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Dianzi xuebao

Measure on the Performance of Belief Propagation Decoding Algorithm

He Yu

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Formulas for updated rules of the belief propagation algorithm applied to low density parity check (LDPC) codes are systematically derived based on the tree and the finite state machine (FSM).The probability measure on the continuous message space is introduced to evaluate the evolution of message densities within iterative decoding rounds,and derivations of the output distributions of common binary symmetric channels and the message density evolution are given in detail.The parameterized estimation of the performance of the algorithm is also discussed.The analytic and calculable means can be used to make systematical analysis for the convergence of BP algorithms independent of channels,and can be of help in designing LDPC codes and analyzing quantization effects for the fast decoding on the purpose of practical applications of LDPC codes to real time communication systems.

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Jounal of Xidian University•Jan 1, 2002•0 citations
Formulas for updated rules of the belief propagation algorithm applied to low density parity check (LDPC) codes are systematically derived based on the tree and the finite state machine (FSM).The probability measure on the continuous message space is introduced to evaluate the evolution of message densities within iterative decoding rounds,and derivations of the output distributions of common binary symmetric channels and the message density evolution are given in detail.The parameterized estimation of the performance of the algorithm is also discussed.The analytic and calculable means can be used to make systematical analysis for the convergence of BP algorithms independent of channels,and can be of help in designing LDPC codes and analyzing quantization effects for the fast decoding on the purpose of practical applications of LDPC codes to real time communication systems.
We consider Gallager's (1963) soft-decoding (belief propagation) algorithm for decoding low-density parity-check (LDPC) codes, when applied to an arbitrary binary-input symmetric-output channel. By considering the expected values of the messages, we derive both lower and upper bounds on the performance of the algorithm. We also derive various properties of the decoding algorithm, such as a certain robustness to the details of the channel noise. Our results apply both to regular and irregular LDPC codes.
We consider Gallager's (1963) soft decoding (belief propagation) algorithm for decoding low density parity check (LDPC) codes, when applied to an arbitrary binary-input symmetric-output channel. By considering the expected values of the messages, we derive both lower and upper bounds on the performance of the algorithm. We also derive various properties of the decoding algorithm, such as robustness to the initial values of the messages. Our results apply both to regular and irregular LDPC codes.
Density evolution is often used to determine the performance of an ensemble of low-density parity-check (LDPC) codes under iterative message-passing algorithms. Conventional density evolution techniques over memoryless channels are based on the independence assumption amongst all the processed messages in variable and check nodes. This assumption is valid for many algorithms such as standard belief propagation (BP) and min-sum (MS) algorithms. However, there are other important iterative algorithms such as successive relaxation (SR) versions of BP and MS, and differential decoding with binary message passing (DD-BMP) algorithm of Mobini et. al., for which this assumption is not valid. The dependence created among messages for these algorithms is due to the introduction of memory in the iterative algorithm. In this work, we propose a model for iterative decoding algorithms with memory which covers SR and DD-BMP algorithms as special cases. Based on this model, we derive a Bayesian network for iterative algorithms with memory over memoryless channels and use this representation to analyze the algorithms using density evolution. The density evolution technique is developed based on truncating the memory of the decoding process and approximating it with a finite order Markov process, and can be implemented efficiently. As an example, we apply our technique to analyze the performance of DD-BMP on regular LDPC code ensembles, and make a number of interesting observations with regard to the performance/complexity trade off of DD-BMP in comparison with BP and MS algorithms.
Summary form only given. The capacity of the message-passing decoder for LDPC codes can be computed by the density evolution algorithm by iteratively computing message densities. The infinite-dimensional problem of iteratively calculating the message densities in the case of the binary input AWGN channel and the belief propagation decoder can be simplified to a one-dimensional problem by using a Gaussian approximation. By assuming Gaussian densities the density evolution simplifies to updating means of Gaussian densities. An improved Gaussian approximation algorithm is suggested for computing the capacity of the BP decoder based on the mutual information measure. An analytical method for computing the mutual information of a transmitted bit with the message corresponding to it is proposed. Using this method we can approximate the non-Gaussian message by a Gaussian message that has the same mutual information with the transmitted bit. Computationally, the algorithm is similar to the Gaussian approximation algorithm proposed in Chung (2001). For various regular LDPC codes that were examined, the algorithm computed threshold values within 0.01dB or less from the exact threshold which is an improvement over the Gaussian approximation. Furthermore, additional insight on the convergence process can be gained from EXIT charts that can be derived from the algorithm. This can assist in designing better irregular LDPC codes.
Density evolution is often used to determine the performance of an ensemble of low-density parity-check (LDPC) codes under iterative message-passing algorithms. Conventional density evolution techniques over memoryless channels are based on the assumption that messages at iteration \ell are only a function of the messages at iteration \ell -1 and possibly the channel output. This assumption is valid for many algorithms such as standard belief propagation (BP) and min-sum (MS) algorithms. However, there are other important iterative algorithms such as successive relaxation (SR) versions of BP and MS, and differential decoding with binary message passing (DD-BMP) algorithm of Mobini et al., for which this assumption is not valid. The reason is the introduction of memory in these algorithms. In this work, we propose a model for iterative decoding algorithms with memory which covers SR and DD-BMP algorithms as special cases. Based on this model, we derive a Bayesian network for iterative algorithms with memory over memoryless channels and use this representation to analyze the performance of the algorithms using density evolution. The density evolution technique is developed based on truncating the memory of the decoding process and approximating it with a finite order Markov process, and can be implemented efficiently. As an example, we apply our technique to analyze the performance of DD-BMP on regular LDPC code ensembles, and make a number of interesting observations with regard to the performance/complexity tradeoff of DD-BMP in comparison with BP and MS algorithms. The model presented in this paper is based on certain simplifying assumptions about the memory structure of iterative algorithms such as the existence of memory only at the output of variable nodes in the code's Tanner graph rather than at both outputs of variable and check nodes. The Bayesian network framework introduced here however, can still be used to analyze the more general scenarios.
A fast quantized belief propagation algorithm is proposed for decoding low-density parity-check(LDPC) codes. With operations performed by addressing tables, the proposed algorithm takes as operands the indices of labels rather than the quantization levels. The intermediate calculations have been contained in operation tables with much higher precision. The required storage can be significantly reduced by applying the symmetry property while keeping low time complexity. Without the increase in time complexity, it is convenient to realize the decoding with uniform quantization or any nonuniform quantizations that is designed for improving the performance. The time-varying decoder also becomes possible. This algorithm makes it possible to apply LDPC codes to practical systems or to realize fast simulation systems. Some simulation results are also given. The 6?bit logarithmic quantization results in gains of about 0.2?dB relative to the uniform one, approaching to the continuous decoding at low signal-to-noise ratios (SNRs) and yielding a loss of about 0.2?dB at high SNRs. Proper quantization of high order can achieve the performance of continuous decoding, and it may be superior to continuous decoding at high SNRs.