sol4 - Solution of Assignment 4 Problem 4.6 By Problem 4.5...

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Solution of Assignment 4 Problem 4.6 By Problem 4.5, let { Y t } and { W t } be two uncorrelated stationary processes with spectral distribution functions F Y ( . ) and F W ( . ). Then, the spectral distribution function of { Y t + W t } is given by F W ( . ) + F Y ( . ) . Now, let W t be the following process W t = Acos ( πt/ 3) + Bsin ( πt/ 3) where A and B are uncorrelated with mean 0 and variance ν 2 . The process Y t is an MA(1) process Y t = Z t + 2 . 5 Z t - 1 where Z t is WN(0, σ 2 ) and the process Z t is uncorrelated with A and B . Therefore Y t and W t are uncorrelated. This implies that F X ( λ ) = F W ( λ ) + F Y ( λ ). Given that γ W ( h ) = ν 2 cos ( πh/ 3) we have that the spectral distribution function of { W t } is F W ( λ ) = 0 if - π λ < - π/ 3 0 . 5 ν 2 if - π/ 3 λ < π/ 3 ν 2 if π/ 3 λ On the other hand, the spectral density of the Y t process ( f Y ( λ ) = σ 2 2 π (1 + 2 . 5 2 + 5 cos ( λ ))) implies that the spectral distribution function of { Y t } is F Y ( λ ) = λ - π f Y ( ω ) = σ 2 2 π λ - π 7 . 25 + 5 cos ( ω ) = σ 2 2 π (7 . 25( λ + π ) + 5 sin ( λ )) Thus, the spectral distribution function of the process { X t } F X ( λ ) = σ 2 2 π (7 . 25( λ + π ) + 5 sin ( λ )) if - π λ < - π/ 3 0 . 5 ν 2 + σ 2 2 π (7 . 25( λ + π ) + 5 sin ( λ )) if - π/ 3 λ < π/ 3 ν 2 + σ 2 2 π (7 . 25( λ + π ) + 5 sin ( λ )) π/ 3 λ .
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