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I am looking at using Matlab to estimate the transfer function of an unknown system using input and output time history data. I found this page:

https://www.mathworks.com/help/dsp/ug/estimate-the-transfer-function-of-an-unknown-system-1.html

It states:

The relationship between the input x and output y is modeled by the linear, time-invariant transfer function Txy. The transfer function is the ratio of the cross power spectral density of x and y, Pyx, to the power spectral density of x, Pxx:

Txy(f)=Pyx(f)/Pxx(f)

Does anyone know why that is true? I found some questions on Stack Exchange that say it would be: Txy(f)=Pyy(f)/Pxx(f). So not sure where the cross PSD comes from.

I am also confused as to how this gives Phase information. Doesn't the PSD just have magnitude?

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  • \$\begingroup\$ You could use the System Identification toolbox. \$\endgroup\$ – Chu Jan 28 at 8:25

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