized degree sequences ever obtained for these channels.
This paper presents a reduced-complexity approximate density evolution (DE) scheme for low-density parity-check (LDPC) codes in channels with memory in the form of a hidden Markov chain. This approximation is used to design degree sequences representing some of the best known LDPC code ensembles for the Gilbert-Elliott channel, and example optimizations are also given for other Markov channels. The problem of approximating the channel estimation is addressed by obtaining a specially constructed message-passing schedule in which the channel messages all approach their stable densities. It is shown that this new schedule is much easier to approximate than the standard schedule, but has the same ultimate performance in the limits of long block length and many decoding iterations. This result is extended to show that all message-passing schedules that satisfy mild conditions will have the same threshold under density evolution
It has been shown using density evolution (DE) that joint estimation-decoding strategies have excellent performance for regular lowdensity parity-check (LDPC) codes in the GilbertElliott (GE) channel. However, DE is computationally intensive, and is therefore not suitable as a tool in searching for good degree sequences for irregular LDPC codes. In this paper, we present a reduced-complexity approximate DE scheme that is particularly suitable for the GE channel, based on the expected value approximation from [7]. The proposed technique takes advantage of a specially constructed decoder for which the ultimate performance is the same as the standard decoder, but for which DE is easier to approximate. We show that this straightforward approximation allows the channel messages to be precalculated, resulting in computational complexity comparable to the memoryless case.
This paper presents the design of block lengths for irregular low-density parity-check (LDPC) codes based on the maximum variable degree d v,max . To design a block length, the performance degradation of belief-propagation (BP) decoding performance from upper bounds on the maximum likelihood (ML) decoding performance is used as an important factor. Since for large block lengths, the performance of irregular LDPC codes is very close to the Shannon limit, we focus on moderate block lengths (5×10²≤N≤4×10³). Given degree distributions, the purpose of our paper is to find proper block lengths based on the maximum variable degree d v,max . We also present some simulation results which show how a block length can be optimized.
This paper investigates the performance of irregular low-density parity-check (LDPC) codes on memoryless BI-AWGN (Binary Input - Additive White Gaussian Noise) channels, with sum-product decoding. Objective of this work is to study the relationship between an LDPC code performance and some parameters specifying the code itself, such as the coefficients of its degree distributions. In fact, these coefficients where shown in Di et al.'s 2006 paper to determine the growth rate of the minimum distance of an LDPC code, which can be only sublinear in the block length in some well defined conditions of the degree distribution pair.
Noise threshold of Low Density Parity Check (LDPC) codes ensemble could be calculated by density evolution of the decoding messages. According to the threshold, performance of the decoding algorithm could be evaluated and degree distribution pairs of irregular LDPC codes also might be optimized. In this paper, for regular LDPC codes on Rice channel, Discrete Density Evolution(DDE) for BP,BP-based and offset BP-based decoding algorithm were compared on different quantization level and step. Furthermore, degree distribution pairs of the irregular LDPC codes were optimized on Rice channel, which was based on DDE for offset BP-based decoding algorithm, the corresponding threshold is also calculated. Finally, the Probability Mass Function (PMF) evolution curve of optimized irregular LDPC codes with code rate 1/2 on Rice channel is given.