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Exact Thresholds for LDPC Codes Transmitted Over Binary Erasure Channels

Thorsten Hehn, Andac Doenmez, Johannes B. Huber

Published Jan 1, 2005• 4 citations
OpenAlex

Abstract

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.

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On the EXIT chart analysis of low-density parity-check codes

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The extrinsic information transfer (EXIT) charts are used to analyze the sum-product decoding algorithm of low-density parity-check (LDPC) codes. The codes are optimized by performing curve fitting on EXIT charts. This technique is developed for binary erasure channels based on area property and is extended to other channels. We show that on channels where the area property is not valid, the curve fitting approach is a sub-optimal method of optimizing for the best convergence threshold. On Gaussian channels, the messages from variable nodes to check nodes are approximated by a symmetric Gaussian distribution. We prove that the check node detector mutual information transfer function derived using this approximation is an upper bound. So the EXIT chart analysis predicts an optimistic value for the convergence threshold. We give an empirical lower bound for the check node detector mutual information transfer function. For few irregular LDPC codes designed for AWGN channel using density evolution, we have examined the accuracy of convergence thresholds predicted by EXIT charts using these bounds for the check node transfer function. The analysis of EXIT chart technique presented in this paper can be extended to other channels for which the distribution of channel messages can be approximated by a symmetric Gaussian/Gaussian mixture distribution.

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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.
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.
Communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding is considered. The average bit error probability of an irregular LDPC code ensemble after a fixed number of iterations converges to a limit, which is calculated via density evolution, as the blocklength n tends to infinity. The difference between the bit error probability with blocklength n and the large-blocklength limit behaves asymptotically like ¿/n, where the coefficient ¿ depends on the ensemble, the number of iterations and the erasure probability of the BEC. In, ¿ is calculated for regular ensembles. In this paper, ¿ for irregular expurgated ensembles is derived. It is demonstrated that convergence of numerical estimates of ¿ to the analytic result is significantly fast for irregular unexpurgated ensembles.
In this paper, we propose the concept of extrinsic information transfer (EXIT) trajectories, and show how to use it to analyze convergence performance of belief propagation (BP) decoding for irregular low-density parity-check (LDPC) codes. The proposed analysis method utilizes the EXIT characteristics in an improved way, and allows us to calculate the convergence threshold quickly and accurately. On the binary phase-shift keying (BPSK) modulated additive white Gaussian noise (AWGN) channel, the achieved threshold values for the enumerated irregular LDPC codes have error to those computed by density evolution only about 0.03 dB.
We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. The bit error probability for infinite block length is known by density evolution and it is well known that a difference between the bit error probability at finite iteration number for finite block length nnn and for infinite block length is asymptotically α/nα/nα/n, where ααα is a specific constant depending on the degree distribution, the iteration number and the erasure probability. Our main result is to derive an efficient algorithm for calculating ααα for regular ensembles. The approximation using ααα is accurate for (2,r)(2,r)(2,r)-regular ensembles even in small block length.
The extrinsic information transfer (EXIT) charts are used to analyze the sum-product decoding algorithm of low-density parity-check (LDPC) codes. The codes are optimized by performing curve fitting on EXIT charts. This technique is developed for binary erasure channels based on area property and is extended to other channels. We show that on channels where the area property is not valid, the curve fitting approach is a sub-optimal method of optimizing for the best convergence threshold. On Gaussian channels, the messages from variable nodes to check nodes are approximated by a symmetric Gaussian distribution. We prove that the check node detector mutual information transfer function derived using this approximation is an upper bound. So the EXIT chart analysis predicts an optimistic value for the convergence threshold. We give an empirical lower bound for the check node detector mutual information transfer function. For few irregular LDPC codes designed for AWGN channel using density evolution, we have examined the accuracy of convergence thresholds predicted by EXIT charts using these bounds for the check node transfer function. The analysis of EXIT chart technique presented in this paper can be extended to other channels for which the distribution of channel messages can be approximated by a symmetric Gaussian/Gaussian mixture distribution.