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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!


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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