# zz-31 - Utah State University ECE 6010 Stochastic Processes...

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Unformatted text preview: Utah State University ECE 6010 Stochastic Processes Homework # 9 Solutions 1. Suppose { X t , t ∈ R } is a ramdom process with power spectral density S X ( ω ) = 1 (1 + ω 2 ) 2 . Find the autocorrelation function of X t . R X ( τ ) = F- 1 { S X ( ω ) } = F- 1 1 (1 + ω 2 ) * F- 1 1 (1 + ω 2 ) = 1 2 e-| τ | * 1 2 e-| τ | = 1 4 Z ∞-∞ e-| t | e-| τ- t | dt = 1 4 Z-∞ e t e t- τ dt + 1 4 Z τ e- t e t- τ dt + 1 4 Z ∞ τ e- t e τ- t dt (for τ ≥ 0) = 1 4 Z-∞ e 2 t- τ dt + Z τ e- τ dt + Z ∞ τ e t- 2 τ dt = 1 4 1 2 e- τ + τe- τ + 1 2 e- τ = 1 4 ( τe- τ + e- τ ) (for τ ≥ 0) Similarly, R X ( τ ) = 1 4 (- τe τ + e τ ) (for τ < 0) Therefore, R X ( τ ) = 1 4 e-| τ | ( | τ | + 1) 2. Suppose that ω is a random variable with p.d.f. f ω and θ is a random variable independent of ω uniformly distributed in (- π, π ) . Define a random process by X t = a cos( ωt + θ ) , t ∈ R where a is a constant. Find the power spectral density of { X t } . E [ X t 1 X t 2 ] = E { a 2 cos( ωt 1 + θ ) cos( ωt 2 + θ ) = 1 2 a 2 E { cos( ωt 1- ωt 2 )- cos( ωt 1 + ωt 2 + 2 θ ) } = 1 2 a 2 E { cos( ωt 1- ωt 2 ) } - E { cos( ωt 1 +...
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## This note was uploaded on 03/01/2012 for the course ECE 6010 taught by Professor Stites,m during the Spring '08 term at Utah State University.

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zz-31 - Utah State University ECE 6010 Stochastic Processes...

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