Mutual Information and Capacity of a Linear Digital Channel

In this paper we analyse a simple model of a digital communications channel. This model proves to be closely related to an iterated function system (IFS) related to the well-known Bernoulli convolution. We derive it from a randomly forced first-order ordinary differential equation. This allows the parameter of the Bernoulli convolution—the contraction rate, λ—to be related to the rate at which symbols are input to the channel. It is shown that for a channel with equiprobable binary inputs the mutual information between input and output distributions is the stationary measure of the complement of the overlap region of the IFS. We show that the mutual information is Hölder continuous with respect to λ and decreases hyperexponentially as λ → 1. We also study the case of non-equiprobable binary inputs and show that the maximum of the mutual information—the channel capacity— does not always correspond to equiprobable inputs.