NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 generator of the mathematicalgroup consisting 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.