Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

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\sv_0\Vert^2} \sv_0
= \frac{\left<[x_0,x_1],[1,1]\right>}{2} \sv_0\\ [5pt]
&=& \frac{(x_0 \cdot \overline{1} + 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\sv_1\Vert^2} \sv_1
= \frac{\left<[x_0,x_1],[1,-1]\right>}{2} \sv_1\\ [5pt]
&=& \frac{(x_0 \cdot \overline{1} - 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}{2}\sv_0 + \frac{x_0 - x_1}{2}\sv_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!


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA