**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## The Discrete Fourier Transform (DFT)

Given a signal , the

spectrumis defined by

or, as is most often written

That is, the th sample of the spectrum of is defined as the inner product of with the th DFT sinusoid . This definition is times thecoefficient of projectionof onto , i.e.,

The projection of onto itself is

The inverse DFT is simply the sum of the projections:

or, as we normally write,

In summary, the DFT is proportional to the set of coefficients of projection onto the sinusoidal basis set, and the inverse DFT is the reconstruction of the original signal as a superposition of its sinusoidal projections.

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