# Average time-domain signal power from correlated Fourier transforms

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?