Hello everybody

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

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

$\displaystyle 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 $\displaystyle E[X(t - \Delta)|\Delta] $

And second, why by averaging the mean of the WSCS process over its period $\displaystyle 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 $\displaystyle Y(t) $ from the autocorrelation function of $\displaystyle X(t) $

Please help me !!