Assume we have the following system (coming from control systems theory, hence in s-domain)
$ Y(s) = H_A (s) \cdot A(s) - H_B (s) \cdot B(s) $
I now wish to consider $a(t)$ and $b(t)$ as white noise of unit variance, and I'm interested in the Power spectral density of $y(t)$ (rather the RMS of y(t) derived via the integral of the PSD of $y(t)$, but regardless).
Intuition tells me, that I should get something along the lines of
$ |Y(j\omega)|^2 = |H_A (j\omega)|^2 \cdot 1 + |H_B (j\omega)|^2 \cdot 1 $
But I cannot show how. Especially the switch from subtraction to addition stumps me.