Changing Coordinates

What's more interesting is when we project a signal $ x$ onto a set of vectors other than the coordinate set. This can be viewed as a change of coordinates in $ {\bf C}^N$. In the case of the DFT, the new vectors will be chosen to be sampled complex sinusoids.

An Example of Changing Coordinates in 2D

As a simple example, let's pick the following pair of new coordinate vectors in 2D:

\sv_0 &\isdef & [1,1] \\
\sv_1 &\isdef & [1,-1]

These happen to be the DFT sinusoids for $ N=2$ having frequencies $ f_0=0$ (``dc'') and $ f_1=f_s/2$ (half the sampling rate). (The sampled complex sinusoids of the DFT reduce to real numbers only for $ N=1$ and $ N=2$.) We already showed in an earlier example that these vectors are orthogonal. However, they are not orthonormal since the norm is $ \sqrt{2}$ in each case. Let's try projecting $ x$ onto these vectors and seeing if we can reconstruct by summing the projections.

The projection of $ x$ onto $ \sv_0$ is, by definition,5.12

{\bf P}_{\sv_0}(x) &\isdef & \frac{\left<x,\sv_0\right>}{\Vert...
...+ x_1 \cdot \overline{1})}{2} \sv_0
= \frac{x_0 + x_1}{2}\sv_0.

Similarly, the projection of $ x$ onto $ \sv_1$ is

{\bf P}_{\sv_1}(x) &\isdef & \frac{\left<x,\sv_1\right>}{\Vert...
...- x_1 \cdot \overline{1})}{2} \sv_1
= \frac{x_0 - x_1}{2}\sv_1.

The sum of these projections is then

{\bf P}_{\sv_0}(x) + {\bf P}_{\sv_1}(x) &=&
\frac{x_0 + x_1}...
...} - \frac{x_0 - x_1}{2}\right) \\ [5pt]
&=& (x_0,x_1) \isdef x.

It worked!

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Projection onto Linearly Dependent Vectors
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