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.

 

[blue_raver22]

I would first clarify the problem by restating it in simpler terms. It seems that we have two signals, X(t) and G(t), and a random process Y(t) that is equal to the product of X(t) and G(t). We are also given that X(t) and G(t) are both wide sense stationary and have expectation values of 0 and 1, respectively. Additionally, G(t) is a cosine function with a uniform distribution of phase values and is independent of X(t).

The problem asks us to calculate the error, denoted by ε, which is the expected value of the squared difference between X(t) and Y(t). To solve this, we can first transform the signals into the frequency domain using the given transfer function, H(Ω). This will give us the frequency domain representation of Y(Ω).

Next, we need to use the law of total expectation to condition the problem. This means that we need to consider all possible values of the independent variable, in this case, the phase value of G(t). Since the phase is uniformly distributed, we can integrate over all possible values of ψ, which is the variable representing the phase.

To integrate over ψ, we can use the fact that the cosine function is periodic with a period of 2π. This means that we can break up the integration into smaller intervals of 2π and then sum up the results.

Once we have the frequency domain representation of Y(Ω) and have integrated over ψ, we can take the inverse Fourier transform to get the time domain representation of Y(t). Then, we can simply calculate the expected value of the squared difference between X(t) and Y(t) using the given formula.

In summary, to calculate the error function, we need to transform the signals into the frequency domain, integrate over all possible values of the phase, and then take the inverse Fourier transform to get the time domain representation. Finally, we can use the given formula to calculate the expected value of the squared difference between X(t) and Y(t).