Noise prediction for channels with side information at the transmitter

The computation of channel capacity with side information at the transmitter side (but not at the receiver side) requires, in general, extension of the input alphabet to a space of strategies, and is often hard. We consider the special case of a discrete memoryless modulo-additive noise channel Y = X + Zs, where the encoder observes causally the random state S G S that governs the distribution of the noise Z_S. We show that the capacity of this channel is given by C = log |X| - min_t:S→X H(Z_S - t(S)). This capacity is realized by a state-independent code, followed by a shift by the "noise prediction" t_min(S) that minimizes the entropy of Z_s -t(S). If the set of conditional noise distributions {p( z \s), s ∈ S} is such that the optimum predictor t_min(·) is independent of the state weights, then C is also the capacity for a noncausal encoder, that observes the entire state sequence in advance. Furthermore, for this case we also derive a simple formula for the capacity when the state process has memory.