## Converting a Wide Sense CycloStationary process into a Wide Sense Stationary process

Hello everybody

I have a bit of a problem with understanding the conversion from a WSCS process $X(t)$ to a WSS process $Y(t) = X(t - \Delta)$. With $\Delta$ the time shift being a uniform random variable on $(0,T)$, independent of $X(t)$ and $T$ being the period of the mean function of $X(t)$

The problem begins with the method to find the mean function of $Y(t)$ :

$m_{Y} = E\{X(t - \Delta)\} = E\{E[X(t - \Delta)|\Delta]\} = E\{m_{X}(t - \Delta)\}$

First, and it might seem very basic, I don't get the syntax $E[X(t - \Delta)|\Delta]$

And second, why by averaging the mean of the WSCS process over its period $T$ would we get the mean function of the WSS process ?

If I understand that I could understand the same kind of process used to find the autocorrelation function of $Y(t)$ from the autocorrelation function of $X(t)$