We are interested in the manipulation and/or the synthesis of audio signals. Because we work with computers, these signals are uniformly sampled, or discretized, in time and/or space. There are various signal characteristics, or signal metrics, that we can make use of to analyze discrete-time signals.
The logarithmic expression is just the inverse of the exponential expression by = x. The value b is referred to as the ``base'', which is chosen to be a positive real number.
The most common bases for logarithms are 10, e, and 2. A logarithm with base e is typically written as and is referred to as a "natural logarithm".
Logarithms can be useful for expressing relationships that occur over very large ranges on a linear scale in more manageable, intuitive terms. For example, the pressure ``threshold of pain'' of our auditory system is 106 times greater than the "threshold of audibility". On a decibel scale, this relationship can be expressed as 120 dB.
A decibel (dB) is defined as one tenth of a bel, which is an amplitude unit defined as the of sound intensity relative to some reference intensity level. Since signal intensity, power, and energy are proportianal to the square of signal amplitude, amplitude relationships in dB are given by of sound amplitude to a reference amplitude.
In signal processing, the maximum signal amplitude is typically chosen as the reference amplitude. In other words, signals are normalized so that the maximum amplitude is 1, or 0 dB.
Logarithms have the following properties and identities:
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:
where is the exponential time constant and A is the peak amplitude. The time constant is the time it takes for the quantity y(t) to decay by 1/e (i.e.
The figure below depicts a normalized exponential decay curve.
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 1/e is not generally significant. A more common measure of audio decay is the time required for a signal to decay by 60 dB, which is represented as t60:
a(t60) / a(0) = 10-60/20 = 0.001
In terms of exponential time constants,
, or about seven time constants.