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Mean, Autocorrelation, and Pdf

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Let X(t) be a zero-mean wide-sense stationary Gaussian white noise process with autocorrelation function RXX(τ) = δ(τ). Suppose that X(t) is the input to a linear time-invariant system with an impulse response h(t) = 1[0,T](t) where T > 0. Let Y(t) be the output of the system, and assume that the input has been applied to the system for all time. What is the mean of Y(t)? What is the autocorrelation function of Y(t)? What is the expression for the second-order pdf fY(t1)Y(t2)(y1,y2) of Y(t)? Under what conditions on t1 and t2 will Y(t1) and Y(t2) be statistically independent?

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The solution determines the mean, autocorrelation and pdf.

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