Signal Reconstruction from Projections

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Signal Reconstruction from Projections

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Signal Reconstruction from Projections

We now know how to project a signal onto other signals. We now need to learn how to reconstruct a signal $\ from its projections onto$N$ different vectors $\, $k=0,1,2,\. This will give us theinverse DFT operation (or the inverse of whatever transform we are working with).

As a simple example, consider the projection of a signal $x\ onto the rectilinear coordinate axes of ${\. The coordinates of the projection onto the $0$th coordinate axis are simply $(x_0,0,\. The projection along coordinate axis $1$ has coordinates $(0,x_1,0,\, and so on. The original signal $x$ is then clearly the vector sum of its projections onto the coordinate axes:

\

To make sure the previous paragraph is understood, let's look at the details for the case $N=2$. We want to project an arbitrary two-sample signal $\ onto the coordinate axes in 2D. A coordinate axis can be represented by any nonzero vector along its length. The horizontal axis can be represented by any vector of the form $(\ while the vertical axis can be represented by any vector of the form $(0,\. For maximum simplicity, let's choose the positive unit-length representatives:

\



The projection of $\ onto $\ is by definition

\

Similarly, the projection of $\ onto $\ is

\

The reconstruction of $x$ from its projections onto the coordinate axes is then the vector sum of the projections:

\

The projection of a vector onto its coordinate axes is in some sense trivial because the very meaning of the coordinates is that they are scalars $x_n$ to be applied to the coordinate vectors $\ in order to form an arbitrary vector $\ as a linear combinationof the coordinate vectors:

\

Note that the coordinate vectors are orthogonal. Since they are also unit length, $\, we say that the coordinate vectors $\ are orthonormal.

What's more interesting is when we project a signal $\ onto a set of vectors other than the coordinate set. This can be viewed as a change of coordinates in ${\. In the case of the DFT, the new vectors will be chosen to be sampled complex sinusoids.


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