**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Dual of the Convolution Theorem

The

Dual^{8.7}of the Convolution Theorem says thatmultiplication in the time domainisconvolution in thefrequency domain:

Theorem:

Proof:The steps are the same as in the Convolution Theorem.This theorem also bears on the use of

FFT windows. It says thatwindowing in the time domaincorresponds tosmoothing in the frequency domain. That is, the spectrum of is simply filtered by , or, . This smoothing reducessidelobesassociated with therectangular window(which is the window one gets implicitly when no window is explicitly used). FFT windows are covered inMusic 420.