Exponentials

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Exponentials

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Exponentials

The canonical form of an exponential function, as typically used in signal processing, is

\

where $\ is called the time constant of the exponential. $A$ is the peak amplitude, as before. The time constant is the time it takes to decay by $1/e$, i.e.,

\

A normalized exponential decay is depicted in Fig. 5.5.

Figure 5.5:The decaying exponential$Ae^{-t/\.
\


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