Projection

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Projection

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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 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 $N=2$ in Fig. 6.9 for $\ and $\, in which case

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Figure 6.9:Projection of $\onto $\ in 2D space.
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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

\



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