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Cross-Correlation Theorem

Let f*g denote the cross-correlation of functions f(t) and g(t). Then

f*g=int_(-infty)^inftyf^_(tau)g(t+tau)dtau
(1)
=int_(-infty)^infty[int_(-infty)^inftyF^_(nu)e^(2piinutau)dnuint_(-infty)^inftyG(nu^')e^(-2piinu^'(t+tau))dnu^']dtau
(2)
=int_(-infty)^inftyint_(-infty)^inftyint_(-infty)^inftyF^_(nu)G(nu^')e^(-2piitau(nu^'-nu))e^(-2piinu^'t)dtaudnudnu^'
(3)
=int_(-infty)^inftyint_(-infty)^inftyF^_(nu)G(nu^')e^(-2piinu^'t)[int_(-infty)^inftye^(-2piitau(nu^'-nu))dtau]dnudnu^'
(4)
=int_(-infty)^inftyint_(-infty)^inftyF^_(nu)G(nu^')e^(-2piinu^'t)delta(nu^'-nu)dnu^'dnu
(5)
=int_(-infty)^inftyF^_(nu)G(nu)e^(-2piinut)dnu
(6)
=F[F^_(nu)G(nu)],
(7)

where F denotes the Fourier transform, z^_ is the complex conjugate, and

f(t)=F_nu[F(nu)](t)=int_(-infty)^inftyF(nu)e^(-2piinut)dnu
(8)
g(t)=F_nu[G(nu)](t)=int_(-infty)^inftyG(nu)e^(-2piinut)dnu.
(9)

Applying a Fourier transform on each side gives the cross-correlation theorem,

f*g=F[F^_(nu)G(nu)].
(10)

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

See also

Fourier Transform, Wiener-Khinchin Theorem

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Cite this as:

Weisstein, Eric W. "Cross-Correlation Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Cross-CorrelationTheorem.html

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