**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## 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 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 noisefloorto just below the threshold of hearing. Since the threshold of hearing is non-uniform, we would also prefer ashapedquantization noise floor (a feat that can be accomplished usingfiltered error feedback^{4.9}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 forheadroomneeded in a digital recording scenario. We also need both headroom andguard bitson 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.