[gwbas1c]

> In particular, this minimal frequency is twice the bandwitdh of the function.

Careful, this is misleading.

If the peaks of the frequency align with your samples, you'll get the full bandwidth.

If the 0-crossings align with your samples, you'll miss the frequency.

These are why people swear by things like HD audio, SACD/DSD, even though "you can't hear over 20khz"

[luma]

You've misunderstood something about Nyquist. A sample rate of, say, 44KHz, will capture ALL information below 22KHz and recreate it perfectly.

There are of course implementation details to consider, for example you probably want to have a steep filter so you don't wind up with aliasing artifacts from content above 22KHz. However it's important to understand: Nyquist isn't an approximation. If your signal is below one half the sample rate, it will be recreated with no signal lost.

[GlenTheMachine]

Nyquist is a mathematical statement. As such, it has two commonly overlooked requirements:

- the signal being sampled has to be stationary

- you have an infinite number of samples

In that case, a sampling frequency of 2N+epsilon will perfectly reproduce the signal. Otherwise there can be issues.

[alanbernstein]

I don't recall seeing Nyquist described with those requirements before. I think it is evident that in the real world, there are many practical signals which do not exactly meet those requirements, but which still yield nearly-exact reproduction.

I wonder, what are some examples of signals that fail to reproduce after sampling in a way that is "nearly Nyquist"?

[GlenTheMachine]

If you look at the Wikipedia entry on the Nyquist Sampling Theorem, you should note that the summations to reconstruct the original signal go from negative infinity to positive infinity. In other words, that sum requires an infinite number of samples.

There are many signals of practical interest that can be approximately reconstructed with a finite truncation of the series. Note, however, that any signal that has only a finite length, eg has a uniformly zero amplitude after some time t_final, does not have a finite bandwidth, and cannot be exactly reconstructed by any sampling scheme. This is the case whenever you stop sampling a signal, eg it is always the case whenever you step outside the mathematical abstraction and start running real code on a real computer. So any signal reconstructed from samples is always approximate, except for some relatively trivial special cases.

[drdeca]

Hm, yes, a function cannot have bounded support in both the time domain and the frequency domain…

What if you take a function that has bounded support in the time domain, and then turn it into a periodic function? Might the resulting function have bounded support in the frequency domain even though the original function did not? I suppose doing this would force the Fourier transform to have discrete support? But under what conditions would it have bounded support?…

I guess technically a low-pass filter applied to a signal with finite support in the time domain, would result in a function which has infinite support in the time domain.

I suppose sinc(f t + c) doesn’t have bounded support, and it is unsurprising that a non-trivial linear combination of finitely many terms of this form would also not have finite support.

Still, such a linear combination could decay rather quickly, I imagine. (Idk if asymptotically faster than (1/t) , but (1/(f t)) is still pretty fast I think, for large f.)

Soon enough the decay should be enough that the amplitude should be smaller than the smallest that the speaker hardware is capable of producing, I suppose.

[GlenTheMachine]

“ What if you take a function that has bounded support in the time domain, and then turn it into a periodic function?”

When you perform a finite sample reconstruction, this is essentially the unstated approximation you’re making.

[ImageXav]

I think it is you who have misunderstood the Nyquist-Shannon theorem. Aliasing and noise are real concerns. Tim Wescott explains it very well [0] (Figures 3, 10 and 11). If your signal is below one half the sample rate but the noise isn't, you'll lose information about the signal. If your signal phase is shifted wrt. the sampling, you'll lose information. If your sampling period isn't representative, you'll lose information. These are not implementation details.

[0] https://www.wescottdesign.com/articles/Sampling/sampling.pdf

[StrangeDoctor]

I was just about to post something saying similar. If I had to guess,

>If the 0-crossings align with your samples, you'll miss the frequency.

This is where the issue is. This isn’t possible with more than double the sampling rate.

[kevin_thibedeau]

It can only happen with a source exactly at N/2 and correlated with your sampling clock. That doesn't happen in the real world for audio.

[mlyle]

Anything close to N/2 is going to have varying magnitude that requires filtering and likely oversampling to remove.

How close to the Nyquist bandwidth you can get depends upon the quality of your filtering.

44.1KHz is a reasonable compromise for a 20KHz passband. 48KHz is arguably better now that bits are cheap-- get a sliver more than 20KHz and be less demanding on your filter. Garbage has to be way up above 28KHz before it starts to fold over into the audible region, too.

[Sesse__]

> Garbage has to be way up above 28KHz before it starts to fold over into the audible region, too.

You brick-wall everything at 20 kHz (with an analogue filter) before you sample it; that's part of the CD standard, and generally also what all other digital CD-quality audio assumes. This ensures there simply is no 28 kHz garbage to fold. The stuff between 20 and 28 in your reconstructed signal then is a known-silent guard band, where your filter is free to do whatever it wants—which in turn means that you can design it only for maximum flatness (and ideally, zero phase) below 20 kHz and maximum dampening above 28 kHz (where you will be seeing the start of your signal's mirror image after digital-to-audio conversion), not worrying about the 20–28 kHz region.

[mlyle]

Your comment is mutually contradictory-- what is it, a brick wall (impossible) analog filter or a more gentle rolloff as things fold over?

What you really do, these days, is you sample at a higher frequency; you can have an exceptionally gentle analog filter (which will help you make it linear, too). E.g. if you sample at 96KHz, you just need to roll to near zero by 75KHz. Then you can digitally downsample/decimate to 44.1KHz or 48KHz.

Also note for CD audio, it's more like 24KHz where you get worried, not 28KHz.

[marcosdumay]

Yep, that's why people do things like 44kHz sampling instead of 40kHz.

[Sesse__]

No, 44 kHz is because you want to reconstruct the (20 kHz) bandlimited signal and it's (much) easier to realize such a filter if you have a bit of a transition band.

[gwbas1c]

> You've misunderstood something about Nyquist. A sample rate of, say, 44KHz, will capture ALL information below 22KHz and recreate it perfectly.

Let's do a thought experiment. Imagine a digital image where the pixels are the exact minimum size that you can see.

If a line is exactly 1-pixel-wide, it'll display perfectly when it aligns perfectly with the pixels.

But, if the 1-pixel-wide image doesn't align with the pixels, what happens?

You can see this in practice when you have a large screen TV, and watch lower-resolution video. Smooth gradients look fine, but narrow lines have artifacts. IE, I recently saw a 1024p movie in the theater and saw pixels occasionally.

The same thing happens in sound, but because a lot of us have trouble hearing high frequencies, we don't miss it as much.

[01HNNWZ0MV43FF]

How bad is it around the frequencies I can hear as a 30-something?