# Phase information from product of complex conjugate transfer functions

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?

• hey, what is $\phi_\cdot$ in your question? And where's the conjugate product of transfer functions from your title (maybe I'm just confused)? Mar 10, 2018 at 14:07
• $\phi$ is power spectral density function. $H(\omega)\cdot H^*(\omega) = {\lvert H(\omega)\rvert}^2$ Mar 10, 2018 at 15:37

The relationship you show is well known and used to relate the input to output of Gaussian stochastic processes through a linear filter. Any filter that has the same magnitude response $\mid H(\omega ) \mid$ will produce the same result.
The easiest way would be $$H(\omega)=\frac{Y(\omega)}{X(\omega)}$$ which gives you both magnitude and phase.