Projection
The
orthogonal projection (or simply ``projection'') of

onto

is defined by

The complex
scalar

is called the
coefficient 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.
5.9 for
![$ x= [4,1]$](http://www.dsprelated.com/josimages_new/mdft/img878.png)
and
![$ y=[2,3]$](http://www.dsprelated.com/josimages_new/mdft/img879.png)
, in which case
Figure:
Projection of
onto
in 2D space.
![\includegraphics[scale=0.7]{eps/proj}](http://www.dsprelated.com/josimages_new/mdft/img881.png) |
Derivation: (1) Since any projection onto

must lie along the
line collinear with

, write the projection as

. (2) Since by definition the
projection error

is orthogonal to

, we must have
Thus,
See §
I.3.3 for illustration of orthogonal projection in
matlab.
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