Exponential functions
are common in audio processing because they represent natural decay patterns in acoustical systems. When a quantity decays exponentially, it decreases in "equal" proportions relative to its current value.
A general exponential function is given by:
(7)
where is the exponential time constant and A is the peak amplitude. The time constant is the time it takes for the quantity to decay by (i.e.
).
The figure below depicts a normalized exponential decay curve.
Figure 11:
An exponential decay curve.
Note that a positive time constant will produce a decaying exponential while a negative time constant will produce a growing exponential.
For audio applications, a decay of is not generally significant. A more common measure of sound decay is the time required for a signal to decay by 60 dB, which is represented as :
or
(8)
In terms of exponential time constants and from Eqs. (7) and (8),
from which
, or about seven time constants.