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Characterizing the Gilbert-Elliott Parameter Space under LDPC Decoding

Andrew W. Eckford, Frank R. Kschischang, Subbarayan Pasupathy

Published Jan 1, 2002• 1 citation
OpenAlex

Abstract

There has been much recent interest in analyzing the performance of LDPC codes in channels with memory.Density evolution (DE) may be used to analyze estimation-decoding algorithms employing the Sum-Product Algorithm for LDPC codes in Gilbert-Elliott (GE) channels.In this paper, we provide theoretical results which mitigate the complexity of characterizing the 4-dimensional GE parameter space using DE.For each point in the parameter space which is shown by DE to converge to P e → 0, we show that a region of convergence is induced, within which all points also converge to P e → 0. Conversely, for points found not to converge, we show that a region of non-convergence is induced.Using these results, the GE parameter space is partitioned into a region of convergence, a region of nonconvergence, and an uncertain region, which guides the selection of new points to test using DE.

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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.

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In this paper, we investigate the design of low-density parity-check (LDPC) codes for the Gilbert-Elliott (GE) channel. Recently, Eckford et al. proposed a design method of irregular LDPC codes using approximate density-evolution (DE) for Markov channels. In the design method proposed by Eckford et al., the probability density function (PDF) of the messages from variable nodes to check nodes is approximated by the Gaussian distribution. In this paper, we first show the method to obtain the accurate PDF of the messages from variable nodes to check nodes by utilizing two DE steps for the Gaussian distribution. We call this method the iterative density approximation (IDA). Using this method, we can design the good LDPC codes. Next, we propose an efficient design method of irregular LDPC codes by using Beta approximation to the PDF of the channel state probability for the GE channel. Consequently, we show that the complexity to calculate PDFs of the channel messages is considerably reduced though the rates of LDPC codes obtained by using the proposed approximation are almost the same as that of the IDA method.

International Symposium on Information Theory and its Applications•Dec 1, 2008•1 citation
There has been much recent interest in analyzing the performance of LDPC codes in channels with memory.Density evolution (DE) may be used to analyze estimation-decoding algorithms employing the Sum-Product Algorithm for LDPC codes in Gilbert-Elliott (GE) channels.In this paper, we provide theoretical results which mitigate the complexity of characterizing the 4-dimensional GE parameter space using DE.For each point in the parameter space which is shown by DE to converge to P e → 0, we show that a region of convergence is induced, within which all points also converge to P e → 0. Conversely, for points found not to converge, we show that a region of non-convergence is induced.Using these results, the GE parameter space is partitioned into a region of convergence, a region of nonconvergence, and an uncertain region, which guides the selection of new points to test using DE.
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.
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.
Low-density parity-check (LDPC) codes perform very close to capacity for long lengths on several channels. However, the amount of memory (fixed-point numbers that need to be stored) required for implementing the message-passing algorithm increases linearly as the number of edges in the graph increases. In this letter, we propose a decoding algorithm for decoding LDPC codes that reduces the memory requirement at the decoder. The proposed decoding algorithm can be analyzed using density evolution; further, we show how to design good LDPC codes using this. Results show that this algorithm provides almost the same performance as the conventional sum-product decoding of LDPC codes.
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
In this paper, we investigate the design of low-density parity-check (LDPC) codes for the Gilbert-Elliott (GE) channel. Recently, Eckford et al. proposed a design method of irregular LDPC codes using approximate density-evolution (DE) for Markov channels. In the design method proposed by Eckford et al., the probability density function (PDF) of the messages from variable nodes to check nodes is approximated by the Gaussian distribution. In this paper, we first show the method to obtain the accurate PDF of the messages from variable nodes to check nodes by utilizing two DE steps for the Gaussian distribution. We call this method the iterative density approximation (IDA). Using this method, we can design the good LDPC codes. Next, we propose an efficient design method of irregular LDPC codes by using Beta approximation to the PDF of the channel state probability for the GE channel. Consequently, we show that the complexity to calculate PDFs of the channel messages is considerably reduced though the rates of LDPC codes obtained by using the proposed approximation are almost the same as that of the IDA method.