  In this paper, we first consider a channel that is contaminated by two
independent Gaussian noises $S ~ N(0,Q)$ and $Z_0 ~ N(0,N_0)$. 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 $0.5log(1+P/(Q(N_1/(Q+N_1))+N_0))$, where $P$ is the input
power constraint and $N_1$ is the power of the noise corrupting $S$. We then
consider a Gaussian cognitive interference channel (IC) and propose a causal
noisy dirty paper coding (DPC) strategy. We compute the achievable region using
this noisy DPC strategy and quantify the regions when it achieves the upper
bound on the rate.
