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 $a$ 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 $M$ 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 $P\{x[n]\}$ 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 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 $n$ 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 $1 / {\sqrt N}$ times the peak amplitude, but never less than that.

Figure 1.2: Root mean square (RMS) and peak amplitudes of signals compared. For a sinusoid (part a), the peak amplitude is higher than RMS by a factor of $\sqrt 2$.

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 $a$ and its RMS amplitude about $a / {\sqrt 2}$. Figure 1.2 shows the peak and RMS amplitudes of two digital audio signals.