Changing Coordinates

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 ${\bf C}^N$ . 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:

$\begin{eqnarray*} \sv_0 &\isdef & [1,1] \\ \sv_1 &\isdef & [1,-1] \end{eqnarray*}$

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 $\sqrt{2}$ in each case. Let's try projecting onto these vectors and seeing if we can reconstruct by summing the projections.

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

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

Similarly, the projection of onto $\sv_1$ is

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

The sum of these projections is then

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

It worked!

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Projection onto Linearly Dependent Vectors
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Projection