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Progress in Natural Science Materials International

Exact thresholds for low-density parity-check codes over the binary erasure channel

Jianjun Mu, Xiaopeng Jiao, Xinmei Wang

Published Apr 29, 2009• 1 citation
Publisher

Abstract

A simple method for determining the threshold of irregular LDPC codes over the binary erasure channel (BEC) under message-passing decoding is proposed. An exact formula for calculating the threshold of irregular LDPC codes over the BEC is proved. This generalizes the known result on the threshold of regular LDPC codes to irregular LDPC codes. Moreover, our new method can avoid the computation of the inverse of the degree distribution function for irregular LDPC codes. Numerical results demonstrate the correctness of our proposed method.

DOI
10.1016/j.pnsc.2008.12.003

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A simple method for determining the threshold of irregular LDPC codes over the binary erasure channel (BEC) under message-passing decoding is proposed. An exact formula for calculating the threshold of irregular LDPC codes over the BEC is proved. This generalizes the known result on the threshold of regular LDPC codes to irregular LDPC codes. Moreover, our new method can avoid the computation of the inverse of the degree distribution function for irregular LDPC codes. Numerical results demonstrate the correctness of our proposed method.
Analytic expressions of decoding thresholds both for regular and irregular low-density parity-check (LDPC) codes over the binary erasure channel (BEC) are given, and low-complexity methods for determining the thresholds are proposed based on the analytic expressions. Specifically, firstly, a fixed-point equation, which represents a constraint relationship between the threshold and erasure probability of transmitted bits, is established. Secondly, by introducing an auxiliary equation, the threshold can be solved as an analytically function of solutions of a polynomial equation. Moreover, this method is extended to the Additive White Gaussian Noise (AWGN) channel. Numerical results show that, our proposed method is more accurate, and the complexity is much lower than that of conventional density evolution (DE). Moreover, our method does not require to compute the inverse of degree distribution function for irregular LDPC codes.
In this paper we present a simple method for determining the exact decoder convergence thresholds of low-density parity-check codes analytically.We assume transmission over the binary erasure channel and belief propagation decoding.To this aim we use EXIT charts and derive analytical formulas for the EXIT functions of regular and irregular LDPC codes transmitted over the binary erasure channel.Our method is equivalent to density evolution but faster and does not utilize a search algorithm.Thus it is not affected by numerical inaccuracy and does not depend on the number of iterations performed.
We provide an analytical expression for the threshold for right regular low-density parity-check (LDPC) codes for the erasure channel (EC), under message passing decoding. We first develop an expression for the threshold for regular LDPC codes, alternative to that already available in the literature; then we generalize it for the class of right regular codes. Thus, we show that the problem of computing the threshold can be reduced to the search for the fixed points of a function depending only on the variable nodes degree distribution, /spl lambda/(x), and on the uniform weight, d/sub c/, of check nodes. Finally, we show the application of our analysis to a class of capacity-approaching right regular LDPC codes.
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.
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.