# Orthogonality

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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

The vectors (signals) and are said to be orthogonal if , denoted . That is to say

Note that if and are real and orthogonal, the cosine of the angle between them is zero. In plane geometry (), the angle between twoperpendicular lines is , and , as expected. More generally, orthogonality corresponds to the fact that two vectors in-space intersect at a right angle and are thus perpendiculargeometrically.

Example ():

Let and , as shown in Fig. 6.8.

The inner product is . This shows that the vectors are orthogonal. As marked in the figure, the lines intersect at a right angle and are therefore perpendicular.

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