Wiener-Khinchin Theorem

Recall the definition of the autocorrelation function of a function ,

(1)

Also recall that the Fourier transform of is defined by

(2)

giving a complex conjugate of

(3)

Plugging and into the autocorrelation function therefore gives

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

so, amazingly, the autocorrelation is simply given by the Fourier transform of the absolute square of .

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