Convolution

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Convolution

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

<< Previous page  TOC  INDEX  Next page >>

Convolution



Definition: The convolution of two signals $x$ and $y$ in ${\ is denoted “$x\” and defined by

\

Note that this is cyclic or “circular” convolution.8.2 The importance of convolution in linear systems theory is discussed in §8.7

Convolution is commutative, i.e.,

\

Proof:

\

where in the first step we made the change of summation variable $l\, and in the second step, we made use of the fact that any sum over all $N$ terms is equivalent to a sum from $0$ to $N-1$.



Subsections

<< Previous page  TOC  INDEX  Next page >>

 

© 1998-2018 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy