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arXiv:1410.4086 · IEEE International Black Sea Conference on Communications and Networking

Design of LDPC code ensembles with fast convergence properties

Ian P. Mulholland, Enrico Paolini, Mark F. Flanagan

Published May 1, 2015• 18 citations
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Abstract

The design of low-density parity-check (LDPC) code ensembles optimized for a finite number of decoder iterations is investigated. Our approach employs EXIT chart analysis and differential evolution to design such ensembles for the binary erasure channel and additive white Gaussian noise channel. The error rates of codes optimized for various numbers of decoder iterations are compared and it is seen that in the cases considered, the best performance for a given number of decoder iterations is achieved by codes which are optimized for this particular number. The design of generalized LDPC (GLDPC) codes is also considered, showing that these structures can offer better performance than LDPC codes for low-iteration-number designs. Finally, it is illustrated that LDPC codes which are optimized for a small number of iterations exhibit significant deviations in terms of degree distribution and weight enumerators with respect to LDPC codes returned by more conventional design tools.

DOI
10.1109/blackseacom.2015.7185085

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IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2007•5 citations
The design of low-density parity-check (LDPC) code ensembles optimized for a finite number of decoder iterations is investigated. Our approach employs EXIT chart analysis and differential evolution to design such ensembles for the binary erasure channel and additive white Gaussian noise channel. The error rates of codes optimized for various numbers of decoder iterations are compared and it is seen that in the cases considered, the best performance for a given number of decoder iterations is achieved by codes which are optimized for this particular number. The design of generalized LDPC (GLDPC) codes is also considered, showing that these structures can offer better performance than LDPC codes for low-iteration-number designs. Finally, it is illustrated that LDPC codes which are optimized for a small number of iterations exhibit significant deviations in terms of degree distribution and weight enumerators with respect to LDPC codes returned by more conventional design tools.
The design of low-density parity-check (LDPC) code ensembles optimized for a finite number of decoder iterations is investigated. Our approach employs EXIT chart analysis and differential evolution to design such ensembles for the binary erasure channel and additive white Gaussian noise channel. The error rates of codes optimized for various numbers of decoder iterations are compared and it is seen that in the cases considered, the best performance for a given number of decoder iterations is achieved by codes which are optimized for this particular number. The design of generalized LDPC (GLDPC) codes is also considered, showing that these structures can offer better performance than LDPC codes for low-iteration-number designs. Finally, it is illustrated that LDPC codes which are optimized for a small number of iterations exhibit significant deviations in terms of degree distribution and weight enumerators with respect to LDPC codes returned by more conventional design tools.
Low-density parity-check (LDPC) codes have received significant attention due to their near-Shannon-limit error performance. The belief propagation (BP) algorithm is the best-known decoding algorithm for LDPC codes. However it often needs several tens or hundreds of iterations to converge. Although fast convergence BP algorithms have been proposed, there is neither an easy tool to analyze their performance nor a standard VLSI architecture to implement them. Moreover, there exists a trade-off between waterfall (i.e., threshold) and error floor (i.e., minimum pseudo distance) performances for LDPC codes. For generalized LDPC (GLDPC) codes, better minimum distance properties can be achieved but at the expense of a rate loss. This research investigates performance analysis and hardware implementation of LDPC codes using fast convergence algorithms. Closed-form extrinsic mutual information transfer (EXIT) functions for several fast convergence BP decoding algorithms are proposed. It is shown theoretically that these algorithms converge faster than standard BP, while the corresponding thresholds remain unchanged. Since these algorithms are of serial nature, their parallelization without compromising convergence speed and error performance is also investigated. This result is useful in hardware implementation because in practice partly parallel decoding schemes are often used. This result also sheds some light on how to group nodes in order to preserve performance. The VLSI architecture of a particular decoding algorithm for LDPC codes is proposed. This approach can be adopted in many standards to achieve a large throughput. Doubly GLDPC (DGLDPC) codes are proposed to compensate the rate loss of GLDPC codes. EXIT charts are used to analyze DGLDPC codes and EXIT functions for variable component codes are thoroughly discussed. Based on EXIT charts, differential evolution is used to optimize their threshold. Ensemble weight enumerators are generalized to protograph-based DGLDPC codes. Simulation results show that DGLDPC codes constructed based on these results can improve the performance of their LDPC and GLDPC counterparts in both the waterfall region and the error floor region.
In this paper, we are concerned with the finite-length analysis of low-density parity-check (LDPC) codes when used over additive white Gaussian noise (AWGN) channels. We present the iterative decoding algorithm of LDPC codes, and we also discuss the encoding methods and the bipartite graph representation of LDPC codes ensemble. Finite-length LDPC code performances are investigated, and the influence of cycle is pointed out. Simulation results show that irregular LDPC codes with variable and check node degree distribution optimized for infinite-length codes are not optimal for finite-length LDPC codes, and the error patterns of both regular and irregular codes are investigated as an effort to improve the performance and design methodology of finite-length LDPC codes.
We derive a new fast convergent Density Evolution algorithm for finding optimal rate Low-Density Parity-Check (LDPC) codes used over the binary erasure channel (BEC). The fast convergence property comes from the modified Density Evolution (DE), a numerical method for analyzing the behavior of iterative decoding convergence of a LDPC code. We have used the method of [16] for designing of a LDPC code with optimal rate. This has been done for a given parity check node degree distribution, erasure probability and specified DE constraint. The fast behavior of DE and found optimal rate with this method compare with the previous DE constraint.
Existing design methods for irregular Low-Density Parity-Check (LDPC) codes over the additive white Gaussian noise channel are based on using asymptotic analysis tools such as density evolution in an optimization process. Such a process is computationally expensive particularly when a large number of constituent variable node degrees are involved in the design. In this paper, we propose a systematic approach for the design of irregular LDPC codes. The proposed method, which is based on a pre-computed upper bound on the fraction of edges connected to variable nodes of degree 3, is considerably less complex than the conventional optimization approach. Through a number of examples, we demonstrate that using our method, ensembles with performance very close to those devised based on optimization, can be designed. In addition to having very good performance, the number of constituent variable node degrees in the designed ensembles is only three or four. This, in some cases, is much smaller than the corresponding number for optimization-based designs with similar performance.
We generalize the results about how to compute iterative decoding thresholds over the binary erasure channel of non-binary LDPC code ensembles of [16] to non-binary TLDPC codes [2], [3]. We show in this case how density evolution can be performed in order to calculate iterative decoding thresholds and find several families with a very simple regular structure and thresholds close to the Shannon limit. To check the performances of these codes over other channels we have tested one of the simplest codes over F4which has rate 1/2 on the Gaussian channel. For the (binary) length 1008 for instance, without any optimization on the permutation structure of the code, it matches the performances of the best binary codes of the same length up to the word-error rate 10-3. We also notice that all LDPC codes (binary or not) having at least two symbols of degree 2 per parity-check equation can be represented as a special kind of TLDPC codes. We show that this representation and the associated decoding algorithm leads in the case of cycle codes to a significant reduction of the number of iterations which are needed for iterative decoding.