Circular Motion

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Circular Motion

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

Since the modulus of the complex sinusoid is constant, it must lie on acircle in the complex plane. For example,

\

traces out counter-clockwise circular motion along the unit circle in the complex plane, while
\

is clockwise circular motion.

We call a complex sinusoid of the form $e^{j\, where $\, apositive-frequency sinusoid. Similarly, we define a complex sinusoid of the form $e^{-j\, with $\, to be a negative-frequency sinusoid. Note that a positive- or negative-frequency sinusoid is necessarily complex.

$\ REMARK: Add figure: circular motion (animation for web version) $\

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