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Question

X(t) is a WSS random process with an average power of 2 W, another random process Y(t) is defined as,
Y(t)=X(t) cos (ωot+θ)
Where, θ is a random variable which is uniformly distributed in the interval [0,2π] and ω0 is a constant. If X(t) and θ are statistically independent, then the average power of the random process Y(t) will be

A
4 W
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B
2 W
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C
zero
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D
1 W
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Solution

The correct option is D 1 W
Y(t)=X(t) cos (ωot+θ)



ACF of Y(t), RY(τ)=E[Y(t)Y(t+τ)]

=E[X(t)cos(ωot+θ)X(t+τ)cos(ωot+θ+ωoτ)]

=E[X(t) X(t+τ)]E[cos (ωot+θ)cos(ωot+θ+ω0τ)]

=RX(τ)[12cosω0τ+12E[cos(2ωot+2θ+ω0τ)]]

E[cos(2ω0t+ω0τ+2θ)]=E[cos (α+2θ)]=cos(α+2θ)fθ(θ)dθ

=2π0cos(α+2θ)12πdθ=0
So, RY(τ)=12RX(τ)cos(ω0τ)
Given that, the average power of X(t)=RX(0)=2W
Average power of Y(t)=RY(0)=12RX(0)=12(2)=1W

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