How Does Time Shifting Convert a WSCS Process to a WSS Process?

[MajorGrubert]

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 $$X(t)$$

Please help me !

 

[Stephen Tashi]

On this forum, the LaTex will appear more gracefully if you use the "itex" tag when you want the expression to appear inline with the text.

the conversion from a WSCS process [itex] X(t) [/itex] to a WSS process [itex] Y(t) = X(t - \Delta) [/itex].

the conversion from a WSCS process $X(t)$ to a WSS process $Y(t) = X(t - \Delta)$.

I don't get the syntax $$E[X(t - \Delta)|\Delta]$$

On my screen the square brackets are hard to distinguish from the vertical bars. My interpretation is that it is a "conditional expectation". Roughly speaking, compute the expected value of $X(t - \Delta)$ for one particular value of $\Delta$.

So:

$$E( E\{X(t-\Delta)| \Delta\}) = \int_{y=0}^{y=T} \bigg( \int_{-\infty}^{\infty} (x) p(X(t-y)=x)| \Delta=y) dx \bigg) p(\Delta=y) dy$$