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.
The process of convolution of two functions and is defined in one dimension, as
Fourier domain equivalent
Replacing and by their Fourier domain representations
where and are the Fourier transforms of and respectively.
Substituting these representations into the original integral representation of convolution yields
We may rearrange the order of integrations
Recognizing the factor in as the frequency domain representation of the Dirac delta function
Permitting the following to be written
The integral may be performed, exploiting the sifting property of the delta function to yield the equivalence of
frequency domain multiplication and convolution