Orthogonality of the DFT Sinusoids

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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

Orthogonality of the DFT Sinusoids

We now show mathematically that the DFT sinusoids are exactly orthogonal. Let

\

denote the $k$th complex DFT sinusoid. Then
\

where the last step made use of the closed-form expression for the sum of ageometric series. If $k\, the denominator is nonzero while the numerator is zero. This proves
\

While we only looked at unit amplitude, zero phase complex sinusoids, as used by the DFT, it is readily verified that the (nonzero) amplitude and phase have no effect on orthogonality.

<< Previous page  TOC  INDEX  Next page >>

 

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