B.1.1 General Form of Channels with Side Information

The class of channels studied by Gel'fand and Pinsker is illustrated in Figure B.1. For each use of the channel, t, a random value, s[ t], is drawn from a set of possible values, . Before the transmitter chooses a code word to transmit, it is told what all values of s[ t] will be during the entire transmission (i.e., it is told that s[ t] = s[ t], for t = 1 ... L, where L is the length of the transmission). The transmitter then chooses a sequence of L values, x, from a set of possible values, , and transmits them. Each of the sequence of values then received by the receiver, y[ t], is a random value dependent on both x[ t] and s[ t]. This dependency is described by a conditional probability distribution, P _{yx= x[ t], s= s[ t]} ( y).

This class of channels includes the dirty-paper channel. Here, s and x are vectors of real numbers, and y is the sum of x,