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Cycle analysis of time-invariant LDPC convolutional codes

Hua Ren Zhou, Norbert Goertz

Published Jan 1, 2010• 13 citations
Publisher

Abstract

Time-invariant low-density parity-check convolutional codes (LDPCccs) can be constructed from a polynomial form of a parity-check matrix that defines quasi-cyclic LDPC block codes based on circulant matrices. Based on this polynomial matrix, we discuss the relationships between the polynomial domain and the time domain parity-check and syndrome former matrices with respect to cycle properties. We present a novel, simple way to describe cycles in the polynomial version of the syndrome former matrix and we exploit this concept in a new cycle counting algorithm.

DOI
10.1109/ictel.2010.5478744

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Time-invariant low-density parity-check convolutional codes (LDPCccs) can be constructed from a polynomial form of a parity-check matrix that defines quasi-cyclic LDPC block codes based on circulant matrices. Based on this polynomial matrix, we discuss the relationships between the polynomial domain and the time domain parity-check and syndrome former matrices with respect to cycle properties. We present a novel, simple way to describe cycles in the polynomial version of the syndrome former matrix and we exploit this concept in a new cycle counting algorithm.
Time-variant Low-Density Parity-Check convolutional codes (LDPCccs) can be derived from unwrapping Quasi-Cyclic (QC) LDPC block codes. Rather than analyzing cycles in a large-scale time-domain parity check matrix, we propose a new way to describe cycle properties in compact form by a “polynomial syndrome former”, i.e., a syndrome former in an equivalent polynomial representation. According to the relationship of cycle structures between time- and polynomial-domain syndrome formers, we present a cycle counter algorithm to analyze cycle properties in time-variant LDPCccs as well as in time-invariant LDPCccs.
Low-Density Parity-Check convolutional codes (LDPCccs) can be efficiently decoded by a pipelined sub-optimal Sum Product Algorithm. The latter may suffer, however, from convergence problems, due to cycles in the Tanner graph. To improve the decoding performance, we analyze the cycle properties, based on the connections between monomials in the polynomial syndrome former (transposed parity-check matrix in polynomial form) of time-invariant LDPCccs. Due to specific structures in the polynomial syndrome former, some cycles are unavoidable no matter what monomials are placed in the polynomial syndrome former. It is shown that large-weight entries in the polynomial syndrome former lead to small girth, while monomial or empty entries, which can break short unavoidable cycles, may result in large girth. A novel algorithm is proposed to generate “good” LDPCccs with respect to their cycle properties: destructive structures in the polynomial syndrome former leading to small girth are systematically avoided.
Time-invariant low-density parity-check convolutional codes (TI LDPC-CCs) can be represented by a polynomial-domain parity-check matrix derived from the corresponding quasi-cyclic (QC) LDPC block codes (LDPC-BCs), while time-varying (TV) LDPC-CCs can be obtained by unwrapping the parity-check matrices of LDPC-BCs. The cycle enumerators for TI and TV LDPC-CCs are compared. Based on the analysis of the graphical structures of short cycles in HT(D), we introduce a method of designing the polynomial syndrome former matrix HCRT(D) for LDPC-CCs. It eliminates short cycles and shows improved decoding performance on an additive white Gaussian noise (AWGN) channel with lower bit error ratio (BER) curves.
Low-Density Parity-Check convolutional codes (LD- PCccs) are very interesting for practical error-correction coding in wireless transmission as they have excellent performance and at the same time they allow for variable block size with low complexity encoding and decoding. As for all LDPC codes that are decoded by the sub-optimal (but highly efficient) Sum Product Algorithm, the cycles in the code graph are very important for the practical performance of the coding scheme. Time-invariant LDPCccs can be defined by a polynomial syndrome former (transposed parity-check matrix in polynomial form), that can be derived from corresponding Quasi-Cyclic (QC) LDPC block codes. Given the polynomial syndrome former with certain structures, unavoidable cycles with lengths ranging from 6 to 12 will be shown to exist. We provide some rules for designing good codes with respect to the shortest cycle in the code-graph, the girth of the code, which is a crucial parameter for its decoding performance.
Time-invariant low-density parity-check convolutional codes (LDPC-CCs) derived from corresponding quasi-cyclic (QC) LDPC block codes (LDPC-BCs) can be described by a polynomial syndrome former matrix (polynomial-domain transposed parity-check matrix). In this paper, an estimation of the distance spectrum of time-invariant LDPC-CCs is obtained by splitting the polynomial syndrome former matrix into submatrices representing “super codes” and then evaluating the linear dependence between codewords of the corresponding super codes. This estimation results in an upper bound on the minimum free distance of the original code and, additionally, a lower bound on the number of codewords Awwith Hamming weight w.
③ Abstract: In this paper, a new ensemble of the polynomial matrix of a time-invariant LDPC convolutional code is proposed. Based on the method of deriving time-invariant LDPC convolutional codes from QC (Quasi-Cyclic)- LDPC block codes, the elements over finite fields are used to generate directly the polynomial parity-check matrices of LDPC convolutional codes. A concrete example of using MDS (Maximum-Distance Separable) convolutional codes to derive the polynomial matrices is given. The proposed method ensures the fast encoding property, maximum encoding memory and designed rate. Simulation results show that the proposed LDPC convolutional codes exhibit low error floor and good decoding performance under BP (Belief Propagation) decoding algorithm over AWGN (Additive White Gaussian Noise) channel.