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Scientia Sinica Informationis

Discrete density evolution and LDPC codes designwith asymmetric channels

涛 袁, 征 马, 琦珂 庞, 小虎 唐

Published Aug 29, 2022• 1 citation
Publisher

Abstract

To design low-density parity-check (LDPC) codes with asymmetric channels and analyze their performance, an achievable asymmetric discrete density evolution method is proposed, which is based on offset min-sum decoding. First, the iterative formula of low-complexity asymmetric channel discrete density evolution is derived by means of combinatorics, difference method and the binomial theorem. The iterative formula is applied to further optimize the design of the LDPC codes using the offset min-sum decoding, and the optimal degree distributions and performance of the LDPC codes under different quantization ranges and step sizes are obtained under the asymmetric channel. Finally, the optimal offset values under high-precision quantization are obtained. The results show that under the asymmetric channel, the complexity of the proposed asymmetric density evolution method based on the offset min-sum decoding is reduced by an order of magnitude than that using the original method. For a fixed code rate, the smaller the variable node degree of the regular codes, the better the performance and the smaller the optimal offset value. For irregular LDPC codes, the optimal degree distributions and offset values are determined according to the differential evolution algorithm. For a fixed code rate, the larger the maximum variable node degree, the better the performance of irregular codes.

DOI
10.1360/ssi-2022-0250

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To design low-density parity-check (LDPC) codes with asymmetric channels and analyze their performance, an achievable asymmetric discrete density evolution method is proposed, which is based on offset min-sum decoding. First, the iterative formula of low-complexity asymmetric channel discrete density evolution is derived by means of combinatorics, difference method and the binomial theorem. The iterative formula is applied to further optimize the design of the LDPC codes using the offset min-sum decoding, and the optimal degree distributions and performance of the LDPC codes under different quantization ranges and step sizes are obtained under the asymmetric channel. Finally, the optimal offset values under high-precision quantization are obtained. The results show that under the asymmetric channel, the complexity of the proposed asymmetric density evolution method based on the offset min-sum decoding is reduced by an order of magnitude than that using the original method. For a fixed code rate, the smaller the variable node degree of the regular codes, the better the performance and the smaller the optimal offset value. For irregular LDPC codes, the optimal degree distributions and offset values are determined according to the differential evolution algorithm. For a fixed code rate, the larger the maximum variable node degree, the better the performance of irregular codes.
This paper analyzes the performance of quantized min-sum decoding algorithms for irregular low-density parity-check (LDPC) codes. For regular LDPC codes, it is known that the normalized or offset min-sum decoding algorithm with quantization bits less than 6 bits achieves good performances over wide range of signal-to-noise ratios (SNR). However, finite precision effects in decoding irregular LDPC codes are different from that in decoding regular LDPC codes which is caused by the difference of convergence speeds between low degree nodes and high degree nodes. This paper proposes a novel method to improve the performance of the conventional normalized or offset min-sum decoding algorithm when it is approximated with finite precision for hardware implementations. The proposed method applies down-scaling factors to intrinsic information which has effects on increasing the reliability of extrinsic information at variable nodes and compensating the quantization errors caused by finite precision. Computer simulation results for irregular LDPC codes show that the proposed normalized and offset min-sum decoding algorithms achieve better performances at high SNR compared to the conventional normalized and offset min-sum algorithms under (6:2) quantization scheme.
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.
In this paper a modified Offset Min-Sum decoding algorithm for Low-Density Parity Check Codes is presented. In this modified algorithm, the offset factor in the original Offset Min-Sum algorithm is adjusted iteratively with the assistance of the check-node computation outputs of the Normalized Min-Sum algorithm. As a result, the offset factor in our modified algorithm can be calculated adaptively and more efficiently according to the previous decoding procedure, which can minimize the decoding degradation compared with the Belief Propagation decoding algorithm. The simulation results show that, compared with the original Offset Min-Sum decoding algorithm, our modified algorithm can achieve noticeable performance improvement with minor extra hardware complexity. For example, when BER is 10−5, our algorithm can achieve 0.1dB and 0.2dB decoding gain over Offset Min-Sum algorithm for regular and irregular LDPC codes respectively.
In this paper, we analyze the performance of quantized offset min-sum (MS) decoding algorithm and propose an optimally quantized offset MS algorithm for a flexible low-density parity-check (LDPC) decoder. It is known that the offset MS decoding algorithm is implemented with simplified hardware complexity and achieves good decoding performance. However, the finite precision effects in decoding LDPC codes result in performance different from floating point. The performance degradation is caused by different dynamic ranges of input data at high signal-to-noise ratio (SNR). The proposed offset MS algorithm uses the received data directly instead of log-likelihood ratio (LLR) data as the intrinsic information. It can achieve better performance than the conventional one since its offset factor is more effective at a wide range of SNR and the intrinsic information is quantized more robustly since it is independent of channel information. Also, it is possible for the proposed scheme to use a same quantization scheme for a flexible LDPC decoder, which can decode several kinds of LDPC codes. Simulation results show that our optimally quantized offset MS algorithms with 5-bits for (1728, 864) and (1728, 1296) irregular LDPC codes achieve better performance compared with the conventional offset MS algorithms with 6-bits quantization scheme.
A statistical-based density evolution algorithm is proposed for nonbinary low-density parity-check (LDPC) codes. It is applicable to both regular and irregular codes under study. The algorithm proposed serves as a tool to analyze the performance limit for iterative decoding of nonbinary LDPC codes, which in turn guides code design. Specifically, it provides approximated evaluations of convergence thresholds for codes given specific degree distributions. It is shown that, for a class of quasi-cyclic (QC) structured extended irregular repeat-accumulate (SeIRA) nonbinary LDPC codes, degree distributions can be optimized via the algorithm proposed. Simulation results exhibit that the nonbinary LDPC codes designed outperform the optimized binary LDPC codes on the AWGN channel with similar code length and rate in terms of bits. Performance gap between the binary and nonbinary LDPC codes is larger in the scenario of higher order modulation.