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

• Nov 16th 2011, 01:16 AM
MajorGrubert
Converting a Wide Sense CycloStationary process into a Wide Sense Stationary process
Hello everybody (Hi)

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] \$ (Thinking)

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 ? (Headbang)

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) \$