On the Capacity Equivalence with Side Information at Transmitter and Receiver

In this paper, a channel that is contaminated by two independent Gaussian noises $S ~ N(0,Q)$ and $Z_0 ~ N(0,N_0)$ is considered. The capacity of this channel is computed when independent noisy versions of $S$ are known to the transmitter and/or receiver. It is shown that the channel capacity is greater then the capacity when $S$ is completely unknown, but is less then the capacity when $S$ is perfectly known at the transmitter or receiver. For example, if there is one noisy version of $S$ known at the transmitter only, the capacity is $0.5\log(1+\frac{P}{Q(N_1/(Q+N_1))+N_0})$, where $P$ is the input power constraint and $N_1$ is the power of the noise corrupting $S$. Further, it is shown that the capacity with knowledge of any independent noisy versions of $S$ at the transmitter is equal to the capacity with knowledge of the statistically equivalent noisy versions of $S$ at the receiver.