Vtyjkyuk

I have this process:
$$X(t)=Acos(2pi f_0t+Phi)ast operatornamerectleft[fract-T/4T/2right]$$
where:

• $A$ and $Phi$ are uniform random variable in $[0,pi]$ statistically independent


• $ast$ is the convolution

I've found the mean and correlation of the process $B(t)=Acos(2pi f_0t+Phi)$:
$$E[X(t)]=E[A]E[cos(2pi f_0t+Phi)]=int_0^pifrac1pia,daint_0^pifrac1picos(2pi f_0t+phi)dphi=-sin(2pi f_0t)$$
so the mean is time dependent and then not WSS.

The autocorrelation is:
$$R_B(t_1,t_2)=E[Acos(2pi f_0t_1+Phi)Acos(2pi f_0t_2+Phi)]=E[A^2(cos(2pi f_0(t_1-t_2)+cos(2pi f_0(t_1+t_2)+2Phi)]=E[A^2]cos(2pi f_0(t_1-t_2)$$
where $E[cos(2pi f_0(t_1+t_2)+2Phi)] = 0$ since $2Phi$ is uniform in $[0,2pi]$.
So the autocorrelation is time independent and depend only from the delay $t_1-t_2$ but sounds strange and I think I've done some error in the calculus but I don't know where. So can a process hs time dependent mean and time independent autocorrelation?. And from this if the calculus are right, the mean of $X(t)$ I know is $E[X(t)]=h(t)ast E[B(t)]$ but since the process has time independent autocorrelation can I calculate $R_X(t,t-tau)=R_X(tau)=R_B(t)ast R_h(t)$ as if $B(t)$ were a WSS process?