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International Conference on Software, Telecommunications and Computer Networks (SoftCOM)

Performance Study of a Class of Irregular LDPC Codes Based on Their Weight Distribution Analysis

F. Vatta, Fulvio Babich, Flavio Ellero, Matteo Noschese, Giulia Buttazzoni, Massimiliano Comisso

Published Sep 1, 2019• 4 citations
Publisher

Abstract

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.

DOI
10.23919/softcom.2019.8903633

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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.
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.
In this paper, we are concerned with the finite-length analysis of low-density parity-check (LDPC) codes when used over additive white Gaussian noise (AWGN) channels. We present the iterative decoding algorithm of LDPC codes, and we also discuss the encoding methods and the bipartite graph representation of LDPC codes ensemble. Finite-length LDPC code performances are investigated, and the influence of cycle is pointed out. Simulation results show that irregular LDPC codes with variable and check node degree distribution optimized for infinite-length codes are not optimal for finite-length LDPC codes, and the error patterns of both regular and irregular codes are investigated as an effort to improve the performance and design methodology of finite-length LDPC codes.
This paper investigates the performance of a class of irregular low-density parity-check (LDPC) codes through a recently published low complexity upper bound on their beliefpropagation decoding thresholds. Moreover, their performance analysis is carried out through a recently published algorithmic method, presented in Babich et al. 2017 paper. In particular, the class considered is characterized by variable node degree distributions λ(x)\lambda(x)λ(x) of minimum degree i1>2i_1 \gt 2i1​>2: being, in this case, λ′(0)=λ2=0\lambda^{'} (0)=\lambda_2=0λ′(0)=λ2​=0, this is useful to design LDPC codes presenting a linear minimum distance growth with the block length with probability 1, as shown in Di et al.’s 2006 paper. These codes unfortunately cannot reach capacity under iterative decoding, since the achievement of capacity requires λ2≠0\lambda_2 \neq 0λ2​=0. However, in this latter case, the block error probability might converge to a constant, as shown in the aforementioned paper.
Density evolution is an algorithm for computing the capacity of low-density parity-check (LDPC) codes under message-passing decoding. For memoryless binary-input continuous-output additive white Gaussian noise (AWGN) channels and sum-product decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinite-dimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a one-dimensional problem of updating the means of the Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is 10. We show that by using the Gaussian approximation, we can visualize the sum-product decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization.
The performance of low-density parity-check (LDPC) codes transmitted over a memoryless binary-input continuous output additive white Gaussian noise (AWGN) channel and decoded with quantized min-sum decoding is strongly influenced by the decoder's quantization scheme. This paper presents an efficient algorithm that determines the best uniform scalar quantizer for a particular code. To maximize performance, it is necessary to determine degree distributions that best match the characteristics of the quantized min-sum decoder. Toward this end, an iterative optimization framework that jointly optimizes the degree distributions and the quantizer is presented