[quickthrower2]

But surely at 22khz you can only oscillate between 2 values if sampling at 44khz, whereas 2khz gets 20 values in the same 180 degrees of the wave? So doesn’t this mean you lose fidelity at higher frequencies?

Edit: thanks for all these great responses!

[lxgr]

A sine wave is a sine wave; it can't contain any "additional information" that would gain anything from getting encoded by more data points.

If it would, it wouldn't be a sine wave anymore, and you could decompose it into its component of phase and amplitude shifted sine waves, which will occupy all kinds of places in the frequency domain, many of them higher than your initial 22 kHz.

That's in fact what we mean when we say "a 22 kHz signal": An ensemble of all of these overlapping waves represented as a single signal. If you're sampling that with an ADC, you're not only sampling the the 22 kHz wave, but you're also sampling it – and that, but only that, can in fact be described with a single bit per ADC readout.

Looking at it from an information theoretical point of view, the Kolgomorov complexity of "a sine wave of 22 kHz and amplitude x" is pretty minimal – just from that sentence, you can perfectly recreate that signal with no sampling whatsoever, and one ADC readout tells you its amplitude and phase. It can carry a bandwidth of 0 Hz.

[tern]

The magical thing about Nyquist is that this isn't a problem. Two values is enough to define a pure sine wave at 20khz, and all you need is the ability to represent sine waves at every frequency (per Fourier analysis).

This reaches the edge of my knowledge, but I think this is basically because the discrete values are 'slewed' (aka integrated aka low-pass filtered) into a continuous signal by the DAC.

See: https://en.wikipedia.org/wiki/Sampling_(signal_processing)

[aidenn0]

A CD is 16 bits, so you have 65536 possible values to oscillate between.

If you mean "does a 44khz only output a square (or triangle) wave at 22kHz, the answer is "no."

If you have two alternating values at 44kHz, the output will be a perfect (to the limits of noise &c.) sine wave, since the DAC is band limited.

If this is still confusing to you, I recommend watching https://www.youtube.com/watch?v=FG9jemV1T7I

[IIAOPSW]

No. Having the other 20 in-between values doesn't help improve the "accuracy" of the 2 khz wave. An x Khz wave is always an x Khz wave. That is to say the x Khz part of the signal is by definition the part of the signal which looks like sin([freq][time]2pi). Your soundwave is a sum of these Eg

a_i sin(2pitf)

where each a_i is the amount of a given frequency present. If every wave sample you have goes [1,-1,1,-1,1,-1...] then 100% of your wave is accounted for as a sin wave at the sampling rate. There is zero additional frequencies present with unknown amounts. If your wave samples went like [0,1,0,-1,0,1,0,-1...] the only frequency of sound present in the signal is the one at half the sampling rate. The whole wave is accounted for.

On a more physical level, its ok if your sample only sees +1/-1 every other sample, because when that electrical signal turns into pushing a magnet the magnet has to accelerate smoothly in accordance to Newtons laws. In other words, flipping the voltage between +-1 as a 40khz square wave becomes a perfect 20khz sine wave when it reaches the speaker.

[tkuraku]

https://en.m.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_samp...

You can only capture all the information if you sample at twice the rate of the highest frequency of interest.