Wiener-Khinchin theorem

The relations can be derived using the Wiener-Khinchin theorem.

Fortunately, the Wiener-Khinchin theorem provides a simple alternative.

A stationary random process does have an autocorrelation function and hence a spectral density according to the Wiener-Khinchin theorem.

The Wiener-Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest 1/f noise.

By an extension of the Wiener-Khinchin theorem, the Fourier transform of the cross-spectral density is the cross-covariance function.

The Wiener-Khinchin theorem relates the autocorrelation function to the power spectral density via the Fourier transform:

For real-valued functions, the symmetric autocorrelation function has a real symmetric transform, so the Wiener-Khinchin theorem can be re-expressed in terms of real cosines only:

The Fourier transform is often applied to spectra of infinite signals via the Wiener-Khinchin theorem even when Fourier transforms of the signals do not exist.

The Wiener-Khinchin theorem states that the Fourier transform of the field autocorrelation is the spectrum of , i.e., the square of the magnitude of the Fourier transform of .

This type of GRF is completely described by its power spectral density, and hence, through the Wiener-Khinchin theorem, by its two-point autocorrelation function, which is related to the power spectral density through a Fourier transformation.

A deep theorem that was worked out by Norbert Wiener and Aleksandr Khinchin (the Wiener-Khinchin theorem) makes sense of this formula for any wide-sense stationary process under weaker hypotheses: does not need to be absolutely integrable, it only needs to exist.

The term Fourier transform spectroscopy reflects the fact that in all these techniques, a Fourier transform is required to turn the raw data into the actual spectrum, and in many of the cases in optics involving interferometers, is based on the Wiener-Khinchin theorem.