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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.

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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.

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  • $\begingroup$ Thanks! Would you happen to know any resources by any chance on this topic? Any good textbook recommendations or something like that? $\endgroup$
    – Inf_E
    Dec 22, 2020 at 18:59
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    $\begingroup$ @Inf_E: A good book is Probability, Random Variables, and Stochastic Processes by Papoulis. Maybe a bit too in-depth for now, but give it a try. $\endgroup$
    – Matt L.
    Dec 22, 2020 at 19:34

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