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_8: We expand Costa's (1983) writing on dirty paper model to consider general distributions on the two sources of additive noise - one known non-causally to the encoder. We show that, under certain conditions, the capacity is unaffected by the known noise if and only if the unknown noise is Gaussian.
  @cite_5: We consider the generalized dirty-paper channel Y=X+S+N,E X sup 2 spl les P sub X , where N is not necessarily Gaussian, and the interference S is known causally or noncausally to the transmitter. We derive worst case capacity formulas and strategies for "strong" or arbitrarily varying interference. In the causal side information (SI) case, we develop a capacity formula based on minimum noise entropy strategies. We then show that strategies associated with entropy-constrained quantizers provide lower and upper bounds on the capacity. At high signal-to-noise ratio (SNR) conditions, i.e., if N is weak relative to the power constraint P sub X , these bounds coincide, the optimum strategies take the form of scalar lattice quantizers, and the capacity loss due to not having S at the receiver is shown to be exactly the "shaping gain" 1 2log(2 spl pi e 12) spl ap 0.254 bit. We extend the schemes to obtain achievable rates at any SNR and to noncausal SI, by incorporating minimum mean-squared error (MMSE) scaling, and by using k-dimensional lattices. For Gaussian N, the capacity loss of this scheme is upper-bounded by 1 2log2 spl pi eG( spl Lambda ), where G( spl Lambda ) is the normalized second moment of the lattice. With a proper choice of lattice, the loss goes to zero as the dimension k goes to infinity, in agreement with the results of Costa. These results provide an information-theoretic framework for the study of common communication problems such as precoding for intersymbol interference (ISI) channels and broadcast channels.
  @cite_9: This paper examines the problems of dirty-paper and dirty-tape coding with some additional side information (SI) at the decoder. In particular we focus on the situation in which the decoder SI is a noisy version of the SI at the encoder. We show that noisy SI at the decoder is advantageous for many channels in terms of capacity and, in general, the optimal coding strategy needs to take it into account. In particular, the capacities of the binary dirty-paper and dirty-tape channels are computed when the decoder SI is modeled as the output of a binary symmetric channel with the encoder SI as input. Moreover, the advantage offered by the decoder SI on the minimum achievable Eb NO in the dirty-tape scenario with Gaussian interference is demonstrated
  @cite_2: A Gaussian channel when corrupted by an additive Gaussian interfering signal that is not necessarily stationary or ergodic, but whose complete sample sequence is known to the transmitter, has the same capacity as if the interfering signal were not present.
summary: Following Costa's work, several extensions of DPC have been studied, , colored Gaussian noise @cite_2 , arbitrary distributions of @math @cite_8 and deterministic sequences @cite_5 . The case when @math is perfectly known to the encoder and a noisy version is known to the decoder is considered in @cite_9 , mainly focusing on discrete memoryless channels. The only result in @cite_9 for Gaussian channel reveals no additional gain due to the presence of the noisy estimate at the decoder, since perfect knowledge is available at the encoder and DPC can be used. In contrast, in this paper we study the case when only noisy knowledge of @math is available at both transmitter and receiver.