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 module-additive noise channel Y=X+Z/sub s/, where the encoder observes causally the random state S/spl isin/S that governs the distribution of the noise Z/sub s/. We show that the capacity of this channel is given by C=log|/spl chi/|-min/sub t:S/spl rarr//spl chi//H(Z/sub S/-t(S)). This capacity is realized by a state-independent code, followed by a shift by the "noise prediction" t/sub min/(S) that minimizes the entropy of Z/sub s/-t(S). If the set of conditional noise distributions {p(z|s),s/spl isin/S} is such that the optimum predictor t/sub min/(/spl middot/) 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.