  In this paper, we consider 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 over the negative
correlation coefficient. 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.
