Channel Coding in the Presence of Side Information
Foundations and Trends in Communications and Information Theory
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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+Zs, where the encoder observes causally the random state S∈S that governs the distribution of the noise Zs. We show that the capacity of this channel is given by C=log|χ|-mint:S→χH(Z S-t(S)). This capacity is realized by a state-independent code, followed by a shift by the “noise prediction” tmin(S) that minimizes the entropy of Zs-t(S). If the set of conditional noise distributions {p(z|s),s∈S} is such that the optimum predictor tmin(·) 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