What's more interesting is when we project a
signal 
onto a set
of vectors
other than the coordinate set. This can be viewed
as a
change of coordinates in

. In the case of the
DFT,
the new vectors will be chosen to be
sampled complex sinusoids.

As a simple example, let's pick the following pair of new coordinate vectors in 2D:
These happen to be the
DFT sinusoids for

having frequencies

(``
dc'') and

(half the
sampling rate). (The sampled
complex
sinusoids of the DFT reduce to
real numbers only for

and

.) We
already showed in an earlier example that these vectors are
orthogonal. However, they are not orthonormal since the
norm is

in each case. Let's try projecting

onto these vectors and
seeing if we can reconstruct by summing the projections.
The projection of

onto

is, by
definition,
5.12
Similarly, the projection of

onto

is
The sum of these projections is then
It worked!
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