Stationary Channels with a Random Parameter Which Is a Completely Singular Process

A stationary channel with a random parameter $U=\{U_j\}$ which is a completely singular stationary process independent of an input signal $X=\{X_j\}$ is considered. It is shown that under rather weak additional conditions, the information rate $\overline{I}(X;Y)$ between the input signal $X=\{X_j\}$ and output signal $Y=\{Y_j\}$ of such a channel coincides with the conditional information rate $\overline{I}(X;Y|U)$. A special case of such a channel is investigated in more detail, where $Y=UX+Z$ and $Z=\{Z_j\}$ is an additive noise independent of $X$ and $U$.