On the Capacity of the Carbon Copy onto Dirty Paper Channel

The "Carbon Copy onto Dirty Paper" (CCDP) channel is the compound "writing on dirty paper" channel in which the channel output is obtained as the sum of the channel input, white Gaussian noise and a Gaussian state sequence randomly selected among a set possible realizations. The transmitter has non-causal knowledge of the set of possible state sequences but does not know which sequence is selected to produce the channel output. We study the capacity of the CCDP channel for two scenarios: (i) the state sequences are independent and identically distributed, and (ii) the state sequences are scaled versions of the same sequence. In the first scenario, we show that a combination of superposition coding, time-sharing and Gel'fand-Pinsker binning is sufficient to approach the capacity to within three bits per channel use for any number of possible state realizations. In the second scenario, we derive capacity to within four bits-per-channel-use for the case of two possible state sequences. This result is extended to the CCDP channel with any number of possible state sequences under certain conditions on the scaling parameters which we denote as "strong fading" regime. We conclude by providing some remarks on the capacity of the CCDP channel in which the state sequences have any jointly Gaussian distribution.