This video linked is very informative:

Nyquist-Shannon; The Backbone of Digital Sound

https://www.youtube.com/watch?v=pWjdWCePgvA
And the comments copied from there might need a more thorough approach:

Jan-Victor Björkqvist2 weeks ago

I don't think it is strictly true that two samples per cycle can reproduce the only signal that lead to those samples. If you sample (with 2 Hz) at 0 s, 0.5 s, and 1 s you will get samples 0, 0, and 0 from a perfect 1 Hz sine wave, i.e. y=sin(2*pi*x). Still, you would obviously also get constant zero samples from having no signal at all, and there is therefore no way to reproduce the original sine wave based on the three samples.

First, I think the guy in the "

*Nyquist-Shannon; The Backbone of Digital Sound*" video is doing a great job explaining the NS Theorem (NST) in general terms for the lay person (like me), but as he says somewhere, he is doing some simplifications and some generalizations.

And this is because the NST involves advanced mathematics that can't be crammed into a few minutes video. It takes years for an average intelligent person to reach a level of understanding of advanced mathematics. I, personally, Im certainly not there.

The NST is more complicated than "just sample at twice the highest frequency and go home and be happy and enjoy your 44.1/16 CD music collection". Im afraid it is more complicated than that.

As far as I remember (I might be wrong as Im writting from my memory) there are certain conditions and constraints for the NST to work and you know it is crazy but the NST even goes as far as to state that there are even some specific constraints under which a sample rate lower than the minimum stated by the NST itself can take you to the exact original analog signal.

Now, the zero values:

If you sample (with 2 Hz) at 0 s, 0.5 s, and 1 s you will get samples 0, 0, and 0 from a perfect 1 Hz sine wave

As for the example of the samples taken at exactly zero value so unable to get any information to get back to the original analog signal... Again,

**as far as I remember**, the NST takes into account signal phase. It is not just get some samples no matter what. There are conditions and constraints and there is this thing about signal phase that is considered within the NST.

Ok, I think this adds little to this discussion because Im far from certain about what Im writting here but I just want to point in the right direction. And for me the right direction is that the NST is far more complicated than "just get the minimum stated sample rate and that's it, just get 44.1Khz sample rate and that's all you need to know". There is more to it.

It is somewhat similar to scientists trying to explain Einstein's general relativity theory in which gravity is exemplified by the famous example of the bowling ball and the sheet.... It is just a generalization and there is more to it.

I think, as stated in the video, different mathematicians arrived at the NST independently.

Yes, independently.

We are talking about serious, professional competent people. Not just aficionados. Not just a bunch of Youtubers or Bloggers.

For me it is hard to imagine that all of them failed to consider a very simple circumstance in which samples are taken at exactly zero value. Im sure they thought about it!!

Now, what HB tells us in his video of why 192 Khz matter is interesting but If I understand correctly, he focuses more on the practical side of making use of the NST, it is, the practical aspects of filters and all that.

He worries from the fact that in real life no signal is perfectly bandlimited. And from there, he considers there are problems with the filters and suggests going to a higher sample rate.

From what he says I believe there is some reason to believe that we "need" a higher sample rate but not because the NST does not work but because to take advantage of it and make it work better in real life then we need to go to a higher sample rate. My (perhaps limited) understanding is that this guy HB focuses more on the practical aspects given current technology.

Im not saying he is wrong. Im saying he is focussing on a different aspect of the same problem.

Yes, given current technology, there might be reasons to go to a higher sample rate (192khz) but the benefits might be negligible for most people.

Actually, in real life he mentions that no signal is perfecly bandlimited. And the NST works for bandlimited signals. But this is a theoretical discussion and he attempts to give a practical solution for a theoretical problem.

But in practical sense, does it makes sense?

For some people yes it makes sense as some people are certain that high resolution audio is the way to go. There is even some research that suggest some people can tell HR audio from CD quality. But the same research tells us that these are a minority of people.

Finally, I suggest the following reading, "what the NST says and what it does not say":

https://www.audiostream.com/content/sam ... n-services
Quotes from the article above:

**It is a common misconception that the Nyquist-Shannon sampling theorem could be used to provide a simple, straight forward way to determine the correct minimum sample rate for a system**. While the theorem does establish some bounds, it does not give easy answers. So before you decide the sampling rate for your system, you have to have a good understanding of the implications of the sampling, and of the information you really want to measure.
*The difficulty with the Nyquist-Shannon sampling theorem is that ***it is based on the notion that the signal to be sampled must be perfectly band limited. This property of the theorem is unfortunate because no real world signal is truly and perfectly band limited.** In fact, if a signal were to be perfectly band limited—if it were to have absolutely no energy outside of some finite frequency band—then it must extend infinitely in time.
^That problem with the

**perfectly bandlimited** signal is the one HB mentions in his video... But it is not a reason to consider the NST sample rate is wrong in theoretical terms.

The article adresses the problem of samples taken at zero value...

I believe one must decide to make this discussion either theoretical or practical.

If this is a theoretical question I think there is nothing to add as the NST is fine.

If this is a practical question then there is a lot of discussion to have.

And for some people high resolution is the way to go and for some other people there is no need to go the high resolution way.

At the end, no one is wrong.