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Convolution Theorem

Let $f(t)$ and $f(t)$ be arbitrary functions of time $t$ with Fourier Transforms. Take

f(t) = {\mathcal F}^{-1}[F(\nu)]=\int_{-\infty}^\infty F(\nu)e^{2\pi i\nu t}\,d\nu
\end{displaymath} (1)

g(t) = {\mathcal F}^{-1}[G(\nu)]=\int_{-\infty}^\infty G(\nu)e^{2\pi i\nu t}\,d\nu,
\end{displaymath} (2)

where ${\mathcal F}^{-1}$ denotes the inverse Fourier Transform (where the transform pair is defined to have constants $A=1$ and $B=-2\pi$). Then the Convolution is
$\displaystyle f*g$ $\textstyle \equiv$ $\displaystyle \int_{-\infty}^\infty g(t')f(t-t')dt'$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty g(t')\left[{\int_{-\infty}^\infty F(\nu)e^{2\pi i\nu(t-t')} \,d\nu}\right]\,dt'.$ (3)

Interchange the order of integration,
$\displaystyle f*g$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty F(\nu)\left[{\int_{-\infty}^\infty g(t')e^{-2\pi i\nu t'}\,dt'}\right]e^{2\pi i\nu t}\,d\nu$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty F(\nu)G(\nu)e^{2\pi i\nu t}\,d\nu$  
  $\textstyle =$ $\displaystyle {\mathcal F}^{-1}[F(\nu)G(\nu)].$ (4)

So, applying a Fourier Transform to each side, we have
{\mathcal F}[f*g] = {\mathcal F}[f]{\mathcal F}[g].
\end{displaymath} (5)

The convolution theorem also takes the alternate forms
$\displaystyle {\mathcal F}[fg]$ $\textstyle =$ $\displaystyle {\mathcal F}[f]*{\mathcal F}[g]$ (6)
$\displaystyle {\mathcal F}({\mathcal F}[f]{\mathcal F}[g])$ $\textstyle =$ $\displaystyle f*g$ (7)
$\displaystyle {\mathcal F}({\mathcal F}[f]*{\mathcal F}[g])$ $\textstyle =$ $\displaystyle fg.$ (8)

See also Autocorrelation, Convolution, Fourier Transform, Wiener-Khintchine Theorem


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. New York: McGraw-Hill, pp. 108-112, 1965.

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© 1996-9 Eric W. Weisstein