**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 SinusoidsConsider , 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 asampled complex sinusoidwith unit amplitude, and zero phase. Defining thesampling intervalas in seconds provides that is theradian frequencyin radians per second.

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