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

1. Nov 16, 2011

### 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 $$X(t)$$

2. Nov 16, 2011

### Stephen Tashi

Re: Converting a Wide Sense CycloStationary process into a Wide Sense Stationary proc

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.

Code (Text):

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

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:

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