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8. Consider a sinusoid with a random frequency X(t) = cos(2?Ft), where F is a random variable uniformly distributed over some interval (0, f0). (12 points) a) Plot three arbitrary member functions of X(t). (3 points) b) Find and plot the mean function ?X(t). (3 points) c) Find the autocorrelation function RXX(t, t + ?) (3 points) d) Is the random process WSS? Why? (3 points) (Hint: ??? cos(??ft)df = sin(??tA)/??t = Asinc(?At) where sinc(x) = sin(?x)/?x) sin A sin B = 1/2(cos(A - B) - cos(A + B)) cos A cos B = 1/2(cos(A - B) + cos(A + B)) sin A cos B = 1/2(sin(A - B) + sin(A + B))

          8. Consider a sinusoid with a random frequency X(t) = cos(2?Ft), where F is a random variable uniformly distributed over some interval (0, f0). (12 points)
a) Plot three arbitrary member functions of X(t). (3 points)
b) Find and plot the mean function ?X(t). (3 points)
c) Find the autocorrelation function RXX(t, t + ?) (3 points)
d) Is the random process WSS? Why? (3 points)
(Hint: ??? cos(??ft)df = sin(??tA)/??t = Asinc(?At) where sinc(x) = sin(?x)/?x)
sin A sin B = 1/2(cos(A - B) - cos(A + B))
cos A cos B = 1/2(cos(A - B) + cos(A + B))
sin A cos B = 1/2(sin(A - B) + sin(A + B))

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8. Consider a sinusoid with a random frequency X(t) = cos(2?Ft), where F is a random variable uniformly distributed over some interval (0, f0). (12 points)
a) Plot three arbitrary member functions of X(t). (3 points)
b) Find and plot the mean function ?X(t). (3 points)
c) Find the autocorrelation function RXX(t, t + ?) (3 points)
d) Is the random process WSS? Why? (3 points)
(Hint: ??? cos(??ft)df = sin(??tA)/??t = Asinc(?At) where sinc(x) = sin(?x)/?x)
sin A sin B = 1/2(cos(A - B) - cos(A + B))
cos A cos B = 1/2(cos(A - B) + cos(A + B))
sin A cos B = 1/2(sin(A - B) + sin(A + B))

Added by Shawn L.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Adi S

07/02/2023

Step 1

For example, if we choose F1 = 0.25fo, F2 = 0.5fo, and F3 = 0.75fo, we get: X1t = cos(0.5πt) X2t = cos(πt) X3t = cos(1.5πt) These are three arbitrary member functions of Xt. b) Show more…

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8. Consider a sinusoid with a random frequency X(t) = cos(2̀πFt), where F is a random variable uniformly distributed over some interval (0, f₀). a) Plot three arbitrary member functions of X(t). b) Find and plot the mean function μₓ(t). c) Find the autocorrelation function Rₓₓ(t, t + τ). d) Is the random process WSS? Why?

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