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IEEE Transactions on Communications

Hybrid Hard-Decision Iterative Decoding of Irregular Low-Density Parity-Check Codes

Pirouz Zarrinkhat, Amir H. Banihashemi

Published Feb 1, 2007• 6 citations
Publisher

Abstract

Time-invariant hybrid (HscrTI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on HscrTIalgorithms with majority-based (MB) binary message-passing constituents, we use density evolution (DE) and finite-length simulation to analyze the performance and the convergence properties of these algorithms over (memoryless) binary symmetric channels. To apply DE, we generalize degree distributions to have the irregularity of both the code and the decoding algorithm embedded in them. A tight upper bound on the threshold of MB HscrTIalgorithms is derived, and it is proven that the asymptotic error probability for these algorithms tends to zero, at least exponentially, with the number of iterations. We devise optimal MB HscrTIalgorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB HscrTIalgorithms are more robust and can better cope with unknown channel conditions, and thus can be practically more attractive.

DOI
10.1109/tcomm.2006.888584

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Time-invariant hybrid (HscrTI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on HscrTIalgorithms with majority-based (MB) binary message-passing constituents, we use density evolution (DE) and finite-length simulation to analyze the performance and the convergence properties of these algorithms over (memoryless) binary symmetric channels. To apply DE, we generalize degree distributions to have the irregularity of both the code and the decoding algorithm embedded in them. A tight upper bound on the threshold of MB HscrTIalgorithms is derived, and it is proven that the asymptotic error probability for these algorithms tends to zero, at least exponentially, with the number of iterations. We devise optimal MB HscrTIalgorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB HscrTIalgorithms are more robust and can better cope with unknown channel conditions, and thus can be practically more attractive.
Time-invariant hybrid (HTI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on HTIalgorithms with majority-based (MB) binary message-passing constituents, we use density evolution and finite-length simulation to analyze the performance and the convergence properties of these algorithms. Tight upper bounds on the threshold of MB HTIalgorithms are derived, and it is proven that the asymptotic error probability for these algorithms tends to zero at least exponentially with the number of iterations. We devise optimal MB HTIalgorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB HTIalgorithms are more robust, and can better cope with unknown channel conditions, and thus can be practically more attractive
Hybrid decoding means to combine different iterative decoding algorithms with the aim of improving error performance or decoding complexity. In this work, we introduce "time-invariant" hybrid (H/sub TI/) algorithms, and using density evolution show that for regular low-density parity-check (LDPC) codes and binary message-passing algorithms, H/sub TI/ algorithms perform remarkably better than their constituent algorithms. We also show that compared to "switch-type" hybrid (H/sub ST/) algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different iterative decoding algorithms, H/sub TI/ algorithms are far less sensitive to channel conditions and thus can be practically more attractive.
Hybrid decoding is to combine different iterative decoding algorithms with the aim of improving error performance or decoding complexity. In this work, we introduce "time-invariant" hybrid (HTI) algorithms, and show that for regular low-density parity-check codes and binary message-passing algorithms, HTIalgorithms perform remarkably better than their constituent algorithms. We also show that compared to "switch-type" hybrid algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different iterative decoding algorithms, HTIalgorithms are far less sensitive to channel conditions and thus can be practically more attractive.
This work presents a detailed study of a family of binary message-passing decoding algorithms for low-density parity-check (LDPC) codes, referred to as "majority-based algorithms." Both Gallager's algorithm A (G/sub A/) and the standard majority decoding algorithm belong to this family. These algorithms, which are, in fact, the building blocks of Gallager's algorithm B (G/sub B/), work based on a generalized majority-decision rule and are particularly attractive for their remarkably simple implementation. We investigate the dynamics of these algorithms using density evolution and compute their (noise) threshold values for regular LDPC codes over the binary symmetric channel. For certain ensembles of codes and certain orders of majority-based algorithms, we show that the threshold value can be characterized as the smallest positive root of a polynomial, and thus can be determined analytically. We also study the convergence properties of majority-based algorithms, including their (convergence) speed. Our analysis shows that the stand-alone version of some of these algorithms provides significantly better performance and/or convergence speed compared with G/sub A/. In particular, it is shown that for channel parameters below threshold, while for G/sub A/ the error probability converges to zero exponentially with iteration number, this convergence for other majority-based algorithms is super-exponential.
The optimal performance-complexity tradeoff for error-correcting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the minimization of decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallel-update schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsic-information transfer chart analysis of the code. A sufficient condition is presented for convexity of the complexity measure in the variable edge-degree distribution; when it is not satisfied, numerical experiments nevertheless suggest that the local minimum is unique. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform threshold-optimized codes at long block lengths, within the ensemble of irregular codes.
This correspondence presents the switch strategies for switch-type hybrid hard decision decoding algorithms for regular low-density parity-check (LDPC) codes. After the piecewise analysis of extrinsic information transfer functions for Gallager decoding algorithm B (GB), the normalized switch scheme (NSS), for which the majority-based algorithms and GB are two examples, is proposed. Then, several other examples of NSSs are presented and their convergence properties are analyzed based on the extrinsic information transfer (EXIT) functions. In simulations, the proposed NSSs show meaningful performance improvements and less sensitivity to channel parameter underestimations compared with GB for small and moderate block length codes.