**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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

We have defined sinusoids and extended the definition to include complex sinusoids. We now extend one more step by allowing for

exponentialamplitude envelopes:

where and arecomplex, and further defined as

When , we obtain

which is the complex sinusoid at amplitude , radian frequency , and phase .More generally, we have

Defining , we see that the generalized complex sinusoid is just the complex sinusoid we had before with an

exponential envelope: