# Dictionary:Convolution theorem

Revision as of 12:35, 5 December 2016 by JohnWStockwellJr (talk | contribs)

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

## Fourier domain equivalent

Replacing and by their Fourier domain representations

and

yeilds

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. }**