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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}

Why Exponentials are Important

Exponential decay occurs naturally when a quantity is decaying at a rate which is proportional to how much is left. In nature, all linear resonators, such as musical instrument strings and woodwind bores, exhibit exponential decay in their response to a momentary excitation. As another example, reverberant energy in a room decays exponentially after the direct sound stops. Essentially all undriven oscillations decay exponentially (provided they are linear and time-invariant). Undriven means there is no ongoing source of driving energy. Examples of undriven oscillations include the vibrations of a tuning fork, struck or plucked strings, a marimba or xylophone bar, and so on. Examples of driven oscillations include horns, woodwinds, bowed strings, and voice. Driven oscillations must be periodic while undriven oscillations normally are not, except in idealized cases.

Exponential growth occurs when a quantity is increasing at a rate proportional to the current amount. Exponential growth is unstable since nothing can grow exponentially forever without running into some kind of limit. Note that a positive time constant corresponds to exponential decay, while a negative time constant corresponds to exponential growth. In signal processing, we almost always deal exclusively with exponential decay (positive time constants).

Exponential growth and decay are illustrated in Fig.4.8.

Figure 4.8: Growing and decaying exponentials.

Audio Decay Time (T60)

In audio, a decay by $ 1/e$ (one time-constant) is not enough to become inaudible, unless the starting amplitude was extremely small. In architectural acoustics (which includes the design of concert halls [4]), a more commonly used measure of decay is ``$ t_{60}$'' (or T60), which is defined as the time to decay by $ 60$ dB.4.7That is, $ t_{60}$ is obtained by solving the equation

$\displaystyle \frac{a(t_{60})}{a(0)} = 10^{-60/20} = 0.001.

Using the definition of the exponential $ a(t) = A e^{-t/\tau}$, we find

$\displaystyle \zbox {t_{60} = \ln(1000) \tau \approx 6.91 \tau}

Thus, $ t_{60}$ is about seven time constants. See where $ t_{60}$ is marked on Fig.4.7 compared with $ \tau$.

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