Question $2$

$[15$ points$]$

$(1)$ Let $x(t)$ and $Y(t)$ be defined by

$x(t)=$Ucos${}_{0}t+$Vsin${}_{0}t$

$Y(t)=$Vcos${}_{0}t+$Usin${}_{0}t$

where ${}_0$ is constant and $U$ and $V$ are independent r$.$v$.\text{'}$s both having zero mean and variance ${}^2$

A$.[5$ points$]$ Find the cross$-$correlation function of $x(t)$ and $Y(t)$

B$.[5$ points$]$ Find the cross power spectral density of $x(t)$ and $Y(t)$

$(2)[5$ points$]$ A WSS random process $x(t)$ is applied to the input of an LTI system with impulse response $h(t)=3e^{-2t}u(t).$ Find the mean value of $Y(t)$ of the system if $E[x(t)]=2$