How Many Bits are Enough for Digital Audio?

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 $20\log_{10}(2) \approx 6$ dB of dynamic range, we find that we need $120/6 = 20$ 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.