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Electronics Letters

New explicitly invertible approximation of the function involved in LDPC codes density evolution analysis using a Gaussian approximation

F. Vatta, Alessandro Soranzo, Massimiliano Comisso, Giulia Buttazzoni, Fulvio Babich

Published Sep 2, 2019• 10 citations
Publisher

Abstract

Low density parity check (LDPC) codes are still intensively studied investigating their iterative decoding convergence performance. Since the probability distribution function of the decoder's log‐likelihood ratio messages was observed to be approximately Gaussian, a variety of low‐complexity approaches to this investigation were proposed. One of them was presented in Chung et al. 's 2001 paper, involving the function , therein specified, and its inverse. In this Letter, a new approximation of the function is given, such that, unlike the other approximations found in the literature, it is defined by a single expression (i.e. it is not piecewise defined), it is explicitly invertible, and it has less relative error in any x than the other approximations.

DOI
10.1049/el.2019.1349

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Low density parity check (LDPC) codes are still intensively studied investigating their iterative decoding convergence performance. Since the probability distribution function of the decoder's log‐likelihood ratio messages was observed to be approximately Gaussian, a variety of low‐complexity approaches to this investigation were proposed. One of them was presented in Chung et al. 's 2001 paper, involving the function , therein specified, and its inverse. In this Letter, a new approximation of the function is given, such that, unlike the other approximations found in the literature, it is defined by a single expression (i.e. it is not piecewise defined), it is explicitly invertible, and it has less relative error in any x than the other approximations.
Low Density Parity Check (LDPC) codes are currently being deeply analyzed through algorithms that require the capability of addressing their iterative decoding convergence performance. Since it has been observed that the probability distribution function of the decoder’s log-likelihood ratio messages is roughly Gaussian, a multiplicity of moderate entanglement strategies to this analysis has been suggested. The first of them was proposed in Chung et al.’s 2001 paper, where the recurrent sequence, characterizing the passage of messages between variable and check nodes, concerns the function ϕ(x), therein specified, and its inverse. In this paper, we review this old approximation to the function ϕ(x), one variant on it obtained in the same period (proposed in Ha et al.’s 2004 paper), and some new ones, recently published in two 2019 papers by Vatta et al. The objective of this review is to analyze the differences among them and their characteristics in terms of accuracy and computational complexity. In particular, the explicitly invertible, not piecewise defined approximation of the function ϕ(x), published in the second of the two abovementioned 2019 papers, is shown to have less relative error in any x than most of the other approximations. Moreover, its use conducts to an important complexity reduction, and allows better Gaussian approximated thresholds to be obtained.
This letter presents a more accurate mathematical analysis, with respect to the one performed in Chung et al.'s 2001 paper, of belief-propagation decoding for low-density parity-check (LDPC) codes on memoryless binary input-additive white Gaussian noise channels, when considering a Gaussian approximation (GA) for message densities under density evolution. The recurrent sequence, defined in Chung et al.'s 2001 paper, describing the message passing between variable and check nodes follows from the GA approach and involves the function φ(x), therein defined, and its inverse. The analysis of this function is here resumed and studied in depth to obtain tighter upper and lower bounds on it. Moreover, unlike the upper bound given in the above cited paper, the tighter upper bound on φ(x) is invertible. This allows a more accurate evaluation of the asymptotical performance of sum-product decoding of LDPC codes when a GA is assumed.
This paper is concerned with density evolution for iterative decoding of low-density parity-check (LDPC) codes. We first study the problem of density evolution computation for regular LDPC codes. For this, we propose a simple computational algorithm based on the ergodicity theory. This method is shown to match very well with explicit calculations of density functions. The second problem we study is about the approach of Gaussian approximation to density evolution. We point out that it is inappropriate to use the mean of the density only to model the iterative decoding process. Instead, both the mean and variance are needed for Gaussian approximation. Finally, we consider the problem of density evolution for irregular LDPC codes. For this, we extend the density evolution algorithm for regular LDPC codes to irregular LDPC codes. We then illustrate that Gaussian approximation is also valid provided that the degree distributions are not wide. A dynamic model is also presented based on Gaussian approximation.
Density evolution (DE) is a technique for tracking the distribution of the log likelihood ratio (LLR) messages exchanged between the variable nodes and the check nodes in a bipartite graph. It is widely assumed that these distributions are close to Gaussian. However, in many scenarios, this assumption is not valid, e.g., the case that the signal to noise ratio (SNR) is low, or the degree of variable nodes exceeds a certain threshold. This article introduces a new (suboptimal) method for DE algorithm in low-density parity-check (LDPC) codes. We provide a more accurate model for the distribution of message bits (as compared to Gaussian) through matching the first n statistical moments. An iterative message passing algorithm is proposed to compute these moments from the graphical representation of the underlying code. We show that the proposed algorithm results in an improved estimate of the underlying EXIT chart as compared to using a Gaussian assumption. In this respect, the proposed method achieves a performance very close to that of the best earlier methods, while it offers a much lower complexity.
In modern communication systems Low-Density Parity Check (LDPC) codes are widely used. It is because of their performance under iterative decoding, hardware-friendly structure and well-known construction methods. For one to construct good LDPC code Density Evolution (DE) can be used, it estimates the decoding threshold, that is it estimates performance of the code. LDPC codes can be further extended to Generalized LDPC (GLDPC) codes, where simple SPC code is replaced with some component code. With more complex structure common DE is no longer correct. This letter presents the upper bound for decoding error and, therefore, an upper bound of a iterative threshold (in dB). It will be shown that the Gaussian approximation gives an upper bound for a decoding error. As one of the results it means that EXIT-charts analysis also gives an upper bound for an iterative threshold for a GLDPC code.