Dual of the Convolution Theorem Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


Dual of the Convolution Theorem

The dual7.13 of the convolution theorem says that multiplication in the time domain is convolution in the frequency domain:



Theorem:

$\displaystyle \zbox {x\cdot y \;\longleftrightarrow\;\frac{1}{N} X\circledast Y} $



Proof: The steps are the same as in the convolution theorem.

This theorem also bears on the use of FFT windows. It implies that windowing in the time domain corresponds to smoothing in the frequency domain. That is, the spectrum of $ w\cdot x$ is simply $ X$ filtered by $ W$, or, $ W\circledast X$. This smoothing reduces sidelobes associated with the rectangular window (which is the window one gets implicitly when no window is explicitly used). See Chapter 8 for further discussion.


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]