On the dirty paper channel with fast fading dirt

Costa's 'writing on dirty paper' result establishes that full state pre-cancellation can be attained in the Gel'fand-Pinsker problem with additive state and additive white Gaussian noise. This result holds under the assumptions that full channel knowledge is available at both the transmitter and the receiver. In this work we consider the scenario in which the state is multiplied by an ergodic fading process which is not known at the encoder. We study both the case in which the receiver has knowledge of the fading and the case in which it does not: for both models we derive inner and outer bounds to capacity and determine the distance between the two bounds when possible. For the channel without fading knowledge at either the transmitter or the receiver, the gap between inner and outer bounds is finite for a class of fading distributions which includes a number of canonical fading models. In the capacity approaching strategy for this class, the transmitter performs Costa's pre-coding against the mean value of the fading times the state while the receiver treats the remaining signal as noise. For the case in which only the receiver has knowledge of the fading, we determine a finite gap between inner and outer bounds for two classes of discrete fading distribution. The first class of distributions is the one in which there exists a probability mass larger than one half while the second class is the one in which the fading is uniformly distributed over values that are exponentially spaced apart. Unfortunately, the capacity in the case of a continuous fading distribution remains very hard to characterize.