NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 different vectors , . 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 onto the rectilinear coordinate axes of . The coordinates of the projection onto the th coordinate axis are simply . The projection along coordinate axis has coordinates , and so on. The original signal 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 . 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 . 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 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 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.