Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books



Chapters

Mathematics of the DFT
    Geometric Signal Theory
       Signal Reconstruction from Projections
          Changing Coordinates
             An Example of Changing Coordinates in 2D

Chapter Contents:

Search Mathematics of the DFT

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

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!

Order a Hardcopy of Mathematics of the DFT

Previous: Changing Coordinates
Next: Projection onto Linearly Dependent Vectors
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )