# Projection

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

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

<< Previous page  TOC  INDEX  Next page >>

## Projection

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

<< Previous page  TOC  INDEX  Next page >>