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IEEE International Symposium on Information Theory (ISIT)

On Generalized EXIT charts of LDPC code ensembles over binary-input output-symmetric memoryless channels

Hossein Mani, Hamid Saeedi, Ali Eslami, Hossein Pishro-Nik

Published Jul 1, 2012• 0 citations

Abstract

Generalized Extrinsic Information Transfer (GEXIT) charts were introduced as an extension of EXIT charts which have an extensive use in analysis and design of many iterative schemes including Low-Density Parity-Check (LDPC) codes. While a powerful as well as an insightful concept, their full potential as a designing tool for LDPC code ensembles has not been realized due to some missing steps. This papers aims at filling these gaps by proving some important properties of GEXIT charts and using them to design capacity-approaching LDPC code ensembles. The primary results on GEXIT charts are limited to regular variable and check node degrees. Moreover, variable node GEXIT curves have only been derived for the case where no physical channel is present. In a recent paper, GEXIT curves for irregular variable node and check node degree distributions have been derived. In this paper, we derive GEXIT charts of LDPC code ensembles over binary-input output-symmetric memoryless channels with any channel parameter. For the case of binary symmetric channel, we derive closed form expression for the GEXIT curve of variable nodes. We also propose to use an alternative representation of GEXIT charts in which we plot the inverse of variable node GEXIT curve together with dual GEXIT curve of the check node. We prove that the area theorem still holds in this case. Using these results, we analyze and design capacity-approaching LDPC codes using GEXIT charts.

DOI: 10.1109/isit.2012.6283930

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The extrinsic information transfer (EXIT) charts are used to analyze the sum-product decoding algorithm of low-density parity-check (LDPC) codes. The codes are optimized by performing curve fitting on EXIT charts. This technique is developed for binary erasure channels based on area property and is extended to other channels. We show that on channels where the area property is not valid, the curve fitting approach is a sub-optimal method of optimizing for the best convergence threshold. On Gaussian channels, the messages from variable nodes to check nodes are approximated by a symmetric Gaussian distribution. We prove that the check node detector mutual information transfer function derived using this approximation is an upper bound. So the EXIT chart analysis predicts an optimistic value for the convergence threshold. We give an empirical lower bound for the check node detector mutual information transfer function. For few irregular LDPC codes designed for AWGN channel using density evolution, we have examined the accuracy of convergence thresholds predicted by EXIT charts using these bounds for the check node transfer function. The analysis of EXIT chart technique presented in this paper can be extended to other channels for which the distribution of channel messages can be approximated by a symmetric Gaussian/Gaussian mixture distribution.

In this paper extrinsic information transfer (EXIT) charts are proposed to design non-binary low-density parity-check (LDPC) codes for the AWGN channel. A new metric is presented to describe the mutual information of the non-binary messages. The a priori information is modelled using a Gaussian mixture distribution. Analytical expressions are given for the EXIT curves of the variable and check node decoders for both regular and irregular LDPC codes. The analytical expressions are shown to agree well with simulation results. Finally, by matching the variable and check node EXIT curves, it is shown that good nonbinary LDPC can be designed for the AWGN channel.

This work proposes a tractable evaluation of the maximum a posteriori (MAP) threshold of sparse-graph ensembles, by using an approximation for the extended belief propagation generalized extrinsic information transfer (EBP-GEXIT) function, first proposed by Measson et al. The approximation allows to find a MAP threshold in such numerically involved cases as the binary-input additive white Gaussian noise (AWGN) channel, graph ensembles with general component codes and/or irregularities. The paper contains examples of estimations of the MAP thresholds in the case of irregular low-density parity-check (LDPC), generalized LDPC, and doubly generalized LDPC codes ensembles. Our estimations are confirmed by numerical simulations.

To efficiently design LDPC code ensembles that perform well in the presence of channel estimation error (also referred to as "SNR mismatch"), it is necessary to generate extrinsic information transfer (EXIT) charts. We have previously observed that the variable node (check node) EXIT curve in the presence of mismatch lies below (above) the original variable node curve in the absence of mismatch in such a way that it makes the tunnel between the two curves tighter. In this paper, we analytically prove this fact for variable node curves.

Non-binary low-density parity-check(LDPC) codes defined over the finite field GF(q=2p) are an extension of binary LDPC codes and have demonstrated improved performance over binary LDPC codes with the same frame length and rate.This paper introduces the principle of extrinsic information transfer(EXIT) charts,analyzes the convergence behavior of non-binary LDPC codes on AWGN channel,and confirms the thresholds of non-binary LDPC codes,which provides the foundation for analysis and design of real communication system using EXIT tool in future work.Simulation result shows that the performance of quaternary LDPC codes is better than that of binary LDPC codes in the same channel,and validates the superiority of using EXIT chart for iterative research.

Low-density parity-check (LDPC) codes are widely recognized for their excellent forward error correction, near-Shannon-limit performance, and support for high data rates with effective hardware parallelization. Their convolutional counterpart, LDPC convolutional codes (LDPC-CCs), offer additional advantages such as variable codeword lengths, unlimited parity-check matrices, and simpler encoding and decoding. These features make LDPC-CCs particularly suitable for practical implementations with varying channel conditions and data frame sizes. This paper investigates the performance of LDPC-CCs using the extrinsic information transfer (EXIT) chart, a graphical tool for analyzing iterative decoding. EXIT charts visualize mutual information exchange and help predict convergence behavior, estimate performance thresholds, and optimize code design. Starting with the EXIT chart principles for LDPC codes, we derived the mutual information functions for variable and check nodes in regular and irregular LDPC-CC tanner graphs. This involved adapting existing EXIT functions to the periodic parity-check matrix of LDPC-CCs. We compare regular and irregular LDPC-CC constructions, examining the impact of degree distributions and the number of periods in the parity-check matrix on convergence behavior. Our simulations show that irregular LDPC-CCs consistently outperform regular ones, and the EXIT chart analysis confirms that LDPC-CCs demonstrate superior bit error rate (BER) performance compared to equivalent LDPC block codes.