# Calculation of the error function.

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 transfered 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 $Z(\Omega)=X(\Omega)H(\Omega)$, and I want to calculate:

$\epsilon = E[(X(t)-Z(t))^2]$

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.

Edit: I changed the abuse of notation.

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In other words, $X(t)-Z(t)=0$ if $|t|\leqslant4\pi$ and $X(t)-Z(t)=X(t)$ if $|t|\gt4\pi$? –  Did Apr 7 at 12:31