## What bit-depth is required for low frequency sine

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If I understand correctly, for mathematically perfect quantization of 20Hz sine wave at 44.1k sampling rate at least 18 bits are required. If we assume that samples of the sine wave around its min/max (the slowest parts) should differ at least with one bit, then it's easy to find number of bits required for quantization of the whole sine period (for full scale signal):

*Bit-depth = log2(1/(1-cos(2*pi*F/Fs)))*, where

*Fs* - sampling freq.

*F* - freq. of sine wave

For F = 20 Hz:

- @44100 Bit-depth = 18 bit (17.91)
- @48000 Bit-depth = 19 bit (18.15)
- @96000 Bit-depth = 21 bit (20.15)
- @192000 Bit-depth = 23 bit (22.15)

Or I miss something?

(yes, I understand that this doesn't affect perceived sound quality )))