Difference between revisions of "Dictionary:Convolution theorem"

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(Fourier domain equivalent)
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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>
 
<center><math> g(t-\tau) = \frac{1}{2 \pi}  \int_{-\infty}^{\infty} G(\Omega) e^{-i \Omega (t - \tau) } \; d \Omega </math> </center>
  
yields
+
where <math> F(\omega) </math> and <math> G(\omega) </math> are the Fourier transforms of <math> f(t) </math> and <math> g(t),  </math>  respectively.
 +
 
 +
Substituting these representations into the original integral representation of convolution yields  
  
 
<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>
 
<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>
  
Rearranging the order of integrations
+
We may rearrange the order of integrations
  
<center> <math> (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]. </math> </center>
+
<center> <math> (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]. </math> </center>
  
 
Recognizing the factor in <math> [ ... ] </math> as the frequency domain representation of the [[Dirac delta function]]
 
Recognizing the factor in <math> [ ... ] </math> as the frequency domain representation of the [[Dirac delta function]]
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Permitting the following to be written
 
Permitting the following to be written
  
<center> <math> (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 ). </math> </center>
+
<center> <math> (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 ). </math> </center>
  
The <math> \Omega </math> integral may be performed, exploiting the [[sifting property]] of the delta function to yield the tradition representation of the equivalence of
+
The <math> \Omega </math> integral may be performed, exploiting the [[sifting property]] of the delta function to yield the equivalence of
 
frequency domain multiplication and convolution
 
frequency domain multiplication and convolution
  
  
 
<center> <math> (f \star g) (t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty}    F(\omega) G(\omega) e^{-i \Omega t} \; d \omega . </math> </center>
 
<center> <math> (f \star g) (t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty}    F(\omega) G(\omega) e^{-i \Omega t} \; d \omega . </math> </center>

Revision as of 13:15, 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 of two functions and is defined in one dimension, as


Fourier domain equivalent

Replacing and by their Fourier domain representations

and

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