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 of two functions $f(t)$ and $g(t)$ 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

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

where $F(\omega )$ and $G(\omega )$ are the Fourier transforms of $f(t)$ and $g(t),$ respectively.

Substituting these representations into the original integral representation of convolution yields

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

We may rearrange the order of integrations

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

Recognizing the factor in $[...]$ as the frequency domain representation of the Dirac delta function,

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

permits us to write the equivalent expression

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

The $\Omega$ integral may be performed, exploiting the sifting property of the delta function to convert the $\Omega$ to $\omega$ yields the equivalence of multiplication in the frequency domain to convolution in the time domain

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

Convolution in the Frequency domain

Similarly, multiplication in the time domain may be interpreted as convolution in the frequency domain.