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IEEE International Symposium on Information Theory (ISIT)

On the thresholds of generalized LDPC convolutional codes based on protographs

Michael Lentmaier, Gerhard Paul Fettweis

Published Jun 1, 2010• 75 citations
Publisher

Abstract

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.

DOI
10.1109/isit.2010.5513595

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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.
Families of generalized spatially-coupled low-density parity-check (GSC-LDPC) code ensembles can be formed by terminating protograph-based generalized LDPC convolutional (GLDPCC) codes. It has previously been shown that ensembles of GSC-LDPC codes constructed from a protograph have better iterative decoding thresholds than their block code counterparts, and that, for large termination lengths, their thresholds coincide with the maximum a-posteriori (MAP) decoding threshold of the underlying generalized LDPC block code ensemble. Here we show that, in addition to their excellent iterative decoding thresholds, ensembles of GSC-LDPC codes are asymptotically good and have large minimum distance growth rates.
In this paper, we analyze the performance of finite-length generalized LDPC (GLDPC) block codes constructed from protographs when transmission takes place over the binary erasure channel (BEC). A generalized peeling decoder is proposed and we derive a system of differential equations that gives the expected evolution of the graph degree distribution during decoding. We then show that the finite-length performance of a GLDPC code can be estimated by means of a simple scaling law, where a single scaling parameter represents the finite-length properties of the code. We also show that, as we consider stronger component codes, both the asymptotic threshold and the finite-length scaling parameter are improved.
It has previously been shown that ensembles of terminated protograph-based low-density parity-check (LDPC) convolutional codes have a typical minimum distance that grows linearly with block length and that they are capable of achieving capacity approaching iterative decoding thresholds on the binary erasure channel (BEC). In this paper, we review a recent result that the dramatic threshold improvement obtained by terminating LDPC convolutional codes extends to the additive white Gaussian noise (AWGN) channel. Also, using a (3,6)-regular protograph-based LDPC convolutional code ensemble as an example, we perform an asymptotic trapping set analysis of terminated LDPC convolutional code ensembles. In addition to capacity approaching iterative decoding thresholds and linearly growing minimum distance, we find that the smallest non-empty trapping set of a terminated ensemble grows linearly with block length.