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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University