Cross-Correlation Theorem

The cross-correlation theorem expresses the Fourier transform of a cross-correlation as the product of the complex conjugate of one Fourier transform with the other. Let denote the cross-correlation of functions and . Then

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

where denotes the Fourier transform, , , and is the complex conjugate. The inverse Fourier transform gives

			(8)
			(9)

The oscillatory integral over in the derivation is interpreted as a distribution, giving the delta function , rather than as a convergent ordinary integral. Applying a Fourier transform on each side gives the cross-correlation theorem,

(10)

If , then the cross-correlation theorem reduces to the Wiener-Khinchin theorem.