Dictionary:Convolution theorem

The Fourier transform of the convolution of two functions is equal to the product of their individual transforms (or multiplying their amplitude spectra and summing their phase spectra). See Figures F-20 and F-22.

Integral definition

The process of convolution is defined, in one dimension, as

$f\star g(t)=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\;d\tau .$

Fourier domain equivalent

Replacing $f(\tau )$ and $g(t-\tau )$ by their Fourier domain representations

$f(\tau )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )e^{-i\omega \tau }\;d\omega$

and

$g(t-\tau )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }G(\Omega )e^{-i\Omega (t-\tau )}\;d\Omega$

yeilds

$f\star g(t)=\int _{-\infty }^{\infty }\left({\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )e^{-i\omega \tau }\;d\omega \right)\left({\frac {1}{2\pi }}\int _{-\infty }^{\infty }G(\Omega )e^{-i\Omega (t-\tau )}\;d\Omega \right)\;d\tau .$

Rearranging the order of integrations

Failed to parse (syntax error): {\displaystyle f \star g (t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \; d \omega \int_{-\infty}^{\infty} \; d \Omega \int_{-\infty}^{\infty} F(\omega) e^{-i \omega \tau } \; d \omega \right) \left(\frac{1}{2 \pi} \int_{-\infty}^{\infty}F(\Omega) e^{-i \Omega (t - \tau) } \; d \Omega \right) \; d \tau. }