A Formula for the Capacity of the General Gel´fand–Pinsker Channel

We consider the Gel´fand-Pinsker problem in which the channel and state are general, i.e., possibly non-stationary, non-memoryless and non-ergodic. Using the information spectrum method and a non-trivial modification of the piggyback coding lemma by Wyner, we prove that the capacity can be expressed as an optimization over the difference of a spectral inf- and a spectral sup-mutual information rate. We consider various specializations, including the case where the channel and the state are memoryless but not necessarily stationary.