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Exponentials

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

$\displaystyle a(t) = A e^{-t/\tau}, \quad t\geq 0
$

where $ \tau$ 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.,

$\displaystyle \frac{a(\tau)}{a(0)} = \frac{1}{e}.
$

A normalized exponential decay is depicted in Fig.4.7.

Figure 4.7: The decaying exponential $ Ae^{-t/\tau }$ , normalized to unit amplitude ( $ e^{-t/\tau }$ ).
\includegraphics[width=0.8 \twidth]{eps/exponential}



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``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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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