Units of Amplitude

Two amplitudes are often best compared using their ratio rather than their difference. For example, saying that one signal's amplitude is greater than another's by a factor of two is more informative than saying it is greater by 30 millivolts. This is true for any measure of amplitude (RMS or peak, for instance). To facilitate this we often express amplitudes in logarithmic units called decibels. If $a$ is the amplitude of a signal (either peak or RMS, as defined above), then we can define the decibel (dB) level $d$ as:

$$d = 20 \cdot {{{\log}_{10}} ( {a / {a_0}} )}$$

where $a_0$ is a reference amplitude. This definition is set up so that, if we increase the signal power by a factor of ten (so that the amplitude increases by a factor of $\sqrt {10}$), the logarithm will increase by $1/2$, and so the value in decibels goes up (additively) by ten. An increase in amplitude by a factor of two corresponds to an increase of about 6.02 decibels; doubling power is an increase of 3.01 dB. In dB, therefore, adding two uncorrelated signals of equal amplitude results in one that is about 3 dB higher, whereas doubling a signal increases its amplitude by 6 dB. The relationship between linear amplitude and amplitude in decibels is graphed in Figure 1.2.

Figure 1.2: The relationship between Decibel and linear scales of amplitude. The linear amplitude 1 is assigned to 0 dB.

Still using $a_0$ to denote the reference amplitude, a signal with linear amplitude smaller than $a_0$ will have a negative amplitude in decibels: ${a_0}/10$ gives -20 dB, ${a_0}/100$ gives -40, and so on. A linear amplitude of zero is smaller than that of any value in dB, so we give it a dB value of $-\infty$.

In digital audio a convenient choice of reference, assuming the hardware has a maximum amplitude of one, is

$${a_0} = {10^{-5}} = 0.00001$$

so that the maximum amplitude possible is 100 dB, and 0 dB is likely to be inaudibly quiet at any reasonable listening level. Conveniently enough, the dynamic range of human hearing--the ratio between a damagingly loud sound and an inaudibly quiet one--is about 100 dB.

Amplitude is related in an inexact way to the perceived loudness of a sound. In general, two signals with the same peak or RMS amplitude won't necessarily have the same loudness at all. But amplifying a signal by 3 dB, say, will fairly reliably make it sound about one ``step" louder. Much has been made of the supposedly logarithmic nature of human hearing (and other senses), which may partially explain why decibels are such a useful scale of amplitude[RMW02, p. 99].

Amplitude is also related in an inexact way to musical dynamic. Dynamic is better thought of as a measure of effort than of loudness or power. It ranges over nine values: rest, ppp, pp, p, mp, mf, f, ff, fff. These correlate in an even looser way with the amplitude of a signal than does loudness [RMW02, pp. 110-111].