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.
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.
@cite_6
1976109068
In Costa's dirty-paper channel, Gaussian random binning is able to eliminate the effect of interference which is known at the transmitter, and thus achieve capacity. We examine a generalization of the dirty-paper problem to a multiple access channel (MAC) setup, where structured (lattice-based) binning seems to be necessary to achieve capacity. In the dirty-MAC, two additive interference signals are present, one known to each transmitter but none to the receiver. The achievable rates using Costa's Gaussian binning vanish if both interference signals are strong. In contrast, it is shown that lattice-strategies (lattice precoding) can achieve positive rates, independent of the interference power. Furthermore, in some cases-which depend on the noise variance and power constraints-high-dimensional lattice strategies are in fact optimal. In particular, they are optimal in the limit of high SNR-where the capacity region of the dirty MAC with strong interference approaches that of a clean MAC whose power is governed by the minimum of the users' powers rather than their sum. The rate gap at high SNR between lattice-strategies and optimum (rather than Gaussian) random binning is conjectured to be 1 2 log2(e 6)  0.254 bit. Thus, the doubly dirty MAC is another instance of a network setting, like the Korner-Marton problem, where (linear) structured coding is potentially better than random binning.