| Generalized low-density parity-check (GLPDC) codes are a class of iteratively decodable codes, which have wide range of coding rates and are close to the Shannon limit. Compared with standard LDPC codes, the species of component codes corresponding to check nodes of GLDPC code are no longer limited to single parity check (SPC) codes, which bring more flexibilites.In this thesis, Hamming codes are used as component codes to construct GLDPC codes. After designing the effective construction method which is applicable to a variety of GLDPC codes with different kinds of component codes, a specific (147,21) Hamming GLDPC code is constructed. Then the structures and numbers of short cycles are shown as well as the code weight distribution and minimum Hamming distance, i.e., ^min=35.The maximum a posteriori probability (MAP) decoding algorithm for Hamming GLDPC codes is proposed according to sum-product algorithm (SPA) and BCJR algorithm. In order to reduce the computation complexity, Log-MAP algorithm in logarithm domain is devised. Utilizing the Jacobian logarithmic function, the simplified version of Log-MAP, namely, Max-Log-MAP algorithm is proposed. By the comparison of decoding performance and complexity among these three algorithms, it is no doubt that Max-Log-MAP algorithm can reduce the calculating costs of by a half in comparison with MAP algorithm, with decoding performance loss less than.0.5 dB.Simulation results show that by contrast with standard LDPC code with the same coding rate, (147,21) Hamming GLDPC code reveals the decoding outperformance and faster convergence speed not only in AWGN but also in flat Rayleigh fading channel. The observation suggests that the decoding algorithm of GLDPC code can effectively alleviate the negative effects of the 4-cycles in the check sub-matrix corresponding to Hamming component codes in decoding. Furthermore, the error floor of this kind of code does not appear even at BER of 10-7, which is not unexpected owing to of the elegant code weight distribution and the large minimum Hamming distance.Extrinsic-information-transfer (EXIT) chart is an effective theoretical tool to obtain the asymptotic performance of LDPC codes. Due to the dual property of EXIT chart, the EXIT functions of simplex code and its dual code, i.e., Hamming code, in binary erasure channel (BEC), are derived. According to the EXIT curves for variable nodes and check nodes of standard LDPC code, the EXIT chart for GLDPC code with Hamming code as check nodes in BEC is drawn. With hill climbing method, we get the decoding threshold and the corresponding degree distribution of GLDPC code with the fixed coding rate of 1/2. Then the factors influencing the asymptotic performance of GLDPC code are furtherly analyzed. At fixed coding rate, compared with the degree distribution of variable nodes, the degree distribution of check nodes is more important to determine the asymptotic performance of Hamming GLDPC codes.Finally, to further approach Shannon limit, the strategy of mixing SPC with Hamming code to act as the component code of GLDPC is proposed. By adjusting the proportion of SPC and Hamming code, the EXIT curve of check nodes can match that of variable nodes better. Numerical results show that in BEC, in comparison with standard LDPC code, Hamming GLDPC codes, and the state-of-the-art GLDPC code with mixed variable nodes (repetition code as well as SPC code as component codes of variable nodes), this GLDPC code has better asymptotic performance, which is only 0.0021 awav from the Shannon limit!... |
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