Problem $3$:

Let $x(t)$ be a random process with autocorrelation function $R_x()=2().$ The random process $x(t)$ is input to an LTI system whose frequency response is given by:

$H()=3$rect$(\frac{-20}{4})+3$rect$(\frac{+20}{4}).$

Let $Y(t)$ be the output random process.

Find the average power of $x(t)$ using its autocorrelation function.

Find the power spectral density $($PSD$)$ of $x(t).$

Find the average power of $x(t)$ using its power spectral density $($PSD$).$

Find the PSD of $Y(t).$

Find the average power of $Y(t)$ using its PSD$.$

Find the autocorrelation function of $Y(t).$

Find the average power of $Y(t)$ using its autocorrelation function.