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IEEE International Symposium on Information Theory (ISIT)

Complexity-Optimized Irregular Decoders

Masoud Ardakani, Pirouz Zarrinkhat, Raman Yazdani

Published Jul 1, 2006• 3 citations
Publisher

Abstract

Irregular decoding of low-density parity-check codes, i.e., using different algorithms in one iteration of the decoding of a single word, is studied. We formulate density evolution for irregular decoders. Using a one-dimensional representation of density evolution, we then jointly optimize irregular codes and irregular soft decoders for minimizing the decoding complexity. More specifically, for a given set of soft algorithms, a given channel, and a given code-rate, we find an irregular code-decoder pair which is capable of achieving a desired error performance with minimal decoding complexity. Robustness of irregular decoders when there exist channel estimation errors is also shown via an example

DOI
10.1109/isit.2006.262017

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Irregular decoding of low-density parity-check codes, i.e., using different algorithms in one iteration of the decoding of a single word, is studied. We formulate density evolution for irregular decoders. Using a one-dimensional representation of density evolution, we then jointly optimize irregular codes and irregular soft decoders for minimizing the decoding complexity. More specifically, for a given set of soft algorithms, a given channel, and a given code-rate, we find an irregular code-decoder pair which is capable of achieving a desired error performance with minimal decoding complexity. Robustness of irregular decoders when there exist channel estimation errors is also shown via an example
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.
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.
Recent advances in coding theory have uncovered the previously forgotten power of Low-Density Parity-Check (LDPC) codes. Their popularity can be related to their relatively simple iterative decoders and their potential to achieve high performance close to shannon limit. These make them an attractive candidate for error correcting application in communication systems. In this thesis, we focus our research on the iterative decoding algorithms for Low-Density Parity-Check codes and present an improved decoding algorithm. First, the graph structure of LDPC codes is studied and a graph-based search algorithm to find the shortest closed walk and shortest cycle for each node of the graph is proposed. Then, the Deterministic schedule is applied on nodes of the graph with the objective of preserving the optimality of the algorithms. Finally, Hybrid Switch-Type technique is applied on the improved algorithms to provide a desirable complexity/performance trade-off. Hybrid Technique and Deterministic schedule are combined for decoding regular and irregular LDPC codes. The performance and complexity of the decoder is studied for Sum-Product and Gallager A algorithms. The result is a flexible decoder for any available LDPC code and any combination of decoding algorithms based on the communication systems need. In this technique, we benefit the high performance of soft-decision algorithms and low complexity of hard-decision algorithms by changing the decoding rule after a few iterations. Hence, a desirable performance can be obtained with less average number of soft iterations. Moreover, all the nodes do not update messages in each iteration. As a result, the total number of computations is reduced considerably
This paper deals with the irregular binary low-density parity-check (LDPC) codes and two iterative low-complexity decoding algorithms. The first one is the majority error-correcting decoding algorithm, and the second one is iterative erasure-correcting decoding algorithm. The lower bounds on correcting capabilities (the guaranteed corrected error and erasure fraction respectively) of irregular LDPC code under decoding (error and erasure correcting respectively) algorithms with low-complexity were represented. These lower bounds were obtained as a result of analysis of Tanner graph representation of irregular LDPC code. The numerical results, obtained at the end of the paper for proposed lower-bounds achieved similar results for the previously known best lower-bounds for regular LDPC codes and were represented for the first time for the irregular LDPC codes.
This paper presents a low complexity Min-Sum algorithm for decoding irregular Low-Density Parity-Check (LDPC) codes. In the proposed algorithm, the significant improvement in the error correcting performance without enhancing the hardware complexity is achieved by employing adaptive and optimized normalization factors for both the extrinsic information and log-likelihood ratio data bits respectively. The experimental results demonstrate that the proposed algorithm with 6-bit non-uniform quantization achieves reasonable decoding performance in terms of Frame Error Rate (FER) and Bit Error Rate (BER) without increasing the hardware complexity by avoiding finite word length effects of soft information.