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Mathematics of the DFT
    Geometric Signal Theory
       The Inner Product
          Projection

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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 =... ...e{1})}{4^2+1^2} x = \frac{11}{17} x= \left[\frac{44}{17},\frac{11}{17}\right]. $

Figure 5.9: 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... ...}{\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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Previous: The Pythagorean Theorem in N-Space
Next: Signal Reconstruction from Projections
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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