Regular low-density parity-check (LDPC) code with normalized and UMP BP-based algorithms on fast Rayleigh fading channel

For the short regular LDPC codes, we derive the formula of the normalization factor (NF) theoretically by using the probability density function (pdf) of the initial likelihood information for the normalized BP-based algorithm. For the long regular LDPC codes, we derive the optimal NF for the normalized BP-based algorithm by using density evolution (DE). We also analyze the performance of the long regular LDPC codes with the UMP BP-based algorithm by using DE. From the numerical and simulation results, we show that the optimum NF for the short regular LDPC codes is different from that on the AWGN channel. We also show that for the short regular LDPC codes, the normalized BP-based algorithm outperforms the UMP BP-based algorithm, and has the performance very close to those of the BP algorithm. Furthermore, for the long regular LDPC codes, we show that the normalized BP-based algorithm outperforms the BP algorithm and the UMP BP-based algorithm. Therefore, for the short and long regular codes, the normalized BP-based algorithm is shown to be suitable as the decoding algorithm on the fast Rayleigh fading channel.