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IEEE Transactions on Broadcasting

Variable LLR Scaling in Min-Sum Decoding for Irregular LDPC Codes

Yin Xu, Leszek Szczeciński, Bo Rong, Fabrice Labeau, Dazhi He, Yiyan Wu, Wenjun Zhang

Published Nov 12, 2014• 26 citations
Publisher

Abstract

Min-sum decoding is a low-complexity alternative to the so-called belief propagation and consists in simplification of the nonlinear operation on the log likelihood ratios (LLRs) in the check nodes. The resulting suboptimality may be tempered via appropriate scaling of the LLRs, e.g., the fixed optimal scaling in the normalized min-sum algorithm, and variable scaling algorithms gradually appearing in the literature. However, up to now, none of the papers studied variable scaling both as per iteration and as per different check node degree, due to the prohibitive complexity of multioptimization over space of too many parameters. In this paper, we propose a generalized mutual information (GMI) of LLRs as the criterion to search for the scaling factors for different check node degrees in every iteration in a 1-D thus low-complexity manner. This approach is first analyzed via density evolution, and in addition can be extended to practical LLRs based formulas via Monte Carlo tools to cope with the mismatch issue. Bit error rate simulation results on two low-density parity-check codes show that our proposed GMI metrics have a noticeable gain over the variable scaling schemes that appeared in the literature.

DOI
10.1109/tbc.2014.2364532

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Min-sum decoding is a low-complexity alternative to the so-called belief propagation and consists in simplification of the nonlinear operation on the log likelihood ratios (LLRs) in the check nodes. The resulting suboptimality may be tempered via appropriate scaling of the LLRs, e.g., the fixed optimal scaling in the normalized min-sum algorithm, and variable scaling algorithms gradually appearing in the literature. However, up to now, none of the papers studied variable scaling both as per iteration and as per different check node degree, due to the prohibitive complexity of multioptimization over space of too many parameters. In this paper, we propose a generalized mutual information (GMI) of LLRs as the criterion to search for the scaling factors for different check node degrees in every iteration in a 1-D thus low-complexity manner. This approach is first analyzed via density evolution, and in addition can be extended to practical LLRs based formulas via Monte Carlo tools to cope with the mismatch issue. Bit error rate simulation results on two low-density parity-check codes show that our proposed GMI metrics have a noticeable gain over the variable scaling schemes that appeared in the literature.
Min Sum algorithm simplifies the non-linear check node operation of Belief Propagation algorithm via linear approximation, which greatly reduces the complexity of realization of decoder but degrades the performance as well. The resulting sub-optimality could be tempered via scaling of LLRs, e.g. fixed optimal scaling applied to Min Sum output resulting in the Normalized Min Sum algorithm, and variable scaling schemes gradually appear in literature. In this paper, we study the variable scaling decoding algorithm, and propose to generate variable scaling sequences via generalized mutual information (GMI) metric. Simulation results on real LDPC codes for different decoding algorithms have shown that our GMI metric performs better than the variable scaling scheme appearing in literature, and meanwhile improves substantially in terms of BER over the conventional Normalized Min Sum algorithm.
Min-Sum decoding (MS) is an alternative to belief propagation decoding with substantially lower complexity. MS often results in an overestimation of the log likelihood ratio (LLR) in particular in the early stage of the iterative process. A linear post-processing is usually performed as a compensation. With regular low density parity check codes (LDPC), a fixed scaling of the LLR yields sufficiently good results. In contrast, adaptive strategies are mandatory with irregular codes. It is well known that the scaling factor is an increasing function of the reliability of the LLR. In most of the publications, the scaling factor is envisioned as a function of both the iteration number and the signal-to-noise ratio. It is proposed here to use the mutual information between extrinsics as a measure of the reliability of the LLR. A practical implementation is derived with reasonable complexity. Compared to the literature, the proposed method yields slightly better results in terms of BER and a significant reduction in the number of iterations.
LDPC was first proposed in 1962, after that considerable achievements have been made both in decoding algorithms and also in the structure of the decoder. In terms of decoding algorithm, from belief propagation (BP) to min-sum (MS), then normalized MS (NMS) and finally to a more performance oriented variable MS (VMS) algorithm. In terms of the decoder structure, there are mainly two, flooding and shuffled structure (FS and SS), and flooding structure is mainly used in the past and the selection of variable factors in FS VMS has already been studied in the literature. As is known, the horizontal shuffled structure (HSS) are more hardware friendly and now widely deployed, thus to apply VMS algorithm to HSS has an important meaning for the industry. However, there is no effective method for the selection of variable factors in HSS VMS. Based on the study of FS VMS, this paper describes how to apply the generalized mutual information (GMI) based metric to HSS VMS reasonably and proposes the modified GMI based formula. Simulation results on a certain DVB-S2's LDPC code show that, by using modified formula for HSS VMS, we can obtain 0.13dB gain over HSS NMS and the performance is only 0.1dB away from that of HSS BP in terms of bit error rate (BER) at level 1e-7. And simulation results on a certain DVB-T2's LDPC code show that, by using modified formula for HSS VMS, we can obtain 0.04dB gain over HSS NMS and the performance is only 0.06dB away from that of HSS BP in terms of bit error rate (BER) at level 1e-7.
In this paper, we propose a novel low complexity scaling strategy of min-sum decoding algorithm for irregular LDPC codes. In the proposed method, we generalize our previously proposed simplified Variable Scaled Min-Sum (SVS-min-sum) by replacing the sub-optimal starting value and heuristic update for the scaling factor sequence by optimized values. Density evolution and Nelder-Mead optimization are used offline, prior to the decoding, to obtain the optimal starting point and per iteration updating step size for the scaling factor sequence of the proposed scaling strategy. The optimization of these parameters proves to be of noticeable positive impact on the decoding performance. We used different DVB-T2 LDPC codes in our simulation. Simulation results show the superior performance (in both WER and latency) of the proposed algorithm to other Min-Sum based algorithms. In addition to that, generalized SVS-min-sum algorithm has very close performance to LLR-SPA with much lower 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.
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.