Entropy Amplification Property and The Rate Loss in Writing on Dirty Paper ∗ Aaron

Costa’s celebrated ”writing on dirty paper” (WDP) shows that the powerconstrained channel Y = X + S + Z, with Gaussian Z, has the same capacity as the standard AWGN channel Y = X + Z, provided that the ”interference” S (no matter how strong it is) is known at the transmitter. While this ability for perfect interference cancellation is very appealing, it relies heavily on the Gaussianity of the (unknown) noise Z. We construct an example of “bad” noise for writing on dirty paper, namely, “difference set noise”. If the interference S is strong, then difference-set noise limits the WDP capacity to at most 2 bits. At the same time, like in the AWGN case, the zero-interference capacity grows without bound with the input constraint. Thus almost 100 % of the available capacity is lost in WDP in the presence of difference-set noise. These results are based on the “entropy amplification property” of noise with aperiodic probability distribution. Using this property, and the duality between WDP and Wyner-Ziv source coding, we also give an example of dramatic rate-loss in quantizing encrypted source.