$5.($a$)$ A random signal is given by

X$($t$)=$ Acos$(2$Ft$)$

where A and F are independent continuous random variables. Suppose A is a uniformly

distributed in $(0,10)$ interval and F is uniformly distributed in $(0,200)$ Hz interval.

Find the mean$-$value function of X$($t$).$

$($b$)$ A linear and time$-$invariant $($LTI$)$ lter has the following impulse response:

h$($t$)=$

V$0$ if $$ T

$2<$ t $<$ T

$2$

$0$ otherwise.

In other words, h$($t$)=$ V$0$

$$ t

T

$,$ where $()$ denotes the rectangular function.

A white noise signal with two$-$sided power$-$spectral density p$0$ W$/$Hz is applied as the

input to this lter.

i$.$ Find the power spectral density function of the lter output.

ii$.$ Find the autocorrelation function of the lter output.

iii. Find the average power of the lter output.

iv$.$ Let V$0=2($volts$),$ T $=10$ ms$,$ and p$0=5$ mW$/$Hz$.$ In a given trial, you observe the

lter output voltage $0$:$625$ at a certain time instant. Calculate the best possible

prediction for the lter output voltage that you will see $10$ ms later.