NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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The orthogonal projection (or simply ''projection'') of onto is defined by
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 in Fig. 6.9 for and , in which case
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