Measures of Amplitude

Strictly speaking, all the samples in a digital audio signal are themselves amplitudes, and we also spoke of the amplitude $a$ of the SINUSOID above. In dealing with general digital audio signals, it is useful to have measures of amplitude for them. Amplitude and other measures are best thought of as applying to a window, a fixed range of samples of the signal. For instance, the window starting at sample $M$ of length $N$ of an audio signal $x[n]$ consists of the samples,

$$x[M], x[M+1], \ldots, x[M+N-1]$$

The two most frequently used measures of amplitude are the peak amplitude, which is simply the greatest sample (in absolute value) over the window:

$${A_{\mathrm{peak}}} \{x[n]\} = \max \vert x[n] \vert , \hspace{0.3in}n = M, \ldots, M+N-1$$

and the root mean square (RMS) amplitude:

$${A_{\mathrm{RMS}}} \{x[n]\} = \sqrt{P\{x[n]\}}$$

where Px[n] is the mean power, defined as:

$${P\{x[n]\}} = {1 \over N} \left ( {{\vert x[M]\vert} ^2} + \cdots + {{\vert x[M+N-1]\vert} ^2} \right )$$

In this last formula, the absolute value signs aren't necessary as long as we're working on real signals, but they are significant if the signals are complex-valued. The peak and RMS amplitude of any signal is at least zero, and is only exactly zero if the signal itself is zero.

The RMS amplitude of a signal may equal the peak amplitude but never exceeds it; and it may be as little as $1 / {\sqrt N}$ 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 is approximately $a$ and its RMS amplitude about $a / {\sqrt 2}$.