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

(B.15) |

The Fourier transform is then

or,

(B.16) |

Exercise:Show that

(B.17) |

when frequency-domain convolution is defined by

(B.18) |

where is in radians per second, and that

(B.19) |

when frequency-domain convolution is defined by

(B.20) |

with in Hertz.

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University