The special deterministic case of the Wiener-Khintchine theorem, written up with a slightly different notation for a slightly different project.
As seen in correlograms.
Consider an signal We overload notation and refer to a signal with implied free argument, say, , so that , for example, refers to denote the signal We write the inner product between signals and as . Where it is not clear that the free argument is, e.g. , we will annotate it . Say that is the Fourier transform of some , i.e.
and that is the autocorrelogram, i.e.
What is the Fourier transform of ? That is what the Wiener-Khintchine-relation (Wiener 1930) tells us. Assuming all terms are well-defined (which is non-trivial in general!),
To see this, assuming various terms are indeed well-defined, we use the list of properties of the Fourier transform from Wikipedia and grind out the identity…
Corollary: If we are interested in the power spectrum of the autocorrelogram, we note that it relates non-linearly to that of the source signal.
References
Wiener. 1930.
“Generalized Harmonic Analysis.” Acta Mathematica.
Yaglom. 1987a. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics.
———. 1987b. Correlation Theory of Stationary and Related Random Functions Volume I.