Let X$($t$)$ denote a $($real$,$ zero$-$mean, WSS$)$ bandpass process with autocorrelation function

RX $(\backslash$tau $)$ and power spectral density SX $($ f $),$ where SX $(0)=0,$ and let X$($t$)$ denote the

Hilbert transform of X$($t$).$ Then X$($t$)$ can be viewed as the output of a filter, with impulse

response $1$

$\backslash$pi t and transfer function $$ jsgn$($ f $),$ whose input is X$($t$).$ Recall that when X$($t$)$

passes through a system with transfer function H$($ f $)$ and the output is Y $($t$),$ we have

SY $($ f $)=$ SX $($ f $)|$H$($ f $)|$

$2$ and SXY $($ f $)=$ SX $($ f $)$H$($ f $).$

$1.$ Prove that RX$(\backslash$tau $)=$ RX $(\backslash$tau $).$

$2.$ Prove that RX X$(\backslash$tau $)=$R$$ X $(\backslash$tau $)$

$3.$ If Z$($t$)=$ X$($t$)+$ j X$($t$),$ determine SZ $($ f $).$

$4.$ Define Xl$($t$)=$ Z$($t$)$e$$ j$2\backslash$pi f$0$t

$.$ Show that Xl$($t$)$ is a lowpass WSS random process, and

determine SXl$($ f $).$ From the expression for SXl$($ f $),$ derive an expression for RXl$(\backslash$tau $).$