Write the integral expression for the two-parameter output correlation function  over the time intervals .

Consider the following mean-square differential equation,

driven by a WSS random process  with psd

The differential equation is subject to the initial condition , where the random variable  has zero-mean, variance 5, and is orthogonal to the input random process .

(a) As a preliminary step, express the deterministic solution to the above differential equation, now regarded as an ordinary differential equation with deterministic input and initial condition , not a random variable. Write your solution as the sum of a zero-input part and a zero-state part.

(b) Now returning to the m.s. differential equation, write the solution random process  as a mean-square convolution integral of the input process  over the time interval  plus a zero-input term due to the random initial condition . Justify the mean-square existence of the terms in your solution.

(c) Write the integral expression for the two-parameter output correlation function  over the time intervals . You do not have to evaluate the integral.

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