## Exponentials

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

*time constant*of the exponential. is the peak amplitude, as before. The time constant is the time it takes to decay by ,

*i.e.*,

### 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.

### Audio Decay Time (T60)

In audio, a decay by (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 ``''
(or `T60`), which is defined as the
*time to decay by dB*.^{4.7}That is, is obtained by solving the equation

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