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Projection

The orthogonal projection (or simply ``projection'') of $ y\in{\bf C}^N$ onto $ x\in{\bf C}^N$ is defined by

$\displaystyle \zbox {{\bf P}_{x}(y) \isdef \frac{\left<y,x\right>}{\Vert x\Vert^2} x.}
$

The complex scalar $ \left<y,x\right>/\Vert x\Vert^2$ is called the coefficient of projection. When projecting $ y$ onto a unit length vector $ x$ , the coefficient of projection is simply the inner product of $ y$ with $ x$ .

Motivation: The basic idea of orthogonal projection of $ y$ onto $ x$ is to ``drop a perpendicular'' from $ y$ onto $ x$ to define a new vector along $ x$ which we call the ``projection'' of $ y$ onto $ x$ . This is illustrated for $ N=2$ in Fig.5.9 for $ x= [4,1]$ and $ y=[2,3]$ , in which case

$\displaystyle {\bf P}_{x}(y) \isdef \frac{\left<y,x\right>}{\Vert x\Vert^2} x
= \frac{(2\cdot \overline{4} + 3\cdot \overline{1})}{4^2+1^2} x
= \frac{11}{17} x= \left[\frac{44}{17},\frac{11}{17}\right].
$

Figure: Projection of $ y$ onto $ x$ in 2D space.
\includegraphics[scale=0.7]{eps/proj}

Derivation: (1) Since any projection onto $ x$ must lie along the line collinear with $ x$ , write the projection as $ {\bf P}_{x}(y)=\alpha
x$ . (2) Since by definition the projection error $ y-{\bf P}_{x}(y)$ is orthogonal to $ x$ , we must have

\begin{eqnarray*}
(y-\alpha x) & \perp & x\\
\;\Leftrightarrow\;\left<y-\alpha x,x\right> &=& 0 \\
\;\Leftrightarrow\;\left<y,x\right> &=& \alpha\left<x,x\right> \\
\;\Leftrightarrow\;\alpha &=& \frac{\left<y,x\right>}{\left<x,x\right>}
= \frac{\left<y,x\right>}{\Vert x\Vert^2}.
\end{eqnarray*}

Thus,

$\displaystyle {\bf P}_{x}(y) = \frac{\left<y,x\right>}{\Vert x\Vert^2} x.
$

See §I.3.3 for illustration of orthogonal projection in matlab.


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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 © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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