On capacity of the dirty paper channel with fading dirt in the strong fading regime

The classic 'writing on dirty paper' capacity result establishes that full state pre-cancellation can be attained in Gelfand-Pinsker problem with additive state and additive white Gaussian noise. This result holds under the assumption that both the transmitter and the receiver have perfect knowledge of the channel. We are interested in characterizing capacity under the more realistic assumption that only partial channel knowledge is available at the transmitter. To this end we study the 'dirty paper channel with slow fading dirt', a variation of the dirty paper channel in which the state sequence is multiplied by a slow fading value known only at the receiver. For this model we establish two approximate characterizations of capacity, one for the case in which fading takes only two values and one for the case in which fading takes M possible values but these values are greatly spaced apart. For both results, a naive strategy in which the encoder pre-codes against different fading realizations in different time slots is sufficient to approach capacity.