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International Symposium on Communications, Control and Signal Processing

λ-min algorithm using TAP approach

Manel Abdelhedi, Omessaad Hamdi, Ammar Bouallègue

Published Mar 1, 2010• 0 citations
Publisher

Abstract

In this paper, we present the decoding algorithms of Low-density parity-check (LDPC) codes using statistical physics arguments. Previous works, have shown that the Belief-Propagation (BP) algorithm, is equivalent to the Thouless-Anderson-Palmer (TAP) [1] approach. In this paper, we develop the log-likelihood ratios-Belief Propagation (LLR-BP) algorithm and its simplification, the λ-min algorithm using this method developed in the study of disordered systems. The obtained results by simulation confirm that the λ-min algorithm gets closer to the LLR-BP algorithm as λ increases.

DOI
10.1109/isccsp.2010.5463402

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In this paper, we present the decoding algorithms of Low-density parity-check (LDPC) codes using statistical physics arguments. Previous works, have shown that the Belief-Propagation (BP) algorithm, is equivalent to the Thouless-Anderson-Palmer (TAP) [1] approach. In this paper, we develop the log-likelihood ratios-Belief Propagation (LLR-BP) algorithm and its simplification, the λ-min algorithm using this method developed in the study of disordered systems. The obtained results by simulation confirm that the λ-min algorithm gets closer to the LLR-BP algorithm as λ increases.
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 information theory. N. Sourlas was the first to point out that LDPC codes have a similarity with Ising spin systems of statistical physics. Besides, it has been shown that the Belief-Propagation algorithm, the LDPC codes decoding algorithm, is equivalent to the Thouless-Anderson-Palmer(TAP) approach. In this paper, we develop the log likelihood ratios-Belief Propagation (LLR-BP) algorithm and its simplifications the BP-Based algorithm and the λ-min algorithm with the TAP approach.
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.
Low-density parity-check (LDPC) codes are very efficient for communicating reliably through a noisy channel. N.Sourlas [1] showed that LDPC codes, which revolutionize the codes domain and used in many communications standards, can be mapped onto an Ising spin systems. Besides, it has been shown that the Belief-Propagation (BP) algorithm, the LDPC codes decoding algorithm, is equivalent to the Thouless- Anderson-Palmer (TAP) approach [2]. Unfortunately, no study has been made for the other decoding algorithms. In this paper, we develop the Log-Likelihood Ratios-Belief Propagation (LLR-BP) algorithm and its simplifications the BP-Based algorithm and the λ-min algorithm with the TAP approach. We present the performance of these decoding algorithms using statistical physics argument i.e., we present the performance as function of the magnetization.
A novel representation, using soft-bit messages, of the belief propagation (BP) decoding algorithm for low-density parity-check codes is derived as an alternative to the log-likelihood-ratio (LLR)-based BP and min-sum decoding algorithms. A simple approximation is also presented. Simulation results demonstrate the functionality of the soft-bit decoding algorithm. Floating-point soft-bit and LLR BP decoding show equivalent performance; the approximation incurs 0.5-dB loss, comparable to min-sum performance loss over BP. Fixed-point results show similar performance to LLR BP decoding; the approximation converges to floating-point results with one less bit of precision.
Several decoding algorithms based on oscillation Log-likelihood ratio (LLR) for Lowdensity parity-check (LDPC) codes are investigated in this paper, based on which, an improved decoding algorithm which uses the accumulated LLR as the final bit decision merit is proposed. This approach improves the decoding performance with only a very small increase in storage and computation. Combined with normalized belief-propagation (BP), two algorithms are given, which are named as accumulate normalized BP (ACC-Nor-BP) and compare-add normalized BP (CA-Nor-BP). The storage requirements and computational complexity of ACC-Nor-BP is lower than that of CA-Nor-BP and CA-BP, but performance is close to the CA-Nor-BP and better than CA-BP; therefore ACC-Nor-BP algorithm is an effectively improved BP algorithm which has a good balance between performance and complexity.
Various log-likelihood-ratio-based belief-propagation (LLR-BP) decoding algorithms and their reduced-complexity derivatives for low-density parity-check (LDPC) codes are presented. Numerically accurate representations of the check-node update computation used in LLR-BP decoding are described. Furthermore, approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check-node update, or symbol-node update, or both. In particular, two main approaches for simplified check-node updates are presented that are based on the so-called min-sum approximation coupled with either a normalization term or an additive offset term. Density evolution is used to analyze the performance of these decoding algorithms, to determine the optimum values of the key parameters, and to evaluate finite quantization effects. Simulation results show that these reduced-complexity decoding algorithms for LDPC codes achieve a performance very close to that of the BP algorithm. The unified treatment of decoding techniques for LDPC codes presented here provides flexibility in selecting the appropriate scheme from performance, latency, computational-complexity, and memory-requirement perspectives.