This short paper explores density evolution (DE) for low-density parity-check (LDPC) codes at signal-to-noise-ratios (SNRs) that are significantly above the decoding threshold. The focus is on the additive white Gaussian noise channel and LDPC codes in which the variable nodes have regular degree. Prior work, using DE, produced results in the error floor region which were asymptotic in the belief-propagation decoder's log-likelihood ratio (LLR) values. We develop expressions which closely approximate the LLR growth behavior at moderate LLR magnitudes. We then produce bounds on the mean extrinsic check-node LLR values required, as a function of SNR, such that the growth rate of the LLRs exceeds that of a particular trapping set's internal LLRs such that its error floor contribution may be eliminated. We find that our predictions for the mean LLRs to be accurate in the error floor region, but the predictions for the LLR variance to be lacking beyond several initial iterations.
This short paper explores density evolution (DE) for low-density parity-check (LDPC) codes at signal-to-noise-ratios (SNRs) that are significantly above the decoding threshold. The focus is on the additive white Gaussian noise channel and LDPC codes in which the variable nodes have regular degree. Prior work, using DE, produced results in the error floor region which were asymptotic in the belief-propagation decoder's log-likelihood ratio (LLR) values. We develop expressions which closely approximate the LLR growth behavior at moderate LLR magnitudes. We then produce bounds on the mean extrinsic check-node LLR values required, as a function of SNR, such that the growth rate of the LLRs exceeds that of a particular trapping set's internal LLRs such that its error floor contribution may be eliminated. We find that our predictions for the mean LLRs to be accurate in the error floor region, but the predictions for the LLR variance to be lacking beyond several initial iterations.
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.
In this paper, we investigate the performance of the belief propagation (BP) algorithm for decoding low-density parity-check codes over the additive white Gaussian noise channel when there is an incorrect estimate of the channel signal-to-noise ratio (SNR) (referred to as "SNR mismatch") at the decoder. At the extremes for over- and underestimation of SNR, the performance of BP tends to that of min-sum algorithm and the channel bit-error rate, respectively. Our results for regular codes indicate that the sensitivity to mismatch increases by increasing the variable-node degree and by decreasing the check-node degree. The effect of variable-node degree, however, appears to be more profound, such that at a given rate, the codes with the smallest variable and check degrees are more robust against SNR mismatch. For irregular codes, by comparing the thresholds of a few ensembles, we demonstrate that the ensemble which performs better in the absence of mismatch can perform worse in the presence of it. To obtain our asymptotic results, we propose a computationally efficient method based on the Gaussian approximation of density evolution in the presence of SNR mismatch. We also show that the asymptotic results are consistent with simulation results for codes with finite block lengths
Density evolution (DE) is a technique for tracking the distribution of the log likelihood ratio (LLR) messages exchanged between the variable nodes and the check nodes in a bipartite graph. It is widely assumed that these distributions are close to Gaussian. However, in many scenarios, this assumption is not valid, e.g., the case that the signal to noise ratio (SNR) is low, or the degree of variable nodes exceeds a certain threshold. This article introduces a new (suboptimal) method for DE algorithm in low-density parity-check (LDPC) codes. We provide a more accurate model for the distribution of message bits (as compared to Gaussian) through matching the first n statistical moments. An iterative message passing algorithm is proposed to compute these moments from the graphical representation of the underlying code. We show that the proposed algorithm results in an improved estimate of the underlying EXIT chart as compared to using a Gaussian assumption. In this respect, the proposed method achieves a performance very close to that of the best earlier methods, while it offers a much lower complexity.
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.
It is well-known that the performance of low-density parity-check codes is compromised by the emergence of an error floor, which is caused by so-called trapping sets. By identifying the dominant trapping sets, this error floor can be estimated analytically using a linear model. In this paper, this linear approach is improved to make the analysis more accurate. Then, guided by the error probability formula, a simple but effective method to improve the code performance in the error floor region is proposed, by introducing a boosting factor to the log-likelihood ratios returned from unsatisfied check nodes during the early decoding stages. It is shown that the effect of the dominant trapping sets can thus be reduced. The dominance of trapping sets can be further reduced by extending the computational range of the LLRs at the decoder. To illustrate these ideas, a short regular LDPC code constructed by Tanner, as well as the IEEE 802.3 LDPC code, are studied.
The goal of error control coding is to encode information in such a way, that in the event that errors occur during transmission over a noisy communication channel or during storage in an unreliable memory, the receiver can correct the errors and recover the original transmitted information. Low-density parity-check (LDPC) codes are a kind of high performance linear block code, which are already used in many recent communication systems. Information rates guaranteed by these codes approach the theoretical Shannon capacity limit. In addition, LDPC codes are now widely used in practice and included in many communication standards. Therefore study of these codes is very important and practical approaches to their design and decoding techniques are of great interest. There exist very good decoding algorithms for LDPC codes with respect to different channel models. However, LDPC codes suffer from the infamous error floor problem at high signal-to-noise ratios. This problem is attributed to the trapping sets by Richardson. It is shown that the dominant trapping sets of regular LDPC codes, so called absorption sets, undergo a two-phase dynamic behavior in the iterative message-passing decoding algorithm. We present a linear dynamic model for the iteration behavior of these sets. It can be seen that they undergo an initial geometric growth phase which stabilizes in a final bit-flipping behavior where the algorithm reaches a fixed point. Our analysis is shown to lead to very accurate numerical calculations of the error floor bit error rates down to error rates that are inaccessible by simulation. The topologies of the dominant absorption sets of two example codes, the IEEE 802.3an [2048,1723] regular (6,32) LDPC code and the Tanner [155,64,20] regular (3,5) LDPC code, are identified and tabulated by using topological features in combination with search algorithms, respectively. To make our analysis more complete, we provide more solid evidence showing that trapping sets are equivalent to absorption sets. In other words, we argue that absorption sets characterize all failure mechanisms. Some insights from our linear analysis can be borrowed to approach this problem. Another insight that this formula provides to us is the means to reduce the error floor, which is another important part of what we hoped to achieve by developing the formula in the first place. By allowing the log-likelihood ratios (LLRs) utilized by the message-passing decoding to grow bigger in precision length, the absorption sets as a trouble-causing structure will be successfully corrected. Therefore, the absorption sets, generically born with the code design, will no longer threaten the error correcting performance as long as the messages have enough room and time to grow. This translates to greater LLR clipping thresholds and more iterations, both of which are preset at the decoder. However, the actual settings are dependent on how much the subgraph of the dominant absorption set resembles that of a non-zero minimum-weight codeword, therefore the settings are empirical. In general more likeness in their topologies, the more effort it costs to correct the absorption set.