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