Measures of Amplitude

The most fundamental property of a digital audio signal is its
amplitude. Unfortunately, a signal's amplitude has no one canonical
definition.
Strictly speaking, all the samples in a digital audio signal are themselves
amplitudes, and we also spoke of the amplitude of the sinusoid as a whole.
It is useful to have measures
of amplitude for digital audio signals in general. Amplitude
is best thought of as applying to a
*window*, a fixed range of samples of the signal. For instance, the
window starting at sample of length of an audio signal consists of the
samples,

The two most frequently used measures of amplitude are the

and the

where is the mean

(In this last formula, the absolute value signs aren't necessary at the moment since we're working on real-valued signals, but they will become important later when we consider complex-valued signals.) Neither the peak nor the RMS amplitude of any signal can be negative, and either one can be exactly zero only if the signal itself is zero for all in the window.

The RMS amplitude of a signal may equal the peak amplitude but never exceeds it; and it may be as little as times the peak amplitude, but never less than that.

Under reasonable conditions--if the window contains at least several periods and if the angular frequency is well under one radian per sample--the peak amplitude of the sinusoid of Page is approximately and its RMS amplitude about . Figure 1.2 shows the peak and RMS amplitudes of two digital audio signals.