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arXiv:1403.3583 · IEEE International Symposium on Information Theory (ISIT)

Threshold analysis of non-binary spatially-coupled LDPC codes with windowed decoding

Lai Wei, Toshiaki Koike–Akino, David G. M. Mitchell, Thomas E. Fuja, Daniel J. Costello

Published Jun 1, 2014• 19 citations
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Abstract

We study the iterative decoding threshold performance of non-binary spatially-coupled low-density parity-check (NB-SC-LDPC) code ensembles for both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC), with particular emphasis on windowed decoding (WD). We consider both (2, 4)-regular and (3, 6)-regular NB-SC-LDPC code ensembles constructed using protographs and compute their thresholds using protograph versions of NB density evolution and NB extrinsic information transfer analysis. For these code ensembles, we show that WD of NB-SC-LDPC codes, which provides a significant decrease in latency and complexity compared to decoding across the entire parity-check matrix, results in a negligible decrease in the near-capacity performance for a sufficiently large window size W on both the BEC and the BIAWGNC. Also, we show that NBSC-LDPC code ensembles exhibit gains in the WD threshold compared to the corresponding block code ensembles decoded across the entire parity-check matrix, and that the gains increase as the finite field size q increases. Moreover, from the viewpoint of decoding complexity, we see that (3, 6)-regular NB-SC-LDPC codes are particularly attractive due to the fact that they achieve near-capacity thresholds even for small q and W.

DOI
10.1109/isit.2014.6874959

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We study the iterative decoding threshold performance of non-binary spatially-coupled low-density parity-check (NB-SC-LDPC) code ensembles for both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC), with particular emphasis on windowed decoding (WD). We consider both (2, 4)-regular and (3, 6)-regular NB-SC-LDPC code ensembles constructed using protographs and compute their thresholds using protograph versions of NB density evolution and NB extrinsic information transfer analysis. For these code ensembles, we show that WD of NB-SC-LDPC codes, which provides a significant decrease in latency and complexity compared to decoding across the entire parity-check matrix, results in a negligible decrease in the near-capacity performance for a sufficiently large window size W on both the BEC and the BIAWGNC. Also, we show that NBSC-LDPC code ensembles exhibit gains in the WD threshold compared to the corresponding block code ensembles decoded across the entire parity-check matrix, and that the gains increase as the finite field size q increases. Moreover, from the viewpoint of decoding complexity, we see that (3, 6)-regular NB-SC-LDPC codes are particularly attractive due to the fact that they achieve near-capacity thresholds even for small q and W.
In this paper, we compare the finite-length performance of non-binary spatially coupled low-density parity-check (NB SC-LDPC) codes constructed from protographs to non-binary LDPC block codes (NB LDPC-BCs). A sliding window decoding architecture with a stopping rule based on a soft bit-error-rate (BER) estimate for the NB SC-LDPC codes is considered. It is demonstrated that NB SC-LDPC codes with sliding window decoding outperform NB LDPC-BCs with no increase in decoding complexity when the decoding latency of the SC-LDPC codes equals the block length of the LDPC-BCs. We also investigate the relationship between the protograph lifting factor, the decoding window size, and the decoding performance of NB SC-LDPC codes when the decoding latency is fixed. Simulation results for several (3,6)-regular NB code examples confirm that NB SC-LDPC codes can significantly outperform both binary LDPC-BCs and binary SC-LDPC codes with the same decoding latency.
Spatially coupled low-density parity-check block codes (SC-LDPC-BCs) are a class of LDPC codes in which certain ensembles achieve capacity under iterative belief propagation decoding. Prior work has considered these codes mostly from a protograph perspective. Following this viewpoint, our work presents new protographs for the design of protograph-based SC-LDPC-BCs. A reciprocal channel approximation (RCA) analysis of the terminated protographs yields the asymptotic performance limits of the ensembles in terms of their iterative decoding thresholds. Our protographs possess the properties of linear minimum distance growth rate and excellent thresholds. We consider the following channels in our work: binary-input additive white Gaussian noise channel (BI-AWGNC), binary erasure channel (BEC), and binary symmetric channel (BSC). Our work focuses on obtaining good designs, with two classes of ensembles being asymptotically regular, for all the three channels while minimizing rate loss.
The newest family of low-density parity-check (LDPC) codes, spatially-coupled (SC) codes, is shown to have several desirable characteristics including low implementation complexity and close-to-optimal performance over a range of channels. Furthermore, because of their ribbon-shaped parity-check matrices, window decoding can be used to decode these codes, which leads to low-delay implementations. Researchers have focused on asymptotically regular SC code ensembles and have examined several aspects of the code construction processes. In this paper, we concentrate on irregular SC code ensembles. We evaluate their decoding thresholds over the binary erasure channel and show that their performance is better than their regular SC counterparts. It is also shown that the gap between asymptotic coding thresholds of irregular SC ensembles and the fundamental Shannon limit gets negligibly small. For the sake of a better comparison, we have also evaluated the finite-length error performance of selected regular and irregular SC codes over the additive white Gaussian channel and it is also observed that finite-length error performance of these irregular SC codes outperforms regular SC codes. To further improve the error performance of these codes and to lower the possible error floors, progressive edge growth algorithm has also been considered in the finite-length performance analysis.
MacKay-Neal (MN) and Hsu-Anastasopoulos (HA) low-density parity-check (LDPC) codes are known to achieve the capacity of memoryless binary-input symmetric-output channels under maximum likelihood (ML) decoding with bounded col- umn and row weight in their associated parity-check matrices. Recently, Kasai and Sakaniwa showed that spatially coupled (SC) versions of the MN and HA LDPC codes have belief propagation (BP) iterative decoding thresholds that approach capacity on the binary erasure channel (BEC) as the coupling length increases. In this paper, we extend the results of Kasai and Sakaniwa to the additive white Gaussian noise (AWGN) channel and show that the thresholds of the SC-MN and SC-HA ensembles approach capacity with bounded density as the coupling length increases, i.e., the number of edges per information bit approaches a finite value as the estimated BP threshold approaches the Shannon limit. We also perform an asymptotic weight enumerator analysis and show that, provided the density parameters are chosen to be sufficiently large, the SC-MN and SC-HA ensembles are asymptotically good. Further, for certain selections of parameters, some of these ensembles are shown to have both excellent thresholds and good distance properties.
MacKay-Neal (MN) and Hsu-Anastasopoulos (HA) low-density parity-check (LDPC) codes are known to achieve the capacity of memoryless binary-input symmetric-output channels under maximum likelihood (ML) decoding with bounded col- umn and row weight in their associated parity-check matrices. Recently, Kasai and Sakaniwa showed that spatially coupled (SC) versions of the MN and HA LDPC codes have belief propagation (BP) iterative decoding thresholds that approach capacity on the binary erasure channel (BEC) as the coupling length increases. In this paper, we extend the results of Kasai and Sakaniwa to the additive white Gaussian noise (AWGN) channel and show that the thresholds of the SC-MN and SC-HA ensembles approach capacity with bounded density as the coupling length increases, i.e., the number of edges per information bit approaches a finite value as the estimated BP threshold approaches the Shannon limit. We also perform an asymptotic weight enumerator analysis and show that, provided the density parameters are chosen to be sufficiently large, the SC-MN and SC-HA ensembles are asymptotically good. Further, for certain selections of parameters, some of these ensembles are shown to have both excellent thresholds and good distance properties. (Less)
In this paper, we provide an efficient way to predict iterative belief propagation (BP) decoding thresholds of randomly punctured low-density parity-check (LDPC) code ensembles on the binary-input additive white Gaussian noise channel (AWGNC), given only the BP threshold of the mother code ensemble on the binary erasure channel (BEC) and the code design rate. We show that the predictions are accurate by comparing them with values calculated by discretized density evolution for a variety of puncturing fractions. We find that the strength and suitability of an LDPC code ensemble for random puncturing over the AWGNC with respect to iterative decoding threshold is completely determined by a single constant θ, and this behavior is demonstrated using both LDPC block code and spatially coupled LDPC code ensembles. Finally, we present simulation results that confirm the excellent decoding performance promised by the asymptotic results.