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International Symposium on Turbo Codes and Iterative Information Processing

A linear model for the iterative decoding of random LDPC codes in the high SNR region

Kuntal Deka, A. Rajesh, Prabin Kumar Bora

Published Sep 1, 2016• 0 citations
Publisher

Abstract

This paper presents a linear model for the sum-product decoding of random low-density parity-check (LDPC) codes, specifically in the high signal-to-noise ratio (SNR) region. The sum-product algorithm is first approximated as the min-sum algorithm and then a linear model is estimated for the latter. To formulate the model, the dominant trapping sets of the particular code are found. We consider a criterion based on the channel log-likelihood-ratio values to estimate the linear model. The bit error rate performance predicted by the linear model is compared with that obtained through importance sampling. This confirms that the linear model can accurately predict the performance of LDPC codes in the high SNR region.

DOI
10.1109/istc.2016.7593100

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arXiv:1202.2826•Feb 13, 2012•3 citations
This paper presents a linear model for the sum-product decoding of random low-density parity-check (LDPC) codes, specifically in the high signal-to-noise ratio (SNR) region. The sum-product algorithm is first approximated as the min-sum algorithm and then a linear model is estimated for the latter. To formulate the model, the dominant trapping sets of the particular code are found. We consider a criterion based on the channel log-likelihood-ratio values to estimate the linear model. The bit error rate performance predicted by the linear model is compared with that obtained through importance sampling. This confirms that the linear model can accurately predict the performance of LDPC codes in the high SNR region.
Many classes of high-performance Low-density parity check codes are based on parity check matrices composed of permutation sub matrices. The emulation-simulation framework further allows the algorithm and implementation to be iteratively redefined to improve the error floor performance of message passing decoder. Log-Like hood-Ratio (LLR) based Belief-Propagation (BP) algorithm is presented for Low Density Parity Check codes. Numerically accurate representation of check node update computation used in LLR-BP decoding is described. The implementation of Sum-Product algorithm (SPA) within Low Density Parity Check Code (LDPC) decoder is described in this paper and the correction term is used to improve the decoding performance of min-sum algorithm (MSA). Quantization and log-tanh function approximation in sum-product decoder strongly affect which absorbing set dominates in error floor region. For LDPC decoder, bit error rate (BER) decreases with increase in the signal to noise ratio.
Low-density parity-check (LDPC) codes achieve outstanding performance by the sum-product decoder. Its performance approaches the maximum-likelihood (ML) decoding performance at high SNR region. Upper bounds on the error probability of the ML decoded LDPC codes can be computed using the weight distribution. Recently, X.-Y. Hu et al. presented an approximate algorithm for estimating the weight distribution of LDPC codes. However, the weight distribution computed by their method depends on the iterative decoding algorithm, and is not always estimated exactly. In this paper, we propose a probabilistic method for computing the low-part weight distribution of LDPC codes. Using our method, the approximate weight distribution of LDPC codes can be computed with high accuracy
The error-correcting performance of low-density parity check (LDPC) codes, when decoded using practical iterative decoding algorithms, is known to be close to Shannon limits for codes with suitably large blocklengths. A substantial limitation to the use of finite-length LDPC codes is the presence of an error floor in the low frame error rate (FER) region. This paper develops a deterministic method of predicting error floors, based on high signal-to-noise ratio (SNR) asymptotics, applied to absorbing sets within structured LDPC codes. The approach is illustrated using a class of array-based LDPC codes, taken as exemplars of high-performance structured LDPC codes. The results are in very good agreement with a stochastic method based on importance sampling which, in turn, matches the hardware-based experimental results. The importance sampling scheme uses a mean-shifted version of the original Gaussian density, appropriately centered between a codeword and a dominant absorbing set, to produce an unbiased estimator of the FER with substantial computational savings over a standard Monte Carlo estimator. Our deterministic estimates are guaranteed to be a lower bound to the error probability in the high SNR regime, and extend the prediction of the error probability to as low as 10-30. By adopting a channel-independent viewpoint, the usefulness of these results is demonstrated for both the standard Gaussian channel and a channel with mixture noise.
This paper addresses the prediction of error floors of variable-regular Low Density Parity Check (LDPC) codes in the Additive White Gaussian Noise (AWGN) channel. Specifically, we focus on the Sum-Product Algorithm (SPA) decoder in the log-domain at high SNRs. We hypothesize that several published error floor levels are due to numerical saturation within their decoders when handling high SNRs. We take care to develop a log-domain SPA decoder that does not saturate near-certain messages and find the error rates of our decoder to be lower by at least several orders of magnitude. We study the behavior of near-codewords / trapping sets that dominate the reported error floors. J. Sun, in his Ph.D. thesis, used a linear system model to show that error floors due to elementary trapping sets don't exist under certain conditions, assuming that the SPA decoder is non-saturating [1]. We develop a refined linear model which we find to be capable of predicting the error floors caused by elementary trapping sets for saturating decoders. Performance results of several codes at several levels of decoder saturation are presented.
This paper proposes a new min-sum algorithm for low-density parity-check decoding. In this paper, we first define the negative and positive effects of the received signal-to-noise ratio (SNR) in the min-sum decoding algorithm. To improve the performance of error correction by considering the negative and positive effects of the received SNR, the proposed algorithm applies adaptive scaling factors not only to extrinsic information but also to a received log-likelihood ratio. We also propose a combined variable and check node architecture to realize the proposed algorithm with low complexity. The simulation results show that the proposed algorithm achieves up to 0.4 dB coding gain with low complexity compared to existing min-sum-based algorithms.
This paper addresses the prediction of error floors of low-density parity-check (LDPC) codes with variable nodes of constant degree in the additive white Gaussian noise (AWGN) channel. Specifically, we focus on the performance of the sum-product algorithm (SPA) decoder formulated in the log-likelihood ratio (LLR) domain. We hypothesize that several published error floor levels are due to the manner in which decoder implementations handled the LLRs at high SNRs. We employ an LLR-domain SPA decoder that does not saturate near-certain messages and find the error rates of our decoder to be lower by at least several orders of magnitude. We study the behavior of trapping sets (or near-codewords) that are the dominant cause of the reported error floors. We develop a refined linear model, based on the work of Sun and others, that accurately predicts error floors caused by elementary tapping sets for saturating decoders. Performance results of several codes at several levels of decoder saturation are presented.