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Comment by Night_Thastus | Hacker News Reader

From my understanding, doesn't Nyquist-Shannon assume you take infinite steps to refine the discrete steps into the final waveform?

Since real-world DACs don't have infinite taps, either increasing the number of samples per second in the original audio or the number of steps done by the filter will improve how close it gets to the original waveform.

Would that apply to the number of bits used to represent the level as well? I thought that was mainly useful to give some additional headroom when editing?

(Obviously there's diminishing returns either way)

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Nyquist-Shannon simply says that you need 2x the sample rate of the highest frequency to perfectly recreate something. So, to recreate a 20 kHz wave that was sampled at 16 bits, you'd need 40 kHz sample rate. Bump that up to 22.05 kHz (for anti-aliasing) and you get CD audio.

The bit depth, OTOH, is your signal-noise ratio (noise floor) when sampling/quantizing said audio. It's unrelated to Nyquist-Shannon. More bits gives you less noise when reproducing it. For CD audio, 16 bits was chosen to give an acceptable SNR of 96 dB.

Monty Montgomery (of xipf.org) has a nice YouTube video related to this: https://www.youtube.com/watch?v=cIQ9IXSUzuM

Yes, you only need that many samples given you know a maximum frequency. But to actually perform the conversion from discrete samples to continuous wave takes infinitely many steps. The number a given DAC actually does varies. Some do hundreds, some thousands, a couple rare (very expensive) ones do 100k or more.

That's what I'm saying. Increase the number of samples in the audio itself, and the DAC can get closer to the original.

Except it's not. The video shows that. In the real world, the DAC doesn't immediately jump from code N to N+1; it slides (the "slew rate" of the opamp). More bits just makes the difference between codes N and N+1 smaller; i.e. where it slides to is more accurate to the original. The "stair step" or "straight line" view you get from programs like Audacity is a lie.

A final low pass filter (tuned to remove anything above half the sample rate) allows the accurate reconstruction of the original analogue waveform from the sampled digital data. ( The vertical 'stair step' of sampled data is very high frequency information which gets removed, 'bending' the output back into it's original shape). The video linked above is well worth a watch.

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