Projection

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

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

The orthogonal projection (or simply ''projection'') of $\ onto $\ is defined by

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The complex scalar $\ is called thecoefficient of projection. When projecting $\ onto a unit length vector $\, the coefficient of projection is simply the inner product of $\ with $\.

Motivation: The basic idea of orthogonal projection of $\ onto$\ is to ''drop a perpendicular'' from $\ onto $\ to define a new vector along $\ which we call the ''projection'' of $\ onto $\. This is illustrated for $N=2$ in Fig. 6.9 for $\ and $\, in which case

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Figure 6.9:Projection of $\onto $\ in 2D space.
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Derivation: (1) Since any projection onto $\ must lie along the line colinear with $\, write the projection as ${\. (2) Since by definition the projection is orthogonal to $\, we must have

\



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