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IEEE Transactions on Communications

Analysis of low-density parity-check codes based on EXIT functions

E. Sharon, Alexei Ashikhmin, Simon N. Litsyn

Published Aug 1, 2006• 38 citations
Publisher

Abstract

We exploit extrinsic information transfer functions of single parity-check and repetition codes over the binary input additive white Gaussian noise (biAWGN) channel, derived by the authors, for asymptotic performance analysis of belief propagation decoding of low-density parity-check codes. The approach is based on a Gaussian approximation (GA) of the density evolution algorithm using the mutual information measure. We show that this method allows more accurate prediction of the decoding threshold in the biAWGN channel than the earlier known GA methods

DOI
10.1109/tcomm.2006.878828

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Analysis of belief-propagation decoding of LDPC codes over the biAWGN channel using improved Gaussian approximation based on the mutual information measure

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Summary form only given. The capacity of the message-passing decoder for LDPC codes can be computed by the density evolution algorithm by iteratively computing message densities. The infinite-dimensional problem of iteratively calculating the message densities in the case of the binary input AWGN channel and the belief propagation decoder can be simplified to a one-dimensional problem by using a Gaussian approximation. By assuming Gaussian densities the density evolution simplifies to updating means of Gaussian densities. An improved Gaussian approximation algorithm is suggested for computing the capacity of the BP decoder based on the mutual information measure. An analytical method for computing the mutual information of a transmitted bit with the message corresponding to it is proposed. Using this method we can approximate the non-Gaussian message by a Gaussian message that has the same mutual information with the transmitted bit. Computationally, the algorithm is similar to the Gaussian approximation algorithm proposed in Chung (2001). For various regular LDPC codes that were examined, the algorithm computed threshold values within 0.01dB or less from the exact threshold which is an improvement over the Gaussian approximation. Furthermore, additional insight on the convergence process can be gained from EXIT charts that can be derived from the algorithm. This can assist in designing better irregular LDPC codes.

IEEE Convention of Electrical and Electronics Engineers in Israel•Jun 26, 2003•4 citations
We exploit extrinsic information transfer functions of single parity-check and repetition codes over the binary input additive white Gaussian noise (biAWGN) channel, derived by the authors, for asymptotic performance analysis of belief propagation decoding of low-density parity-check codes. The approach is based on a Gaussian approximation (GA) of the density evolution algorithm using the mutual information measure. We show that this method allows more accurate prediction of the decoding threshold in the biAWGN channel than the earlier known GA methods
We exploit extrinsic information tranfer functions of single parity-check and prepetition codes over the binay input additive white Gaussian noise (biAWGN) channel, for asymptotic performance analysis of belief propagation decoding of low-density parity-check codes. The approach is based on a Gaussian approximation (GA) of the density evolution algorithm using the mutual information measure. We show that our method allows more accurate prediction of the decoding threshold in the biAWGN channel than the earlier known GA methods.
In this paper, we study the density evolution analysis of spatially coupled low-density parity-check (SC-LDPC) codes over binary input additive white Gaussian noise (BIAWGN) channels under the belief propagation (BP) decoding algorithm. Using reciprocal channel approximation and Gaussian approximation, we propose averaging techniques for the density evolution of SC-LDPC codes over BIAWGN channels. We show that the proposed techniques can closely predict the decoding threshold while offering reduced complexity compared to the existing multi-edge-type density evolution.
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.
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.
This letter presents a more accurate mathematical analysis, with respect to the one performed in Chung et al.'s 2001 paper, of belief-propagation decoding for low-density parity-check (LDPC) codes on memoryless binary input-additive white Gaussian noise channels, when considering a Gaussian approximation (GA) for message densities under density evolution. The recurrent sequence, defined in Chung et al.'s 2001 paper, describing the message passing between variable and check nodes follows from the GA approach and involves the function φ(x), therein defined, and its inverse. The analysis of this function is here resumed and studied in depth to obtain tighter upper and lower bounds on it. Moreover, unlike the upper bound given in the above cited paper, the tighter upper bound on φ(x) is invertible. This allows a more accurate evaluation of the asymptotical performance of sum-product decoding of LDPC codes when a GA is assumed.
Summary form only given. The capacity of the message-passing decoder for LDPC codes can be computed by the density evolution algorithm by iteratively computing message densities. The infinite-dimensional problem of iteratively calculating the message densities in the case of the binary input AWGN channel and the belief propagation decoder can be simplified to a one-dimensional problem by using a Gaussian approximation. By assuming Gaussian densities the density evolution simplifies to updating means of Gaussian densities. An improved Gaussian approximation algorithm is suggested for computing the capacity of the BP decoder based on the mutual information measure. An analytical method for computing the mutual information of a transmitted bit with the message corresponding to it is proposed. Using this method we can approximate the non-Gaussian message by a Gaussian message that has the same mutual information with the transmitted bit. Computationally, the algorithm is similar to the Gaussian approximation algorithm proposed in Chung (2001). For various regular LDPC codes that were examined, the algorithm computed threshold values within 0.01dB or less from the exact threshold which is an improvement over the Gaussian approximation. Furthermore, additional insight on the convergence process can be gained from EXIT charts that can be derived from the algorithm. This can assist in designing better irregular LDPC codes.