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IEEE International Symposium on Information Theory (ISIT)

Exact erasure channel density evolution for protograph-based generalized LDPC codes

Michael Lentmaier, Marcos B. S. Tavares, Gerhard Paul Fettweis

Published Jun 1, 2009• 23 citations
Publisher

Abstract

We derive explicit density evolution equations for protograph-based generalized LDPC codes on the binary erasure channel. They are obtained from an analysis of multi-dimensional input/output transfer functions of the component decoders. Belief propagation decoding with optimal component APP decoders is considered. Based on the resulting transfer functions, a threshold analysis is performed for some protograph examples.

DOI
10.1109/isit.2009.5205688

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We derive explicit density evolution equations for protograph-based generalized LDPC codes on the binary erasure channel. They are obtained from an analysis of multi-dimensional input/output transfer functions of the component decoders. Belief propagation decoding with optimal component APP decoders is considered. Based on the resulting transfer functions, a threshold analysis is performed for some protograph examples.
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Exact density evolution of protograph Low Density Parity Check (LDPC) codes over the Binary Erasure Channel (BEC) is considered. Upper bounds on the threshold are derived and expressed as single-variable minimizations. A simplified version of the upper bound is expressed in closed form in terms of the degrees of the nodes in the protograph. By maximizing the upper bound, useful conditions are derived for optimizing protographs to get thresholds close to capacity bounds. Using these conditions, a randomized construction method for good small-sized protographs is presented.
In this paper we present a simple method for determining the exact decoder convergence thresholds of low-density parity-check codes analytically.We assume transmission over the binary erasure channel and belief propagation decoding.To this aim we use EXIT charts and derive analytical formulas for the EXIT functions of regular and irregular LDPC codes transmitted over the binary erasure channel.Our method is equivalent to density evolution but faster and does not utilize a search algorithm.Thus it is not affected by numerical inaccuracy and does not depend on the number of iterations performed.
A threshold analysis of terminated generalized LDPC convolutional codes (GLDPC CCs) is presented for the binary erasure channel. Different ensembles of protograph-based GLDPC CCs are considered, including braided block codes (BBCs). It is shown that the terminated PG-GLDPC CCs have better thresholds than their block code counterparts. Surprisingly, our numerical analysis suggests that for large termination factors the belief propagation decoding thresholds of PG-GLDPC CCs coincide with the ML decoding thresholds of the corresponding PG-GLDPC block codes.
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Spatially coupled low-density parity-check block codes (SC-LDPC-BCs) are a class of LDPC codes in which certain ensembles achieve capacity under iterative belief propagation decoding. Prior work has considered these codes mostly from a protograph perspective. Following this viewpoint, our work presents new protographs for the design of protograph-based SC-LDPC-BCs. A reciprocal channel approximation (RCA) analysis of the terminated protographs yields the asymptotic performance limits of the ensembles in terms of their iterative decoding thresholds. Our protographs possess the properties of linear minimum distance growth rate and excellent thresholds. We consider the following channels in our work: binary-input additive white Gaussian noise channel (BI-AWGNC), binary erasure channel (BEC), and binary symmetric channel (BSC). Our work focuses on obtaining good designs, with two classes of ensembles being asymptotically regular, for all the three channels while minimizing rate loss.