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On the EXIT chart analysis of low-density parity-check codes

S.R. Kollu, Hamid Jafarkhani

Published Jan 1, 2005• 14 citations
Publisher

Abstract

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.

DOI
10.1109/glocom.2005.1577830

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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.
Low density parity check (LDPC) codes are known to achieve a performance very close to the Shannon capacity limit on Additive White Gaussian Noise (AWGN) and erasure channels. The extrinsic information transfer (EXIT) chart is a powerful method for analyzing iteratively decoded codes such as Turbo codes. EXIT charts were introduced as a method for representing how the mutual information between the decoder output and the transmitted bits changes over turbo decoding iterations. It is possible to apply the EXIT chart method to LDPC codes by treating the LDPC decoder as a concatenation of variable and check nodes. However the achieved convergence threshold values obtained with the EXIT charts are (plusmn0.1) dB away from the more precise results derived from the density evolution (DE) method. Subsequently, the EXIT trajectories method was proposed as an improved performance-analysis method for LDPC codes under belief propagation (BP) decoding and achieved more accurate convergence threshold values than the EXIT chart method. In this paper, the EXIT chart and the EXIT trajectories methods are proposed as analysis tools to compare between the BP, log likelihood and max log likelihood decoding algorithms for LDPC codes in terms of convergence thresholds. Furthermore, a comparison between EXIT chart, EXIT trajectories and DE methods is introduced for the mentioned LDPC codes. Simulations and numerical calculations on convergence thresholds for various ensembles of (dv, dc) regular LDPC codes for binary input AWGN channels are performed which results that the BP is the best decoding algorithm for LDPC code followed by the log likelihood and the max log likelihood algorithms, respectively. Also, the EXIT trajectory method yields more accurate results than EXIT charts which come very close to the DE method, while being more convenient numerically as a code research tool than the latter which require many Fourier transform operations.
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.
In this paper, we propose the concept of extrinsic information transfer (EXIT) trajectories, and show how to use it to analyze convergence performance of belief propagation (BP) decoding for irregular low-density parity-check (LDPC) codes. The proposed analysis method utilizes the EXIT characteristics in an improved way, and allows us to calculate the convergence threshold quickly and accurately. On the binary phase-shift keying (BPSK) modulated additive white Gaussian noise (AWGN) channel, the achieved threshold values for the enumerated irregular LDPC codes have error to those computed by density evolution only about 0.03 dB.
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.
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.