Figuring Out Sampling Theory by Playing Around with Complex Sinusoids

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Figuring Out Sampling Theory by Playing Around with Complex Sinusoids

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Figuring Out Sampling Theory by Playing Around with Complex Sinusoids

Consider $z_0\, with $\. Then we can write$z_0$ in polar form as

\

where $\, $\, and $T_s$ are real numbers.

Forming a geometric sequence based on $z_0$ yields the sequence

\

where $t_n\. Thus, successive integer powers of $z_0$produce a sampled complex sinusoid with unit amplitude, and zero phase. Defining the sampling interval as $T_s$ in seconds provides that $\ is the radian frequency in radians per second.


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