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# zz-32 - Utah State University ECE 6010 Stochastic Processes...

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Utah State University ECE 6010 Stochastic Processes Homework # 8 Solutions 1. Suppose { X t , t 0 } is a Wiener process. Define a process { Y t , t 0 } by Y t = X t + D - X t for a fixed positive number D . (a) Find the mean and autocorrelation functions of { Y t } . Mean : μ Y ( t ) = E [ Y t ] = E [ X t + D ] - E [ X t ] = μ ( t + D ) - μ ( t ) = μD Autocorrelation : R Y ( t, s ) = E [( X t + D - X t )( X s + D - X s )] = E [ X t + D X s + D ] - E [ X t + D X s ] - E [ X t X s + D ] + E [ X t X s ] = [ σ 2 min( t + D, s + D ) + μ 2 ( t + D )( s + D )] - [ σ 2 min( t + D, s ) + μ 2 ( t + D ) s ] - [ σ 2 min( t, s + D ) + μ 2 ( S + D ) t ] + [ σ 2 min( t, s ) + μ 2 st ] = σ 2 [min( t + D, s + D ) - min( t + D, s ) - min( t, s + D ) + min( t, s )] + μ 2 [ ts + Dt + sD + D 2 - st - sD - st - Dt + st ] = σ 2 [ s + D - s - min( t, s + D ) + s ] + μD 2 t s σ 2 [ t + D - t - min( t, s + D ) + t ] + μD 2 t < s = μ 2 D 2 t s , t - s - D 0 σ 2 [ s - t + D ] + μ 2 D 2 t s , t - s - D < 0 μ 2 D 2 t < s , t - s + D < 0 σ 2 [ t - s + D ] + μ 2 D 2 t < s , t - s + D 0 = μ 2 D 2 | t - s | ≥ D σ 2 [ D - | t - s | ] + μ 2 D 2 | t - s | < D So R Y ( t, s ) = σ 2 ax (0 , D - | t - s | ) + μ 2 D 2 . (b) Show that { Y t } is a stationary and find its spectrum. μ y ( t ) is a constant and R Y ( t, s ) = R Y ( t - s ) W.S.S. Since Gaussian too Strictly Stationary. R Y ( τ ) = μ 2 D 2 + rect 2 τ D * rect 2 τ D σ 2 D So, S Y ( ω ) = F{ R Y ( τ ) } = 2 πμ 2 D 2 δ ( ω ) + σ 2 D sin 2 ( ωD/ 2) ( ωD/ 2) 2 2. Suppose { X t } and { Y t } are zero mean and individually and jointly W.S.S. Show that the mean-

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