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Orthogonality

The vectors (signals) $ x$ and $ y$ 5.11are said to be orthogonal if $ \left<x,y\right>=0$ , denoted $ x\perp y$ . That is to say

$\displaystyle \zbox {x\perp y \Leftrightarrow \left<x,y\right>=0.}
$

Note that if $ x$ and $ y$ are real and orthogonal, the cosine of the angle between them is zero. In plane geometry ($ N=2$ ), the angle between two perpendicular lines is $ \pi/2$ , and $ \cos(\pi/2)=0$ , as expected. More generally, orthogonality corresponds to the fact that two vectors in $ N$ -space intersect at a right angle and are thus perpendicular geometrically.

Example ($ N=2$ ):

Let $ x=[1,1]$ and $ y=[1,-1]$ , as shown in Fig.5.8.

Figure 5.8: Example of two orthogonal vectors for $ N=2$ .
\includegraphics[scale=0.7]{eps/ip}

The inner product is $ \left<x,y\right>=1\cdot \overline{1} + 1\cdot\overline{(-1)} = 0$ . 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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``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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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