Cross-Correlation Theorem

Let $f\star g$ denote the Cross-Correlation of functions $f(t)$ and $g(t)$. Then

$f\star g = \int_{-\infty}^\infty f^*(\tau)g(t+\tau)\,d\tau$
$= \int_{-\infty}^\infty \left[{\int_{-\infty}^\infty F^*(\nu)e^{2\pi i\nu\tau}\... ... \int_{-\infty}^\infty G(\nu') e^{-2\pi i\nu'(t+\tau)}\,d\nu'}\right]\!\! d\tau$
$= \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty F^*(\nu)G(\nu') e^{-2\pi i\tau(\nu'-\nu)}e^{-2\pi i\nu't} \,d\tau\,d\nu\,d\nu'$
$= \int_{-\infty}^\infty \int_{-\infty}^\infty \! F^*(\nu)G(\nu')e^{-2\pi i\nu't... ...int_{-\infty}^\infty \! e^{-2\pi i\tau(\nu'-\nu)}\,d\tau}\right]\!\!d\nu\,d\nu'$
$= \int_{-\infty}^\infty \int_{-\infty}^\infty F^*(\nu)G(\nu')e^{-2\pi i\nu't}\delta(\nu'-\nu)\,d\nu'\,d\nu$
$= \int_{-\infty}^\infty F^*(\nu)G(\nu)e^{-2\pi i\nu t}\,d\nu$
$= {\mathcal F}[F^*(\nu)G(\nu)],$	(1)

where ${\mathcal F}$ denotes the Fourier Transform and

$$f(t)\equiv {\mathcal F}[F(\nu)] = \int_{-\infty}^\infty F(\nu)e^{-2\pi i\nu t}\,dt$$ (2)

$$g(t)\equiv {\mathcal F}[G(\nu)] = \int_{-\infty}^\infty G(\nu)e^{-2\pi i\nu t}\,dt.$$ (3)

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

$$f\star g={\mathcal F}[F^*(\nu)G(\nu)].$$ (4)

If $F=G$, then the cross-correlation theorem reduces to the Wiener-Khintchine Theorem.

See also Fourier Transform, Wiener-Khintchine Theorem