Convolution Theorem

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Convolution Theorem

The convolution theorem for Fourier transforms states that convolution in the time domain equals multiplication in the frequency domain. The continuous-time convolution of two signals and is defined by

$\displaystyle (x\ast y)(t) \isdef \ensuremath{\int_{-\infty}^{\infty}}x(\tau)y(t-\tau)d\tau.$

The Fourier transform is then

$\begin{eqnarray*} \hbox{\sc FT}_\omega(x\ast y) &\isdef & \int_{-\infty}^\infty... ...d\mbox{(by the \emph{shift theorem})}\\ &=& X(\omega)Y(\omega), \end{eqnarray*}$

or,

$\displaystyle \zbox {x\ast y \longleftrightarrow X\cdot Y.}$

(3.5)

Exercise: Show that

$\displaystyle \zbox {x\cdot y \longleftrightarrow \frac{1}{2\pi}X\ast Y}$

when frequency-domain convolution is defined by

$\displaystyle (X\ast Y)(\omega) \isdef \ensuremath{\int_{-\infty}^{\infty}}X(\nu) Y(\omega-\nu) d\nu,$

where $\nu$ is in radians per second, and that

$\displaystyle \zbox {x\cdot y \longleftrightarrow X\ast Y}$

when frequency-domain convolution is defined by

$\displaystyle (X\ast Y)(f) \isdef \ensuremath{\int_{-\infty}^{\infty}}X(\nu) Y(f-\nu) d\nu,$

with $\nu$ in Hertz.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Chapters

Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Selected Continuous Fourier Theorems
          Convolution Theorem