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International Conference on Communications, Circuits and Systems

Another derivation process of belief propagation algorithm

Jingli Lin, Long-jiang Jing, Weile Zhu, Jijun Zheng

Published Jul 1, 2007• 0 citations
Publisher

Abstract

Generally, soft decoding of low-density parity-check codes can be implemented by iterative decoding based on belief propagation (BP) algorithm. In this paper, the BP algorithm is presented and derived in a way which differs from that by Gallager, which helps to understand iterative decoding of LDPC codes more clearly.
DOI
10.1109/icccas.2007.6247590

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In this work, reliability based decoding is combined with belief propagation (BP) decoding for low-density parity check (LDPC) codes. At each iteration, the soft output values delivered by the BP algorithm are used as reliability values to perform reduced complexity soft decision decoding of the code considered. This approach allows to bridge the error performance gap between BP decoding which remains suboptimum, and maximum likelihood decoding (MLD) which is too complex to be implemented for the codes considered. A stopping criterion which reduces the average number of iterations at the expense of very little performance degradation is also proposed for this combined decoding approach. Simulations results for several Gallager LDPC codes and different set cyclic (DSC) codes of hundreds of information bits are given and elaborated.
In this paper, reliability based decoding is combined with belief propagation (BP) decoding for low-density parity check (LDPC) codes. At each iteration, the soft output values delivered by the BP algorithm are used as reliability values to perform reduced complexity soft decision decoding of the code considered. This approach allows to bridge the error performance gap between belief propagation decoding which remains suboptimum, and maximum likelihood decoding which is too complex to be implemented for the codes considered. Trade-offs between decoding complexity and error performance are also investigated. In particular, a stopping criterion which reduces the average number of iterations at the expense of very little performance degradation is proposed for this combined decoding approach. Simulation results for several Gallager (1963, 1968) LDPC codes and different set cyclic codes of hundreds of information bits are given and elaborated.
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.
At the heart of modern coding theory lies the fact that low-density parity-check (LDPC) codes can be efficiently decoded by belief propagation (BP). The BP is an inference algorithm which operates on a graphical model of a code, and lends itself to low-complexity and high-speed implementations, making it the algorithm of choice in many applications. It has unprecedentedly good error rate performance, so good that when decoded by the BP, LDPC codes approach theoretical limits of channel capacity. However, this capacity approaching property holds only in the asymptotic limit of code length, while codes of practical lengths suffer abrupt performance degradation in the low noise regime known as the error floor phenomenon. Our study of error floor has led to an interesting and surprising finding that it is possible to design iterative decoders which are much simpler yet better than belief propagation! These decoders do not propagate beliefs but a rather different kind of messages that reflect the local structure of the code graph. This has opened a plethora of exciting theoretical problems and applications. This paper introduces this new paradigm.
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.
Low-density parity-check (LDPC) codes are based on random construction. Because of this randomness, it is not easy to analyze them with the traditional methods of communication theory. N.Sourlas [6] was the first to point out that LDPC codes have a similarity to Ising spin systems in statistical physics. Besides, it has been shown that the Belief-Propagation algorithm, the decoding algorithm of LDPC codes, is equivalent to the Thouless-Anderson-Palmer(TAP) [9] approach. In this paper, we develop the log-likelihood ratios-Belief Propagation (LLR-BP) and the BP-Based decoding algorithms with the TAP approach.