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Convolution Theorem

Let f(t) and g(t) be arbitrary functions of time t with Fourier transforms. Take

f(t)=F_nu^(-1)[F(nu)](t)=int_(-infty)^inftyF(nu)e^(2piinut)dnu
(1)
g(t)=F_nu^(-1)[G(nu)](t)=int_(-infty)^inftyG(nu)e^(2piinut)dnu,
(2)

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=int_(-infty)^inftyg(t^')f(t-t^')dt^'
(3)
=int_(-infty)^inftyg(t^')[int_(-infty)^inftyF(nu)e^(2piinu(t-t^'))dnu]dt^'.
(4)

Interchange the order of integration,

f*g=int_(-infty)^inftyF(nu)[int_(-infty)^inftyg(t^')e^(-2piinut^')dt^']e^(2piinut)dnu
(5)
=int_(-infty)^inftyF(nu)G(nu)e^(2piinut)dnu
(6)
=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 Theorem

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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 Theorem

Cite this as:

Weisstein, Eric W. "Convolution Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConvolutionTheorem.html

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