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JOURNAL OF ELECTRONICS INFORMATION TECHNOLOGY

Research of Density Evolution of LDPC Codes over Rice Channels

Hua Xu, Chengqi Xu

Published Oct 19, 2006• 0 citations
OpenAlex

Abstract

Noise threshold of Low Density Parity Check (LDPC) codes ensemble could be calculated by density evolution of the decoding messages. According to the threshold, performance of the decoding algorithm could be evaluated and degree distribution pairs of irregular LDPC codes also might be optimized. In this paper, for regular LDPC codes on Rice channel, Discrete Density Evolution(DDE) for BP,BP-based and offset BP-based decoding algorithm were compared on different quantization level and step. Furthermore, degree distribution pairs of the irregular LDPC codes were optimized on Rice channel, which was based on DDE for offset BP-based decoding algorithm, the corresponding threshold is also calculated. Finally, the Probability Mass Function (PMF) evolution curve of optimized irregular LDPC codes with code rate 1/2 on Rice channel is given.

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Noise threshold of Low Density Parity Check (LDPC) codes ensemble could be calculated by density evolution of the decoding messages. According to the threshold, performance of the decoding algorithm could be evaluated and degree distribution pairs of irregular LDPC codes also might be optimized. In this paper, for regular LDPC codes on Rice channel, Discrete Density Evolution(DDE) for BP,BP-based and offset BP-based decoding algorithm were compared on different quantization level and step. Furthermore, degree distribution pairs of the irregular LDPC codes were optimized on Rice channel, which was based on DDE for offset BP-based decoding algorithm, the corresponding threshold is also calculated. Finally, the Probability Mass Function (PMF) evolution curve of optimized irregular LDPC codes with code rate 1/2 on Rice channel is given.
Research of the fixed point which appears in decoding process of LDPC codes is important to further analyze the decoding mechanism and speed up decoding convergence.Firstly,a new optimization scheme which combines DDE(Discrete Density Evolution)for offset BP-based decoding algorithm with differential evolution is presented for optimizing irregular LDPC codes.Then the optimized degree distribution pair and threshold of LDPC codes over Rice channel are given,with the maximum bit node degree 10 and code rate R=1/2,1/3.Furthermore,we analyze a series of simulated curves related to LDPC decoding fixed point and threshold phenomenon.Simulation results show that the node with higher degree contributes more to the decoding for LDPC codes.Therefore,we tentatively put forward two kinds of thinking to improve LDPC decoding algorithm.
For irregular LDPC codes,the density evolution method can track the messages to find out the noise threshold, enabling optimization of the degree distribution. A new optimized scheme for irregular LDPC codes is proposed in this paper, which is combining Zhengya He's modified particle swarm optimization (PSO) algorithm with discrete density evolution (DDE) for the offset BP-based decoding algorithm. Several optimized degree distribution pairs and corresponding threshold of LDPC codes over Rician channel were given using the proposed scheme. The performance of the optimized LDPC codes was also analyzed. Simulation results shows that the proposed scheme is effective.
To design low-density parity-check (LDPC) codes with asymmetric channels and analyze their performance, an achievable asymmetric discrete density evolution method is proposed, which is based on offset min-sum decoding. First, the iterative formula of low-complexity asymmetric channel discrete density evolution is derived by means of combinatorics, difference method and the binomial theorem. The iterative formula is applied to further optimize the design of the LDPC codes using the offset min-sum decoding, and the optimal degree distributions and performance of the LDPC codes under different quantization ranges and step sizes are obtained under the asymmetric channel. Finally, the optimal offset values under high-precision quantization are obtained. The results show that under the asymmetric channel, the complexity of the proposed asymmetric density evolution method based on the offset min-sum decoding is reduced by an order of magnitude than that using the original method. For a fixed code rate, the smaller the variable node degree of the regular codes, the better the performance and the smaller the optimal offset value. For irregular LDPC codes, the optimal degree distributions and offset values are determined according to the differential evolution algorithm. For a fixed code rate, the larger the maximum variable node degree, the better the performance of irregular codes.
A statistical-based density evolution algorithm is proposed for nonbinary low-density parity-check (LDPC) codes. It is applicable to both regular and irregular codes under study. The algorithm proposed serves as a tool to analyze the performance limit for iterative decoding of nonbinary LDPC codes, which in turn guides code design. Specifically, it provides approximated evaluations of convergence thresholds for codes given specific degree distributions. It is shown that, for a class of quasi-cyclic (QC) structured extended irregular repeat-accumulate (SeIRA) nonbinary LDPC codes, degree distributions can be optimized via the algorithm proposed. Simulation results exhibit that the nonbinary LDPC codes designed outperform the optimized binary LDPC codes on the AWGN channel with similar code length and rate in terms of bits. Performance gap between the binary and nonbinary LDPC codes is larger in the scenario of higher order modulation.
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.