Wiener-Khinchin Theorem

The Wiener-Khinchin theorem relates the autocorrelation of a function to the absolute square of its Fourier transform. Recall the definition of the autocorrelation function of a function ,

(1)

Let denote the Fourier transform of . The inverse Fourier transform gives

(2)

giving a complex conjugate of

(3)

Plugging and into the autocorrelation function therefore gives

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

where the integral over giving the delta function is interpreted as a distribution. Thus the autocorrelation is the inverse Fourier transform of the absolute square of .

The Wiener-Khinchin theorem is a special case of the cross-correlation theorem with .