# 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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

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

where , , and are real numbers.

Forming a geometric sequence based on yields the sequence

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

Subsections

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