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 $ 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

We may replace $ 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. Here we have used the symbol $ \Omega $ 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

$ (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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Omega to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega yields the equivalence of multiplication in the frequency domain to convolution in the time domain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (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, 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (F \star G )(\omega) = \int_{-\infty}^{\infty}F(\Omega) G(\omega - \Omega) \; d \Omega

As above, we substitute the Fourier representations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F(\omega) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G(\omega)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F(\Omega) = \int_{-\infty}^{\infty} f(t) e^{i \Omega t} \; d t

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G(\omega - \Omega) = \int_{-\infty}^{\infty} g(\tau) e^{i (\omega - \Omega) \tau } \; d \tau.

As in the derivation above, we substitute the Fourier representations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F(\omega) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G(\Omega) and rearrange the terms to yield

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (F \star G) (\omega) = K \int_{-\infty}^{\infty} \; d t \int_{-\infty}^{\infty} \; d \tau \; f(t) g(\tau) \; \left[ \int_{-\infty}^{\infty} e^{i \Omega (t-\tau)} \; d \Omega \right].

We recognize the term in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): [ ... ] as the Fourier form of the Dirac delta function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2 \pi \delta ( t- \tau ) = 2 \pi \delta( \tau - t ) = \int_{-\infty}^{\infty} e^{i \Omega (t -\tau ) } \; d \Omega .

As before, we apply the sifting property of the delta function, in this case to perform the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tau integration to yield

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (F \star G) (\omega) = (2 \pi) \int_{-\infty}^{\infty} f(t) g(t) e^{i \omega t } \; dt

There is an extra factor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2 \pi. Thus, if we were representing the Fourier transform of the product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(t)g(t) as the Frequency domain convolution of their respective Fourier transforms, we would need to include a factor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1/2 \pi in the convolution

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{1}{2 \pi}(F \star G )(\omega) = \int_{-\infty}^{\infty}f(t) g(t) e^{i \omega t } \; dt.

(The extra factor of $ 1/2\pi $ 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F(\omega), 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(t), given the definition of the Fourier transform we are using here. </translate>