I'm almost definitely misunderstanding this, but as far as I can tell we can equate the Fourier transform of the auto-correlation function $R_{xx}(\tau)$ with power spectral density (PSD) like so:

$\hat{R}(\omega) = S_{xx}(\omega) = \lim_{T \to \infty} \textbf{E}[|\hat{x}(\omega)|^{2}]$

Whereas the Fourier transform of the cross correlation is just the complex product of the two functions in frequency space:

$\mathcal{F}\{x(t)∗y(t)\}=\mathcal{F}\{x(t)\}\mathcal{F}\{y(t)\}^{*} $

Why does the expected value/mean come into play with the auto-correlation/PSD?

  • 2
    $\begingroup$ There is a notion of crosscorrelation function of two jointly wide-sense-stationary processes and its Fourier transform is called the cross-power spectral density. These notions bring in the expectations that you expect to find. $\endgroup$ Feb 27 '20 at 15:56
  • $\begingroup$ @DilipSarwate What is the difference between that concept and the cross-correlation cited here? $\endgroup$
    – Petra
    Feb 27 '20 at 15:59
  • $\begingroup$ @Petra your statement "Is just the product of two functions" is the answer here: you're confusing a property of a stochastic process (the PSD) with something that you apply to a deterministic (because observed) signal. Can't generally do that, so "the cross-correlation is just the product of two functions" is wrong, IMHO. $\endgroup$ Feb 27 '20 at 16:24
  • $\begingroup$ I'm not quite sure I understand (sorry!! my knowledge of signal processing is fairly shaky haha). Does it not follow from the convolution theorem that the Fourier transform of a cross-correlation is the complex product of the Fourier transforms of the functions? $\endgroup$
    – Petra
    Feb 27 '20 at 16:35
  • $\begingroup$ no, since the (strictly speaking) cross-correlation is a property of two random processes, not of deterministic functions; in this situation it's good to remember that you can make a difference between sample correlation, which you calculate on two actual signals that you have, and correlation, which is an expectation of two random processes with an infinite amount of realizations (of which you can only observe a finite amount, if at all) $\endgroup$ Feb 27 '20 at 17:52

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