Calculation of the error function.

[MathematicalPhysicist]

I have the next two signals:

X(t) and G(t) and a random process Y(t)=G(t)X(t) where X(t) and G(t) are wide sense stationary with expectation values: E(X)=0, E(G)=1.

Now, it's also given that $G(t)=\cos(3t+\psi)$ where $\psi$ is uniformly distributed on the interval $(0,2\pi]$ and is statistically independent of X(t).

The signal X(t) is transferred through a low pass filter, given in the frequency domain as $H(\Omega)=1$ when $\Omega \leq 4\pi$ and otherwise zero.

I am given that $Y(\Omega)=X(\Omega)H(\Omega)$, and I want to calculate:

$\epsilon = E((X(t)-Y(t))^2)$

I guess I can go to the frequency domain, but I also need to use the http://en.wikipedia.org/wiki/Law_of_total_expectation

But I am not sure how exactly to condition this, thanks in advance.

 

[Mark44]

For LaTeX, use either $$ on each end or ## (for inline LaTeX) on each end of your expressions. The $ pair is pretty much equivalent to [ tex ] and the # pair is equivalent to [ itex ] (all without extra spaces inside the brackets).

 

[Omega0]

Hmmmmm... I am not that expert but isn't it

epsilon = E((X(t)-Y(t))^2) = E((X-G X)^2)= E((X(1-G))^2) = E(X^2 (1-G)^2) =
E (X^2)*E((1-G)^2) = E^2(X)*E^2(1-G) = 0

 

[MathematicalPhysicist]

Sorry, I have abuse of notation, I should have denoted:
$$Z(\Omega)=Y(\Omega)H(\Omega)$$

And I am looking for $$E((X(t)-Z(t))^2)$$

I was tired yesterday evening, a long exam that day.