Sometimes we are interested in the statistical behavior of an LTI system when the input...

Sometimes we are interested in the statistical behavior of an LTI system when the input is a suddenly applied random signal. Such a situation is depicted in Figure P2.94.

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

is a nonstationary 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 φ_yy[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.