For an input signal $x(t)$ and output signal $y(t)$ through an LTI system I $H(t)$ I have found and interesting property that concerns signals' power spectral densities:
$${\lvert H(\omega)\rvert}^2 = \frac{\phi_y(\omega)}{\phi_x(\omega)}$$
This relation gives me information on the magnitude of $H(\omega)$ but no phase information. Is there a way to obtain phase information of $H(\omega)$ from this relation?