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IEEE Wireless Communications and Networking Conference

Thresholds Calculation of LDPC Code Ensembles Using a Novel Gaussian Approximation Algorithm

Piming Ma, Kyungsup Kwak

Published Apr 1, 2009• 3 citations
Publisher

Abstract

A novel Gaussian approximation algorithm is proposed for calculating the thresholds of low-density parity- check (LDPC) code ensembles under sum-product decoding. The updating rules of means and error probability are derived. Numerical results show that the thresholds obtained by the proposed algorithm closely coincide with the values computed by density evolution (DE). Moreover, our algorithm can calculate the thresholds of irregular LDPC codes with higher accuracy than the algorithm proposed by Lehmann and Maggio.

DOI
10.1109/wcnc.2009.4917989

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A novel Gaussian approximation algorithm is proposed for calculating the thresholds of low-density parity- check (LDPC) code ensembles under sum-product decoding. The updating rules of means and error probability are derived. Numerical results show that the thresholds obtained by the proposed algorithm closely coincide with the values computed by density evolution (DE). Moreover, our algorithm can calculate the thresholds of irregular LDPC codes with higher accuracy than the algorithm proposed by Lehmann and Maggio.
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.
This paper is concerned with density evolution for iterative decoding of low-density parity-check (LDPC) codes. We first study the problem of density evolution computation for regular LDPC codes. For this, we propose a simple computational algorithm based on the ergodicity theory. This method is shown to match very well with explicit calculations of density functions. The second problem we study is about the approach of Gaussian approximation to density evolution. We point out that it is inappropriate to use the mean of the density only to model the iterative decoding process. Instead, both the mean and variance are needed for Gaussian approximation. Finally, we consider the problem of density evolution for irregular LDPC codes. For this, we extend the density evolution algorithm for regular LDPC codes to irregular LDPC codes. We then illustrate that Gaussian approximation is also valid provided that the degree distributions are not wide. A dynamic model is also presented based on Gaussian approximation.
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.
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.