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

  • #1
Hello everybody

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

The problem begins with the method to find the mean function of [tex] Y(t) [/tex] :

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

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

And second, why by averaging the mean of the WSCS process over its period [tex] T [/tex] 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 [tex] Y(t) [/tex] from the autocorrelation function of [tex] X(t) [/tex]

Please help me !!
 

Answers and Replies

  • #2
Stephen Tashi
Science Advisor
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1,465


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:
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 [itex] X(t) [/itex] to a WSS process [itex] Y(t) = X(t - \Delta) [/itex].
I don't get the syntax [tex] E[X(t - \Delta)|\Delta] [/tex]
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 [itex] X(t - \Delta) [/itex] for one particular value of [itex]\Delta[/itex].

So:

[tex]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 [/tex]
 

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