**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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

The complex scalar is called thecoefficient of projection. When projecting onto aunit lengthvector , 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