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 0 db SPL, the ``threshold of pain'' is around 120 dB SPL,
and each bit in a linear PCM format is worth about
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 simplistic 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 noise floor*
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 feedback*^{G.3}.) 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 for
*headroom* 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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