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IEEE Journal on Selected Areas in Communications

Randomly Punctured LDPC Codes

David G. M. Mitchell, Michael Lentmaier, Alí Emre Pusane, Daniel J. Costello

Published Dec 11, 2015• 48 citations
Publisher

Abstract

In this paper, we present a random puncturing analysis of low-density parity-check (LDPC) code ensembles. We derive a simple analytic expression for the iterative belief propagation (BP) decoding threshold of a randomly punctured LDPC code ensemble on the binary erasure channel (BEC) and show that, with respect to the BP threshold, the strength and suitability of an LDPC code ensemble for random puncturing is completely determined by a single constant that depends only on the rate and the BP threshold of the mother code ensemble. We then provide an efficient way to accurately predict BP thresholds of randomly punctured LDPC code ensembles on the binary-input additive white Gaussian noise channel (BI-AWGNC), given only the BP threshold of the mother code ensemble on the BEC and the design rate, and we show how the prediction can be improved with knowledge of the BI-AWGNC threshold. We also perform an asymptotic minimum distance analysis of randomly punctured code ensembles and present simulation results that confirm the robust decoding performance promised by the asymptotic results. Protograph-based LDPC block code and spatially coupled LDPC code ensembles are used throughout as examples to demonstrate the results.

DOI
10.1109/jsac.2015.2507758

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In this paper, we present a random puncturing analysis of low-density parity-check (LDPC) code ensembles. We derive a simple analytic expression for the iterative belief propagation (BP) decoding threshold of a randomly punctured LDPC code ensemble on the binary erasure channel (BEC) and show that, with respect to the BP threshold, the strength and suitability of an LDPC code ensemble for random puncturing is completely determined by a single constant that depends only on the rate and the BP threshold of the mother code ensemble. We then provide an efficient way to accurately predict BP thresholds of randomly punctured LDPC code ensembles on the binary-input additive white Gaussian noise channel (BI-AWGNC), given only the BP threshold of the mother code ensemble on the BEC and the design rate, and we show how the prediction can be improved with knowledge of the BI-AWGNC threshold. We also perform an asymptotic minimum distance analysis of randomly punctured code ensembles and present simulation results that confirm the robust decoding performance promised by the asymptotic results. Protograph-based LDPC block code and spatially coupled LDPC code ensembles are used throughout as examples to demonstrate the results.
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.
In this paper, we study rate-compatible puncturing of finite-length Low-Density Parity-Check (LDPC) codes. First, we derive simple and yet good bounds on the expected performance of punctured codes (constructed by random puncturing) over Binary Erasure Channel (BEC) as a function of the performance of their parent LDPC code. We then present a novel rate-compatible puncturing scheme that is very easy to implement. Our scheme uses the idea that a more uniform distribution of punctured bits across the Tanner graph results in punctured codes with better performance. Although the puncturing scheme tailored to regular codes is presented, it is also directly applicable to irregular parent ensembles. By simulations, the proposed rate-compatible puncturing scheme is shown to be superior to the existing puncturing methods for both regular and irregular LDPC codes over BEC and Additive White Gaussian Noise (AWGN) Channel.
This paper first introduces an improved decoding algorithm for low-density parity-check (LDPC) codes over binary-input-output-symmetric memoryless channels. Then some fundamental properties of punctured LDPC codes are presented. It is proved that for any ensemble of LDPC codes, there exists a puncturing threshold. It is then proved that for any rates R1and R2satisfying 0121to R2resulting in asymptotically good codes for all rates R1lesRlesR2. Specifically, this implies that rates arbitrarily close to one are achievable via puncturing. Bounds on the performance of punctured LDPC codes are also presented. It is also shown that punctured LDPC codes are as good as ordinary LDPC codes. For BEC and arbitrary positive numbers R121lesRlesR2is shown. Based on the above observations, a method is proposed to design good punctured LDPC codes over a broad range of rates. Finally, it is shown that the results of this paper may be used for the proof of the existence of the capacity-achieving LDPC codes over binary-input-output-symmetric memoryless channels
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.
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.
Communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding is considered. The average bit error probability of an irregular LDPC code ensemble after a fixed number of iterations converges to a limit, which is calculated via density evolution, as the blocklength n tends to infinity. The difference between the bit error probability with blocklength n and the large-blocklength limit behaves asymptotically like ¿/n, where the coefficient ¿ depends on the ensemble, the number of iterations and the erasure probability of the BEC. In, ¿ is calculated for regular ensembles. In this paper, ¿ for irregular expurgated ensembles is derived. It is demonstrated that convergence of numerical estimates of ¿ to the analytic result is significantly fast for irregular unexpurgated ensembles.