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Density evolution for BP-based decoding algorithms of LDPC codes and their quantized versions

Jinghu Chen, P.M.C. Fossorier

Published Mar 22, 2004• 51 citations
Publisher

Abstract

In this paper, we analyze the performance of two improved BP-based decoding algorithms for LDPC codes, namely the normalized BP-based and the offset BP-based algorithms, by means of density evolution. The numerical calculations show that with one properly chosen parameter for each of these two improved BP-based algorithms, performances very close to that of the BP algorithm can be achieved. Simulation results for LDPC codes with code length moderately long validate the proposed optimization. Finite quantization effects on the BP-based and the offset BP-based decoding algorithms are evaluated.

DOI
10.1109/glocom.2002.1188424

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In this paper, we analyze the performance of two improved BP-based decoding algorithms for LDPC codes, namely the normalized BP-based and the offset BP-based algorithms, by means of density evolution. The numerical calculations show that with one properly chosen parameter for each of these two improved BP-based algorithms, performances very close to that of the BP algorithm can be achieved. Simulation results for LDPC codes with code length moderately long validate the proposed optimization. Finite quantization effects on the BP-based and the offset BP-based decoding algorithms are evaluated.
In this letter, we analyze the performance of two improved belief propagation (BP) based decoding algorithms for LDPC codes, namely the normalized BP-based and the offset BP-based algorithms, by means of density evolution. The numerical calculations show that with one properly chosen parameter for each of these two improved BP-based algorithms, performances very close to that of the BP algorithm can be achieved. Simulation results for LDPC codes with code length moderately long validate the proposed optimization.
Low-density parity-check (LDPC) codes deliver very good performance when decoded with the BP algorithm. Unfortunately, this algorithm has the disadvantage to be complex for implementation. In this paper, we investigate the BP algorithm and its simplifications (BP-Based, normalized BP-Based, offset BP-Based). Previous works on normalized and offset BP-Based proposed theoretical values for the offset and normalization factors, we perform simulations which allow to validate these theoretical results.
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.
Noise threshold of Low Density Parity Check (LDPC) codes ensemble could be calculated by density evolution of the decoding messages. According to the threshold, performance of the decoding algorithm could be evaluated and degree distribution pairs of irregular LDPC codes also might be optimized. In this paper, for regular LDPC codes on Rice channel, Discrete Density Evolution(DDE) for BP,BP-based and offset BP-based decoding algorithm were compared on different quantization level and step. Furthermore, degree distribution pairs of the irregular LDPC codes were optimized on Rice channel, which was based on DDE for offset BP-based decoding algorithm, the corresponding threshold is also calculated. Finally, the Probability Mass Function (PMF) evolution curve of optimized irregular LDPC codes with code rate 1/2 on Rice channel is given.
Reduced-Complexity Decoding of LDPC Codes 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 representation 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 a performance, latency, computational complexity, and memory-requirement perspective.
The quantization scheme for the iterative belief propagation (BP) algorithm of low density parity check (LDPC) codes is studied.The effects of the bound,resolution,and other parameters for uniform and nonuniform quantization schemes are discussed.Good performance of decoding under properly designed quantization can be obtained as shown by simulations.Furthermore,we found that the quantization of high order for the mid variables of BP decoding algorithm can improve the performance and reduce the error floor of LDPC codes at regions of high signal to noise ratio.