**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Special Case: The Mth Roots of Unity

If , we have

The th roots of unity play an important role in the Discrete Fourier Transform. They are so important that they often have a special notation in the signal processing literature:

In this notation, the roots of unity can be expressed simply as, for .We may also call the

generatorof the mathematicalgroupconsisting of the th roots of unity and their products.We will learn later that the th roots of unity are used to generate all the sinusoids used by the DFT and its inverse. The th sinusoid is given by

where , , and is the sampling interval in seconds.