An Example of Changing Coordinates in 2D

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

$$\begin{eqnarray*} \underline{s}_0 &\isdef & [1,1] \\ \underline{s}_1 &\isdef & [1,-1] \end{eqnarray*}$$

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 $\underline{s}_0$ is, by definition,^5.12

$$\begin{eqnarray*} {\bf P}_{\underline{s}_0}(x) &\isdef & \frac{\left<x,\underline{s}_0\right>}{\Vert\underline{s}_0\Vert^2} \underline{s}_0 = \frac{\left<[x_0,x_1],[1,1]\right>}{2} \underline{s}_0\\ [5pt] &=& \frac{(x_0 \cdot \overline{1} + x_1 \cdot \overline{1})}{2} \underline{s}_0 = \frac{x_0 + x_1}{2}\underline{s}_0. \end{eqnarray*}$$

Similarly, the projection of $x$ onto $\underline{s}_1$ is

$$\begin{eqnarray*} {\bf P}_{\underline{s}_1}(x) &\isdef & \frac{\left<x,\underline{s}_1\right>}{\Vert\underline{s}_1\Vert^2} \underline{s}_1 = \frac{\left<[x_0,x_1],[1,-1]\right>}{2} \underline{s}_1\\ [5pt] &=& \frac{(x_0 \cdot \overline{1} - x_1 \cdot \overline{1})}{2} \underline{s}_1 = \frac{x_0 - x_1}{2}\underline{s}_1. \end{eqnarray*}$$

The sum of these projections is then

$$\begin{eqnarray*} {\bf P}_{\underline{s}_0}(x) + {\bf P}_{\underline{s}_1}(x) &=& \frac{x_0 + x_1}{2}\underline{s}_0 + \frac{x_0 - x_1}{2}\underline{s}_1 \\ [5pt] &\isdef & \frac{x_0 + x_1}{2}(1,1) + \frac{x_0 - x_1}{2} (1,-1) \\ [5pt] &=& \left(\frac{x_0 + x_1}{2} + \frac{x_0 - x_1}{2}, \frac{x_0 + x_1}{2} - \frac{x_0 - x_1}{2}\right) \\ [5pt] &=& (x_0,x_1) \isdef x. \end{eqnarray*}$$

It worked!