Digital Sound & Music: Concepts, Applications, & Science, Chapter 5, last updated 6/25/2013
have full amplitude (all bits on). The quietest possible passage is the
quantization noise itself. Any sound quieter than that would simply be masked
by the noise or rounded down to silence. The difference between the loudest
and quietest parts is therefore the highest (possible) amplitude of the audio
signal compared to the amplitude of the quantization noise, or the ratio
between the signal level to the noise level. This is what is known as signal-to-
quantization-noise-ratio (SQNR), and in this context, dynamic range is the
same thing. This definition is given in Equation 5.2.
Given a bit depth of n, the dynamic range of a digital audio recording is equal to
( ) dB.
You can see from the equation that dynamic range as SQNR is measured in decibels.
Decibels are a dimensionless unit derived from the logarithm of the ratio between two values.
For sound, decibels are based on the ratio between the air pressure amplitude of a given sound
and the air pressure amplitude of the threshold of hearing. For dynamic range, decibels are
derived from the ratio between the maximum and minimum amplitudes of an analog waveform
quantized with n bits. When values are quantized by rounding down to the nearest integer, the
maximum magnitude amplitude is | | . The minimum amplitude of an
analog wave that would be converted to a non-zero value when it is quantized is ½. Signal-to-
quantization-noise is based on the ratio between these maximum and minimum values for a
given bit depth. It turns out that this is exactly the same value as the dynamic range.
Equation 5.2 can be simplified as shown in Equation 5.3.
( ) ( ) ( )
Equation 5.3 gives us a method for determining the possible dynamic range of a
digital recording as a function of the bit depth. For bit depth n, the possible
dynamic range is approximately 6n dB. A bit depth of 8 gives a dynamic range
of approximately 48 dB, a bit depth of 16 gives a dynamic range of about 96
dB, and so forth.
When we introduced this section, we added the adjective “potential” to
“dynamic range” to emphasize that it is the maximum possible range that can be
used as a function of the bit depth. But not all of this dynamic range is used if the amplitude of a
digital recording is relatively low, never reaching its maximum. Thus, we it is important to
consider the actual dynamic range (or actual SQNR) as opposed to the potential dynamic
range (or potential SQNR, just defined). Take, for example, the situation illustrated in Figure
5.15. A bit depth of 7 has been chosen. The amplitude of the wave is 24 dB below the
maximum possible. Because the sound uses so little of its potential dynamic range, the actual
dynamic range is small. We’ve used just a simple sine wave in this example so that you can
easily see the error wave in proportion to the sine wave, but you can imagine a music recording
that has a low actual dynamic range because the recording was done at a low level of amplitude.