**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Normalized DFT

A more “theoretically clean” DFT is obtained by projecting onto the

normalized DFT sinusoids

In this case, the normalized DFT of is

which is also precisely the coefficient of projection of onto . The inverse normalized DFT is then more simply

While this definition is much cleaner from a “geometric signal theory“ point of view, it is rarely used in practice since it requires more computation than the typical definition. However, note that the only difference between the forward and inverse transforms in this case is the sign of the exponent in the kernel.