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Journal of Signal and Information Processing

Tree Matrix Algorithm of LDPC Codes

Huanming Zhang

Published Jan 1, 2014• 0 citations
Publisher

Abstract

LDPC codes are finding increasing use in applications requiring reliable and highly efficient information transfer over bandwidth. An LDPC code is defined by a sparse parity-check matrix and can be described by a bipartite graph called Tanner graph. Loops in Tanner graph prevent the sum-product algorithm from converging. Further, loops, especially short loops, degrade the performance of LDPC decoder, because they affect the independence of the extrinsic information exchanged in the iterative decoding. This paper, by graph theory, deduces cut-node tree graph of LDPC code, and depicts it with matrix. On the basis of tree matrix algorithm, whole depictions of loops can be figured out, providing of foundation for further research of relations between loops and LDPC codes’ performance.

DOI
10.4236/jsip.2014.54020

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LDPC codes are finding increasing use in applications requiring reliable and highly efficient information transfer over bandwidth. An LDPC code is defined by a sparse parity-check matrix and can be described by a bipartite graph called Tanner graph. Loops in Tanner graph prevent the sum-product algorithm from converging. Further, loops, especially short loops, degrade the performance of LDPC decoder, because they affect the independence of the extrinsic information exchanged in the iterative decoding. This paper, by graph theory, deduces cut-node tree graph of LDPC code, and depicts it with matrix. On the basis of tree matrix algorithm, whole depictions of loops can be figured out, providing of foundation for further research of relations between loops and LDPC codes’ performance.
Low-density parity-check (LDPC) codes are forward error-correction and linear block codes. An LDPC code can be described by a bipartite graph called Tanner graph(1). Loops, especially short loops in tanner graph, degrade the performance of LDPC decoder, because they affect the independence of the extrinsic information exchanged in the iterative decoding. In this paper, based on graph theory and Tanner graph, the loop structure in LDPC codes are studied carefully, a new notion, cut-node tree, is proposed to describe LDPC codes. Cut-node tree has full information of Tanner graph. So all loop features in LDPCs can be calculated relatively easy by a computer. Traditional message passing in graph is improved to avoid repeated iteration of information, a new decoding schemes for LDPC codes is proposed and can suppress repeated iteration of information in SPA. The results help to further research on related field.
The linear block code is called a binary Low Density Parity Check code if its parity-check matrix is a sparse matrix.For the features of parity-check matrix,a LDPC code can be illustrated by a Tanner graph.Cycles,especially short cycles in the Tanner graph,which lead to inefficient decoding and prevent the decoding algorithm from converging to the optimal decoding result.In this paper,we demonstrate that a detection method of short cycle presents generation algorithm of check matrix without cycle 4 and simulates the check-matrix.
The linear block code is called a binary Low-Density Parity-Check code if whose parity-check matrix is a sparse matrix. For the features of parity-check matrix, a LDPC code can be illustrated by a Tanner graph. Cycles, especially short cycles in the Tanner graph, lead to inefficient decoding and prevent the decoding algorithm from converging to the optimal decoding result. In this paper, a simple algorithm of constructing four-cycle-free regular A(n,d_v,d_c)LDPC codes, d_v3, is introduced. And we analyse the relationship between the performance and the cycle.
Low Density Parity Check(LDPC) codes is one of the high topics of the channel coding theory and has been paid more and more attention.This paper begins with a brief review of channel coding theories.Then the iterative decoding algorithm of LDPC codes,so-called as Sum-Product algorithm or Belief Propagation algorithm,is represented in detail.And the construction of LDPC codes by means of random bipartite graph and algebraic constructions,and the performance analysis by Density Evolution theory,Gaussian Approximation method and Extrinsic Information Transfer(EXIT) charts etc.,are respectively discussed.Finally,the latest study results of LDPC codes are presented and some open questions about LDPC codes are provided.