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Equations:for X() = X,, Y(+0)=% ELXO) Ya+0)]Rxr ()-LLxfwsJdrdy:Problem 3 (13 points) Two stationary random processes have sample functions of the form: X() = [ cos ...

Question

Equations:for X() = X,, Y(+0)=% ELXO) Ya+0)]Rxr ()-LLxfwsJdrdy:Problem 3 (13 points) Two stationary random processes have sample functions of the form: X() = [ cos ( 1Ot) . Y(t) = N+Mcos (1Ot + 0) Where 0.M and N are random variablesfor 0s0<t fu(m) elsewherefor -28m<2 {4) fv(n) = =€ Jor v8a elsewherefo (0) =OsnsSet up the equation for the cross correlation Ry(r) with all of the correct values substituted in, Do NOT simplify or solve: (Do NOT assume that the process is ergodic )

Equations: for X() = X,, Y(+0)=% ELXO) Ya+0)] Rxr ()-LLxfwsJdrdy: Problem 3 (13 points) Two stationary random processes have sample functions of the form: X() = [ cos ( 1Ot) . Y(t) = N+Mcos (1Ot + 0) Where 0.M and N are random variables for 0s0<t fu(m) elsewhere for -28m<2 {4) fv(n) = =€ Jor v8a elsewhere fo (0) = Osns Set up the equation for the cross correlation Ry(r) with all of the correct values substituted in, Do NOT simplify or solve: (Do NOT assume that the process is ergodic )



Answers

Define a random process $X(t)=A \cos \left(\omega_{0} t+\Theta\right),$ where $A$ and $\Theta$ are independent random variables; $\Theta \sim \operatorname{Unif(}(-\pi \pi, \pi] ;$ and $A$ has mean $\mu_{A}$ and variance $\sigma_{A}^{2} .$ (That is, $X(t)$ models a signal with both phase and amplitude variation.)
(a) Find the mean function of $X(t)$ .
(b) Find the autocorrelation function of $X(t) .$
(c) Is $X(t)$ wide-sense stationary?

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