Difference between revisions of "Dictionary:Convolution theorem"

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

Revision as of 13:08, 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

yields

Rearranging 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 tradition representation of the equivalence of frequency domain multiplication and convolution