**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Complex Sinusoids

Recall Euler’s Identity,

Multiplying this equation by and setting , we obtain the definition of thecomplex sinusoid:

Thus, a complex sinusoid consists of an in-phase component for its real part, and a phase-quadrature component for its imaginary part. Since , we have

That is, the complex sinusoid isconstant modulus. (The symbol “” means “identically equal to,” i.e., for all .) The phase of the complex sinusoid is

The derivative of the phase of the complex sinusoid gives itsfrequency

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