Difference between revisions of "Dictionary:Convolution theorem"

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(Convolution in the Frequency domain)
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Similarly, multiplication in the time domain may be interpreted as convolution in the frequency domain, to within a constant factor.
 
Similarly, multiplication in the time domain may be interpreted as convolution in the frequency domain, to within a constant factor.
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Paralleling the derivation above, we write the convolution in the frequency domain
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<center> <math> (F \star G )(\omega) =  K \int_{-\infty}^{\infty}F(\Omega) G(\omega - \Omega) \; d \Omega </math> </center>
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where the constant <math> K </math> remains to be determined.  As above, we substitute the Fourier representations of <math> F(\omega) </math> and <math> G(\omega) </math>
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<center><math> F(\Omega) = \int_{-\infty}^{\infty} f(t) e^{i \Omega t} \; d t </math> </center>
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and
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<center><math> G(\omega - \Omega) =  \int_{-\infty}^{\infty} g(\tau)  e^{-i  (\omega - \Omega) \tau } \; d \tau.</math> </center>
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As in the derivation above, we substitute the Fourier representations of <math> F(\omega) </math> and <math> G(\Omega) </math> and rearrange the terms to yield
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<center> <math> (F \star G) (\omega) = \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]. </math> </center>
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We recognize the term in <math> [ ... ] </math> as the Fourier form of the Dirac delta function
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<center> <math> 2 \pi \delta ( t- \tau )  = 2 \pi \delta( \tau - t ) =  \int_{-\infty}^{\infty}  e^{i \Omega (t -\tau } \; d \Omega . </math> </center>
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As before, we apply the sifting property of the delta function, in this case to perform the <math> \tau </math> integration to yield
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<center> <math> (F \star G) (\omega) = K (2 \pi) \int_{-\infty}^{\infty} f(t) g(t) e^{i \omega t } \; dt </math> </center>
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Hence, the factor of <math> K = 1/2 \pi </math> indicating that the correct representation of convolution in the frequency domain, with these Fourier transform
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definitions is
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<center> <math> (F \star G )(\omega) =  \frac{1}{2 \pi} \int_{-\infty}^{\infty}F(\Omega) G(\omega - \Omega) \; d \Omega. </math> </center>

Revision as of 13:41, 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,


permits us to write the equivalent expression

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


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.

Paralleling the derivation above, we write the convolution in the frequency domain

where the constant remains to be determined. As above, we substitute the Fourier representations of and

and

As in the derivation above, we substitute the Fourier representations of and and rearrange the terms to yield


We recognize the term in as the Fourier form of the Dirac delta function

As before, we apply the sifting property of the delta function, in this case to perform the integration to yield

Hence, the factor of indicating that the correct representation of convolution in the frequency domain, with these Fourier transform definitions is