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
We may replace and by their Fourier domain representations
where and are the Fourier transforms of and respectively. Here we have used the symbol to represent frequency in the second integral as a dummy
variable in the integration, to avoid ambiguity when combining the integral representations in the next step.
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,
permits us to write the equivalent expression
The integral may be performed, exploiting the sifting property of the delta function to convert the to yields the equivalence of
multiplication in the frequency domain to convolution in the time domain
Convolution in the Frequency domain
Similarly, multiplication in the time domain may be interpreted as convolution in the frequency domain, to within a constant factor. Such
a frequency domain convolution representation is useful, for example, if we were interested in finding the Fourier transform of the product of functions
of known Fourier transform.
Paralleling the derivation above, we write the convolution in the frequency domain
As above, we substitute the Fourier representations of and
As in the derivation above, we substitute the Fourier representations of and and rearrange the terms to yield
We recognize the term in as the Fourier form of the Dirac delta function
As before, we apply the sifting property of the delta function, in this case to perform the integration to yield
There is an extra factor of Thus, if we were representing the Fourier transform of the product
as the Frequency domain convolution of their respective Fourier transforms, we
would need to include a factor of in the convolution
(The extra factor of echoes what is seen in other results involving the Fourier transform, such as Parseval's relation.
To come full circle, if we were to perform the inverse Fourier transform of considered as a purely causal function,
we would need to treat this as a contour integral, and choose
the contour along the real axis to pass over any poles that would be on the real axis, to obtain the causal function
given the definition of the Fourier transform we are using here.