Sometimes we are interested in the statistical behavior of a linear time-invariant system when the input is a suddenly applied random signal. Such a situation is depicted in Figure.

Let x[n] be a stationary white-noise process. The input to the system, w[n], given by

is a non stationary process, as is the output y [n].

(a) Derive an expression for the mean of the output in terms of the mean of the input.

(b) Derive an expression for the autocorrelation sequence Фyy [n1, n2] of the output.

(c) Show that, for large n, the formulas derived in parts (a) and (b) approach the results for stationary inputs.

(d) Assume that h[n] = anu[n]. Find the mean and mean-square values of the output in terms of the mean and mean-square values of the input. Sketch these parameters as a function of n.