# What is the relation between input and output PSDs given system transfer function $H(s)$

If I have the system transfer function $$H(s)$$ in the complex frequency domain, how would I relate the input/output power spectral densities?

I have come across the relation $$P_{out}(f) = |H(f)|^2P_{in}(f)$$ in the frequency domain, where $$P_{out/in}(f)$$ refer to the input and output PSDs. Would I be able to use this same relation in the complex frequency domain as $$P_{out} = |H(i\omega)|^2P_{in}$$? Although I suppose that would mean the PSD would be in complex frequency domain as well?

This is all very new to me so any clarification or resources that I could look at would be greatly appreciated.

If the system described by the transfer function $$H(s)$$ is stable, you can obtain its frequency response by substituting $$s=j\omega$$, and use the relation that you found:
$$S_Y(\omega)=S_X(\omega)\big|H(j\omega)\big|^2\tag{1}$$
where $$S_X(\omega)$$ and $$S_Y(\omega)$$ denote the power spectra of the system's input and its output, respectively.