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Electronics Letters

Low‐complexity bound on irregular LDPC belief‐propagation decoding thresholds using a Gaussian approximation

F. Vatta, Alessandro Soranzo, Fulvio Babich

Published Jul 11, 2018• 22 citations
Publisher

Abstract

Since irregular low‐density parity‐check (LDPC) codes are known to perform better than regular ones, and to exhibit, like them, the so‐called ‘threshold phenomenon’, this Letter investigates a low‐complexity upper bound on belief‐propagation decoding thresholds for this class of codes on memoryless binary input additive white Gaussian noise channels, with sum‐product decoding. A simplified analysis of the belief‐propagation decoding algorithm is used, i.e. consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes.

DOI
10.1049/el.2018.0478

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Since irregular low‐density parity‐check (LDPC) codes are known to perform better than regular ones, and to exhibit, like them, the so‐called ‘threshold phenomenon’, this Letter investigates a low‐complexity upper bound on belief‐propagation decoding thresholds for this class of codes on memoryless binary input additive white Gaussian noise channels, with sum‐product decoding. A simplified analysis of the belief‐propagation decoding algorithm is used, i.e. consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes.
This paper investigates about the usefulness of some recently published low complexity upper bounds on belief-propagation decoding thresholds for irregular low-density parity-check (LDPC) codes to analyze their performance on memoryless binary input - additive white Gaussian noise (BI-AWGN) channels, with sum-product decoding. Irregular LDPC codes are known to perform better than regular ones, and to exhibit, like them, the so called “threshold phenomenon”, being the threshold 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 use a simplified analysis of the belief-propagation decoding algorithm, i.e., consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes on BI-AWGN channels.
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.
We consider low-density parity-check (LDPC) code ensembles over non-binary Galois fields when used for transmission over arbitrary discrete memoryless channels. Belief propagation decoding for these codes has been shown to achieve excellent results. However, computing the decoding threshold using density evolution is usually impractical, since one needs to propagate multi-dimensional probability distributions, and Monte Carlo simulations are required instead. By considering the evolution of the message Bhattacharyya parameter and the message expected value parameter, we derive a simple lower bound on the performance of the algorithm. This bound applies for both regular and irregular non-binary LDPC ensembles.
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.
We consider LDPC code ensembles over non-binary Galois fields, when used for transmission over arbitrary memoryless channels. Belief propagation decoding for these codes has been shown to achieve excellent results. However, computing the decoding threshold using density evolution is usually impractical, since one needs to propagate multi-dimensional probability distributions, and Monte Carlo simulations are required instead. By considering the evolution of the message Bhattacharyya parameter and the message expected value parameter we derive a simple lower bound on the performance of the algorithm. This bound applies for both regular and irregular non-binary LDPC ensembles.
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.