text: In this paper, we first consider a channel that is contaminated by two independent Gaussian noises @math and @math . The capacity of this channel is computed when independent noisy versions of @math are known to the transmitter and or receiver. It is shown that the channel capacity is greater then the capacity when @math is completely unknown, but is less then the capacity when @math is perfectly known at the transmitter or receiver. For example, if there is one noisy version of @math known at the transmitter only, the capacity is @math , where @math is the input power constraint and @math is the power of the noise corrupting @math . 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.
  @cite_6: A channel with output Y = X + S + Z is examined, The state S N(0, QI) and the noise Z N(0, NI) are multivariate Gaussian random variables ( I is the identity matrix.). The input X R^ n satisfies the power constraint (l n) i=1 ^ n X_ i ^ 2 P . If S is unknown to both transmitter and receiver then the capacity is 1 2 (1 + P ( N + Q)) nats per channel use. However, if the state S is known to the encoder, the capacity is shown to be C^ = 1 2 (1 + P N) , independent of Q . This is also the capacity of a standard Gaussian channel with signal-to-noise power ratio P N . Therefore, the state S does not affect the capacity of the channel, even though S is unknown to the receiver. It is shown that the optimal transmitter adapts its signal to the state S rather than attempting to cancel it.
summary: One special case of ) is when a noisy version of @math is known only to the transmitter; our result in this case is a generalization of Costa's celebrated result @cite_6 . @cite_6 , it is shown that the achievable rate when the noise @math is perfectly known at the transmitter is equivalent to the rate when @math is known at the receiver, and this rate does not depend on the variance of @math . A new coding strategy to achieve this capacity was also introduced in @cite_6 and is popularly referred to as dirty paper coding (DPC). We generalize Costa's result to the case of noisy interference knowledge. We show that the capacity with knowledge of a noisy version of @math at the transmitter is equal to the capacity with knowledge of a statistically equivalent noisy version of @math at the receiver. However, unlike @cite_6 where the capacity does not depend on the variance of @math , in the general noisy side information case, the capacity decreases as the variance of @math increases.