I have an input signal x (assumed to be iid Gaussian with $\mu=0$, $\sigma^2$) which is fed into two linear systems:
- $y_1 = h_1 * x$
- $y_2 = h_2 * x$
Now I would like to calculate $\mathbb{E}[y_1 y_2]$. What is the proper way to do this - preferably in the frequency domain?
Background: I want to calculate something like $\operatorname{var}(y_1 + y_2)$. This expands to $\operatorname{var}(y_1)+\operatorname{var}(y_2)+2\mathbb{E}[y_1 y_2]$. Now I am very familar with the first two components - the variance. I know that the variance of a stochastic signal is given by the autocorrelation sequence at position 0. With the Wiener-Kinchin theorem this translates to:
$$ \operatorname{var}(y_1) = r_{yy}(0) = \int_0^{\infty} \Phi_{yy}(f) df $$
and then:
$$ \cdots = \sigma_x^2 \int_0^{\infty} |H_1(f)|^2 df $$
Now for my problem - with $\mathbb{E}[y_1 y_2]$ I arrive at $\mathbb{E}[x^2 \cdots]$ (giving a variance) but the filter is not square magnitude.