Difference between revisions of "Dictionary:Convolution theorem"

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{{lowercase}}{{#category_index:C|convolution theorem}}
 
{{lowercase}}{{#category_index:C|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 [[Dictionary:Fig_F-20|F-20]] and [[Dictionary:Fig_F-22|F-22]].
 
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 [[Dictionary:Fig_F-20|F-20]] and [[Dictionary:Fig_F-22|F-22]].
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== Integral definition ==
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The process of convolution is defined, in one dimension, as
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<center> <math> f \star g (t) =  \int_{-\infty}^{\infty}f(\tau) g(t - \tau) \; d \tau. </math> </center>
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== Fourier domain equivalent ==
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Replacing <math> f(\tau) </math> and <math> g(t - \tau) </math> by their [[Fourier transform|Fourier domain]] representations
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<center><math> f(\tau) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) e^{-i \omega \tau } \; d \omega </math> </center>
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and
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<center><math> g(t-\tau) = \frac{1}{2 \pi}  \int_{-\infty}^{\infty} G(\Omega) e^{-i \Omega (t - \tau) } \; d \Omega </math> </center>
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yeilds
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<center> <math> 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. </math> </center>
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Rearranging the order of integrations
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<center> <math> f \star g (t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \; d \omega  \int_{-\infty}^{\infty} \; d \Omega \int_{-\infty}^{\infty} F(\omega) e^{-i \omega \tau } \; d \omega \right) \left(\frac{1}{2 \pi}  \int_{-\infty}^{\infty}F(\Omega) e^{-i \Omega (t - \tau) } \; d \Omega  \right)  \; d \tau. </math> </center>

Revision as of 12:35, 5 December 2016

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 is defined, in one dimension, as


Fourier domain equivalent

Replacing and by their Fourier domain representations

and

yeilds

Rearranging the order of integrations

Failed to parse (syntax error): {\displaystyle f \star g (t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \; d \omega \int_{-\infty}^{\infty} \; d \Omega \int_{-\infty}^{\infty} F(\omega) e^{-i \omega \tau } \; d \omega \right) \left(\frac{1}{2 \pi} \int_{-\infty}^{\infty}F(\Omega) e^{-i \Omega (t - \tau) } \; d \Omega \right) \; d \tau. }