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A Survey on Old and New Approximations to the Function ϕ(x) Involved in LDPC Codes Density Evolution Analysis Using a Gaussian Approximation

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

Published May 17, 2021• 2 citations
Publisher

Abstract

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.

DOI
10.3390/info12050212

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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.
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.
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.
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.
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.
We present the formulas for searching for the thresholds of regular and irregular low-density parity-check (LDPC) codes under message-passing (MP) algorithm. A Gaussian approximation is applied to studying the evolution of the means of the messages of the variable nodes and the check nodes. Accurate numerical integration methods by using transformations are shown for evaluating the expected values of the message of the check nodes. Tables are built first and interpolations are used for further evaluations. Two curves are used to locate the threshold. We utilize an iterative decoding tunnel between these two curves and study the decoding performance by evaluating conditions of the derivatives of these two curves. Using this method the performance of both regular and irregular LDPC codes can be studied in a unified manner without using simulation.