NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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Armed with the above knowledge, we can visit the question ''how
many bits are enough'' for digital audio. Since the threshold of
hearing
is around 0db
SPL, the ''threshold of pain'' is around 120 dB SPL, and
each bit in a linear PCM
format is worth about 6 dB of dynamic
range, we find that we need 
bits to represent the full dynamic range of audio in a linear
fixed-point format. This is a simplestic analysis because it is
not quite right to equate the least-significant bit with the
threshold of hearing; instead, we would like to adjust the
quantization noisefloor
to just below the threshold of hearing. Since the threshold of
hearing is non-uniform, we would also prefer a shaped
quantization noise floor (a feat that can be accomplished using
filtered
error feedback4.9 Nevertheless, the simplistic result
gives an answer similar to the more careful analysis, and 20 bits
is a good number. However, this still does not provide
forheadroom needed in a digital recording scenario. We also
need both headroom and guard bits on the lower end when we
plan to carry out a lot of signal processing operations, especially
digital
filtering. As an example, a 1024-point FFT (Fast
Fourier Transform) can give amplitudes
1024 times the input amplitude (such as in the case of a constant
''dc'' input signal), thus requiring 10 headroom bits. In general,
24 fixed-point bits are pretty reasonable to work with, although
you still have to scale very carefully, and 32 bits are
preferable.
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