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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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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