Suppose I have real-valued continuous-time sequences $x_1(t), x_2(t), x_3(t)$ with Fourier transforms $X_1(\omega), X_2(\omega), X_3(\omega)$. I know that $X_3(\omega)=H(\omega)( aX_1(\omega)+bX_2(\omega))$, where $H(\omega), a,b$ are complex, and my ultimate goal is to find $\overline{x_3^2(t)}$, i.e. the average value of $(x_3(t))^2$. Furthermore, $x_1, x_2$ are correlated.
I begin by defining $X_2=X_c+X_u$ where $X_1, X_c$ are fully correlated ($X_c=y_cX_1$, where $y_c$ is complex) and $X_1, X_u$ are completely uncorrelated (also define $x_u(t)$ as the inverse transform of $X_u$). I can then write $X_3=H(\omega)(X_1(a+by_c)+bX_u)$. Now, is the following step justified in general?
$$\overline{x_3(t)}=|H(\omega)|^2 (\overline{x_1^2(t)}|a+by_c|^2+\overline{x_u^2(t)}|b|^2)$$
I think this is skipping a couple steps in going directly from a Fourier-domain description to the time domain. For one thing, I think that the proper intermediate step should be:
$$S_3(\omega)=|H(\omega)|^2 (S_1(\omega)|a+by_c|^2+S_u(\omega)|b|^2)$$
Where the $S(\omega)$ variables are PSDs for the corresponding transforms in units of, e.g. Volt^2/Hz if the $x(t)$ signals are voltages. Then there would be an integration of the above equation over the frequency ranges of interest.
Is this description accurate?
