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Circular Motion

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

$\displaystyle x(t) = e^{j\omega t}
$

traces out counter-clockwise circular motion along the unit circle in the complex plane as $ t$ increases, while

$\displaystyle \overline{x(t)} = e^{-j\omega t}
$

gives clockwise circular motion.

We may call a complex sinusoid $ e^{j\omega t}$ a positive-frequency sinusoid when $ \omega>0$ . Similarly, we may define a complex sinusoid of the form $ e^{-j\omega t}$ , with $ \omega>0$ , to be a negative-frequency sinusoid. Note that a positive- or negative-frequency sinusoid is necessarily complex.


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``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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA