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Performance Analysis of Protograph LDPC Codes for Nakagami-$m$ Fading Relay Channels

In this paper, we investigate the error performance of the protograph (LDPC) codes over Nakagami-$m$ fading relay channels. We first calculate the decoding thresholds of the protograph codes over such channels with different fading depths (i.e., different values of $m$) by exploiting the modified protograph extrinsic information transfer (PEXIT) algorithm. Furthermore, based on the PEXIT analysis and using Gaussian approximation, we derive the bit-error-rate (BER) expressions for the error-free (EF) relaying protocol and decode-and-forward (DF) relaying protocol. We finally compare the threshold with the theoretical BER and the simulated BER results of the protograph codes. It reveals that the performance of DF protocol is approximately the same as that of EF protocol. Moreover, the theoretical BER expressions, which are shown to be reasonably consistent with the decoding thresholds and the simulated BERs, are able to evaluate the system performance and predict the decoding threshold with lower complexity as compared to the modified PEXIT algorithm. As a result, this work can facilitate the design of the protograph codes for the wireless communication systems.