Dictionary:Convolution theorem

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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 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