Information transmission over channels with additive-multiplicative noise

Information transmission over a stationary channel whose output signal Y={Y_j} is equal to the sum Y_j=U_jX_j+Z_j,j=O,±1, …, where X={X_j}, U={U_j}, and Z={Z_j } are the input signal, a multiplicative noise, and an additive noise, respectively, is considered. Under the assumption that the processes U, X, and Z are mutually independent and U is a completely singular process, it is shown that for the information rate I¯(X;Y) in such a channel the equality I¯(X;Y)=I¯(X;Y|U) holds. The asymptotic behavior of I¯(X;Y) in the case of a weak input signal is also investigated