Compute the correlation function rhotau for the output of

Question

Consider a second-order system with dynamics

that is forced by Gaussian white noise w with zero mean and variance σ². Assume a,b > 0.

(i) Compute the correlation function ρ(τ) for the output of the system. Your answer should be an explicit formula in terms of a, b and σ.

(ii) Assuming that the input transients have died out, compute the mean and variance of the output.