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Iterative Decoding of LDPC Codes over Arbitrary Discrete-memoryless Channels

Amir Bennatan, David Burshtein

Published Oct 1, 2003• 10 citations
OpenAlex

Abstract

Dept. of Electrical Engineering – SystemsTel-Aviv UniversityTel-Aviv 69978, Israelabn@eng.tau.ac.il, burstyn@eng.tau.ac.ilAbstractWe examine iterative decoding of quantized coset (QC) LDPC codes, designedfor use over arbitrary channels, in particular nonbinary and asymmetric channels.An important element of QC-LDPC codes is the addition of a fixed, random vec-tor, known to the transmitter and the receiver, to all codewords. We show thatthis addition produces symmetry that significantly simplifies mathematical analy-sis. We extend several results from the analysis of LDPC codes over binary-inputsymmetric-output channels, to QC-LDPC codes with arbitrary alphabets over ar-bitrary channels, including the symmetry property and the stability condition.

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Dept. of Electrical Engineering – SystemsTel-Aviv UniversityTel-Aviv 69978, Israelabn@eng.tau.ac.il, burstyn@eng.tau.ac.ilAbstractWe examine iterative decoding of quantized coset (QC) LDPC codes, designedfor use over arbitrary channels, in particular nonbinary and asymmetric channels.An important element of QC-LDPC codes is the addition of a fixed, random vec-tor, known to the transmitter and the receiver, to all codewords. We show thatthis addition produces symmetry that significantly simplifies mathematical analy-sis. We extend several results from the analysis of LDPC codes over binary-inputsymmetric-output channels, to QC-LDPC codes with arbitrary alphabets over ar-bitrary channels, including the symmetry property and the stability condition.
Introduction Sequences of irregular low-density parity-check (LDPC) codes that achieve the capacity of the binary erasure channel (BEC) under iterative decoding were first constructed by Luby, et al. in [11]. This was followed by the work of Chung, et al., which provided evidence suggesting that sequences of iteratively decoded LDPC codes can also achieve the channel capacity of the binary-input additive white Gaussian noise (AWGN) channel [3]. Since then, density evolution (DE) [13] has been used to optimize irregular LDPC codes for a variety of memoryless channels (e.g., [6]), and the results suggest, for each channel, that sequences of iteratively decoded LDPC codes can indeed achieve the channel capacity. In fact, the discovery of a channel whose capacity cannot be approached by LDPC codes would be more surprising than a proof that iteratively decoded LDPC codes can achieve the capacity of any binary-input symmetric channel (BISC). The idea of decoding a code transmitted over a c
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.
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