Generalized Complex Sinusoids

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Generalized 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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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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 $s$ are complex, and further defined as
\


When $\, we obtain

\

which is the complex sinusoid at amplitude $A$, 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:

\


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