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In this paper, we consider the full-duplex and half-duplex Gaussian relay channels where the noises at the relay and destination are arbitrarily correlated. We first derive the capacity upper bound and the achievable rates with three existing schemes: Decode-and-Forward (DF), Compress-and-Forward (CF), and Amplify-and-Forward (AF). We present two capacity results under specific noise correlation coefficients, one being achieved by DF and the other being achieved by direct link transmission (or a special case of CF). The channel for the former capacity result is equivalent to the traditional Gaussian degraded relay channel and the latter corresponds to the Gaussian reversely-degraded relay channel. For CF and AF schemes, we show that their achievable rates are strictly decreasing functions of the correlation coefficient when the correlation coefficient is negative. Moreover, when the noise correlation coefficient is positive, the CF achievable rate may also outperform the independent-noise case if the noise correlation coefficient is within a certain range. Through numerical comparisons under different channel settings, we observe that although DF completely disregards the noise correlation while the other two can potentially exploit such extra information, none of the three relay schemes always outperforms the others over different correlation coefficients. Moreover, the exploitation of noise correlation by CF and AF accrues more benefit when the source-relay link is weak. This paper also considers the optimal power allocation problem under the correlated-noise channel setting. With individual power constraints at the relay and the source, it is shown that the relay should use all its available power to maximize the achievable rates under any correlation coefficient. With a total power constraint across the source and the relay, the achievable rates are proved to be concave functions over the power allocation factor for AF and CF under full-duplex mode, where the closed-form power allocation strategy is derived.

Study of Gaussian Relay Channels with Correlated Noises