Convolution Theorem
Let f(t) and g(t) be arbitrary functions of time t with Fourier transforms. Take
=int_(-infty)^inftyF(nu)e^(2piinut)dnu](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ConvolutionTheorem/Inline6.svg){kind=link}
=int_(-infty)^inftyG(nu)e^(2piinut)dnu,](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ConvolutionTheorem/Inline9.svg){kind=link}
where F_nu^(-1)(t) denotes the inverse Fourier transform (where the transform pair is defined to have constants A=1 and B=-2pi). Then the convolution is
f*g = [画像:int_(-infty)^inftyg(t^')f(t-t^')dt^']
(3)
![画像:int_(-infty)^inftyg(t^')[int_(-infty)^inftyF(nu)e^(2piinu(t-t^'))dnu]dt^'.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ConvolutionTheorem/Inline18.svg){kind=link}
Interchange the order of integration,
![画像:int_(-infty)^inftyF(nu)[int_(-infty)^inftyg(t^')e^(-2piinut^')dt^']e^(2piinut)dnu](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ConvolutionTheorem/Inline21.svg){kind=link}
= F_nu^(-1)[F(nu)G(nu)](t).
(7)
So, applying a Fourier transform to each side, we have
| F[f*g]=F[f]F[g]. |
(8)
|
The convolution theorem also takes the alternate forms
F[fg] = F[f]*F[g]
(9)
F^(-1)(F[f]F[g]) = f*g
(10)
F^(-1)(F[f]*F[g]) = fg.
(11)
See also
Autocorrelation, Convolution, Fourier Transform, Wiener-Khinchin TheoremExplore with Wolfram|Alpha
WolframAlpha
References
Arfken, G. "Convolution Theorem." §15.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 810-814, 1985.Bracewell, R. "Convolution Theorem." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 108-112, 1999.Referenced on Wolfram|Alpha
Convolution TheoremCite this as:
Weisstein, Eric W. "Convolution Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConvolutionTheorem.html