**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 the

inverse DFToperation (or the inverse of whatever transform we are working with).As a simple example, consider the projection of a signal onto the rectilinear

coordinate axesof . 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 thevector sumof 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

Thereconstructionof from its projections onto the coordinate axes is then thevector sum of the projections:

The projection of a vector onto its coordinate axes is in some sense trivial because the very meaning of the

coordinatesis that they are scalars to be applied to thecoordinate vectorsin order to form an arbitrary vector as alinear combinationof the coordinate vectors:

Note that the coordinate vectors areorthogonal. Since they are also unit length, , we say that the coordinate vectors areorthonormal.What's more interesting is when we project a signal onto a set of vectors

other thanthe coordinate set. This can be viewed as achange of coordinatesin . In the case of the DFT, the new vectors will be chosen to besampled complex sinusoids.

Subsections