steve_1979 said:
Andrew17321 said:
davedotco said:
The mathematics of digital sampling are known and understood, the whole technology is built on it.
Sampling at 16 bit 44.1 kHz will reconstruct any signal up to 22.05kHz precisely, no ifs or buts but with 100% accuracy, this is a proven fact.
That is true provided the conditions are met precicely. That includes the signal not changing (a single sustained note) and the sampling done exactly (ie not using digital approximations). I am a mathematician!
Hi Andrew.
Would you mind explaining these two points in a bit more detail please?
p.s. I'm not trolling for the sake of argument. I am genuinely interested to learn more about how and why "the signal not changing" and "the sampling done exactly (not using digital approximations)" can effect Nyquist-Shannon sampling theorem.
I'd also be interested to learn what effect (if any) these two things could have on real world applications of digital audio.
At the risk of diving in (and also being called Andrew) - "Nyquist-Shannon sampling theorem" - is a theorem, not a theory. A theory is a belief, generally accepted by the scientific community - in many cases supported by experimental data - but not proved absolutely. The 'big bang theory' is a good example of this, widely believed, lots of evidence to support it, but no definitive proof. A theorem is derived mathematically from a set of basic mathematical principles, and absent an error in the working out, is a definitive proof.
Nyquist-Shannon imposes constraints in order to make the maths work - and remember the theorem proves that the output of a sampled system is
exactly the same as the input within these constraints. These constraints have implications - one not unreasonable constraint is that there are no errors introduced by the sampling process, that each sample is an exact measurement of the waveform. Another constraint is that the waveform to be sampled is 'band limited' - has no spectral energy outside of a defined upper frequency. This has non-obvious Fourier implications regarding infinite length samples and invariate signals.
Now unlike the other Andrew, I am not a mathematician, I was an engineer - and in the great tradition of engineering, it doesn't have to be exactly right, it just has to be good enough.
So, your home audio system is violating two fundatmental constraints of Nyquist / Shannon - it uses discrete sample levels introducing quantisation noise, and the original sampling process wasn't perfectly band limited which has aliasing implications.
Mathematically a disaster, engineering-wise it works just fine.