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IEEE International Symposium on Information Theory (ISIT)

EXIT charts for non-binary LDPC codes over arbitrary discrete-memoryless channels

Amir Bennatan, David Burshtein

Published Jan 1, 2005• 1 citation
Publisher

Abstract

We consider coset LDPC codes over GF(q), designed for use over arbitrary channels (particularly nonbinary and asymmetric channels). We show that the random selection of the nonzero elements of the GF(q) parity-check matrix induces a permutation-invariance property on the densities of the messages produced by the decoder. We use this property to show that under a Gaussian approximation, the entire q - 1 dimensional distribution of the vector messages is described by a single scalar parameter. We apply this result to develop EXIT charts for our codes. We use appropriately designed signal constellations to obtain substantial shaping gains. Simulation results indicate that our codes outperform multilevel codes at short block lengths. We also present results for the AWGN channel at 0.56 dB of the unconstrained Shannon limit (i.e. not restricted to any signal constellation) at a spectral efficiency of 6 bits/s/Hz

DOI
10.1109/isit.2005.1523288

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We consider coset LDPC codes over GF(q), designed for use over arbitrary channels (particularly nonbinary and asymmetric channels). We show that the random selection of the nonzero elements of the GF(q) parity-check matrix induces a permutation-invariance property on the densities of the messages produced by the decoder. We use this property to show that under a Gaussian approximation, the entire q - 1 dimensional distribution of the vector messages is described by a single scalar parameter. We apply this result to develop EXIT charts for our codes. We use appropriately designed signal constellations to obtain substantial shaping gains. Simulation results indicate that our codes outperform multilevel codes at short block lengths. We also present results for the AWGN channel at 0.56 dB of the unconstrained Shannon limit (i.e. not restricted to any signal constellation) at a spectral efficiency of 6 bits/s/Hz
We discuss three structures of modified low-density parity-check (LDPC) code ensembles designed for transmission over arbitrary discrete memoryless channels. The first structure is based on the well-known binary LDPC codes following constructions proposed by Gallager and McEliece, the second is based on LDPC codes of arbitrary (q-ary) alphabets employing modulo-q addition, as presented by Gallager, and the third is based on LDPC codes defined over the field GF(q). All structures are obtained by applying a quantization mapping on a coset LDPC ensemble. We present tools for the analysis of nonbinary codes and show that all configurations, under maximum-likelihood (ML) decoding, are capable of reliable communication at rates arbitrarily close to the capacity of any discrete memoryless channel. We discuss practical iterative decoding of our structures and present simulation results for the additive white Gaussian noise (AWGN) channel confirming the effectiveness of the codes.
This paper extends the work on density evolution for binary low-density parity-check (LDPC) codes with Gaussian approximation to LDPC codes over GF(q) . We first generalize the definition of channel symmetry for nonbinary inputs to include q-ary phase-shift keying (PSK) modulated channels for prime q and binary-modulated channels for q that is a power of 2. For the well-defined q-ary-input symmetric-output channel, we prove that under the Gaussian assumption, the density distribution for messages undergoing decoding is fully characterized by (q-1) quantities. Assuming uniform edge weights, we further show that the density of messages computed by the check node decoder (CND) is fully defined by a single number. We then present the approximate density evolution for regular and irregular LDPC codes, and show that the (q-1) -dimensional integration involved can be simplified using a dimensionality reduction algorithm for the important case of q=2p. Through application of approximate density evolution and linear programming, we optimize the degree distribution of LDPC codes over GF(3) and GF(4). The optimized irregular LDPC codes demonstrate performance close to the Shannon capacity for long codewords. We also design GF(q) codes for high-order modulation by using the idea of a channel adapter. We find that codes designed in this fashion outperform those optimized specifically for the binary additive white Gaussian noise (AWGN) channel for a short codewords and a spectral efficiency of 2 bits per channel use (b/cu).
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.
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 known to perform well in the presence of additive white Gaussian noise (AWGN) but for very large block lengths. It has been proposed to define the codes over high order Galois fields to overcome this limitation. In this paper we construct new quasi-cyclic non-binary LDPC codes with moderate code lengths from Reed-Solomon codes with two message symbols proposed by Lin et al defined over large finite fields. We evaluate the performance of these codes on the AWGN channel by computer simulation and show that they outperform binary LDPC codes of the same length in binary bits.
A message-passing decoder for GF(q) low-density parity-check codes is defined, which uses discrete messages from a subset of all possible binary vectors of length q. The proposed algorithm is a generalization to GF(q) of Richardson and Urbanke's decoding "Algorithm E" for binary codes. Density evolution requires a mapping between the probability distribution spaces for the channel, variable and check messages, and under the proposed algorithm, exact density evolution is possible. Symmetries in the message densities permit reduction in the size of the probability distribution space. Noise thresholds are obtained for LDPC codes on discrete memoryless channels, and as with Algorithm E, are remarkably close to noise thresholds under more complex belief propagation decoding.