Generalized Writing on Dirty Paper

We consider a generalization of Costa’s writing on dirty paper model in which a powerlimited communicator encounters two independent sources of additive noise, one of which is known non-causally to the encoder. We seek to characterize when the capacity for this channel would not change if the source of noise known to the encoder were also known to the decoder; we call this property private-public equivalence (PPE). Costa showed that this model has PPE if both sources of noise are IID Gaussian. We show that this model has PPE as long as the unknown noise is Gaussian (but not necessarily IID) for any distribution on the known noise. We also conjecture that for a general class of coding strategies and for IID noise sequences (with some additional assumptions), if this model has PPE then the unknown noise must be Gaussian. This result relies on the Darmois-Skitovič Theorem, which states that linear combinations of independent random variables can be independent only for Gaussian random variables.