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Convergence analysis for a class of LDPC convolutional codes on the erasure channel

Arvind Sridharan, Michael Lentmaier, Daniel J. Costello, Kamil Sh. Zigangirov

Published Jan 1, 2004• 52 citations
OpenAlex

Abstract

An ensemble of LDPC convolutional codes with parity-check matrices composed of permutation matrices is introduced.The convergence of the iterative belief propagation based decoder for terminated convolutional codes in the ensemble when operating on the erasure channel is analyzed.The structured irregularity in the Tanner graph of the codes leads to significantly better thresholds when compared to the corresponding LDPC block codes.

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IEEE International Symposium on Information Theory (ISIT)•Dec 23, 2004•18 citations
An ensemble of LDPC convolutional codes with parity-check matrices composed of permutation matrices is introduced.The convergence of the iterative belief propagation based decoder for terminated convolutional codes in the ensemble when operating on the erasure channel is analyzed.The structured irregularity in the Tanner graph of the codes leads to significantly better thresholds when compared to the corresponding LDPC block codes.
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.
Low-density parity-check (LDPC) block codes invented by Gallager in 1962 achieve exceptional error performance on a wide variety of communication channels. LDPC convolutional codes are the convolutional counterparts of LDPC block codes. This dissertation describes techniques for the design of LDPC convolutional codes, and analyzes their distance properties and iterative decoding convergence thresholds. The construction of time invariant LDPC convolutional codes by unwrapping the Tanner graph of algebraically constructed quasi-cyclic LDPC codes is described. The convolutional codes outperform the quasi-cyclic codes from which they are derived. The design of parity-check matrices for time invariant LDPC convolutional codes by the polynomial extension of a base matrix is proposed. An upper bound on free distance, proving that time invariant LDPC codes are not asymptotically good, is obtained. The Tanner graph is used to describe a pipelined message passing based iterative decoder for LDPC convolutional codes that outputs decoding results continuously. The performance of LDPC block and convolutional codes are compared for fixed decoding parameters like computational complexity, processor complexity, and decoding delay. In each case, the LDPC convolutional code performance is at least as good as that of LDPC block codes. An analog circuit to implement pipelined decoding of LDPC convolutional codes is proposed. The distance properties of a permutation matrix based (time varying) ensemble of (J,K) regular LDPC convolutional codes is investigated. It is proved that these codes (for J > 2) have free distance increasing linearly with constraint length, i.e., they are asymptotically good. Further, the asymptotic free distance to constraint length ratio for the convolutional codes is several times larger than the minimum distance to block length ratio for corresponding LDPC block codes. Iterative decoding of terminated LDPC convolutional codes, based on the ensemble mentioned above, is analyzed, assuming transmission over a binary erasure channel. The structured irregularity of the codes leads to significantly better convergence thresholds compared to corresponding LDPC block codes. At the calculated thresholds, both the bit and block error probability can be made arbitrarily small. The results obtained suggest that the thresholds approach capacity with increasing J.
Abstract—We consider a windowed decoding scheme for LDPC convolutional codes that is based on the belief-propagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed decoder. We will consider the performance of these ensembles and codes over erasure channels with and without memory. We show that the structure of LDPC convolutional code ensembles is suitable to obtain performance close to the theoretical limits over the memoryless erasure channel, both for the BP decoder and windowed decoding. However, the same structure imposes limitations on the performance over erasure channels with memory. Index Terms—Low-density parity-check codes, Convolutional
This work presents an ensemble of irregular low-density parity check (LDPC) codes based on an ensemble of bipartite graphs, which formulated the average weight distribution. An analysis of the error performance of LDPC codes over a binary erasure channel together with an iterative decoding algorithm based on belief propagation has been clarified that a notion of stopping sets and their distribution with the asymptotic expression are derived. An irregular LDPC code ensemble that exhibits better performance in the sense of threshold is obtained by density evolution and explicitly formulates weight and stopping set distributions are defined.