Representing frequency of n Hz needs sampling rate >2n Hz, not =2n, From: Badly drawn waveforms vs. the audio that’s actually output/93496 Options

[icstm]

Hehe, shortly after posting here there was a nice picture posted that tells the whole story

where is the nice picture? That is really important to share!
The whole point of FFTs is that using if I am drawing what the FFT has stored in the time domain (ie if I am drawing sine waves) then I only need a couple of points to draw a PERFECT reproduction!
That is one of the biggest issues that people need to first get their head around when understanding digital audio.

when you see an irregular wave, what you need to ask your self is "what regular wave would I have to add together (ie what frequencies would have to be simultaneously present) to create the irregular wave I see”

so an irregular sine wave is in effect a regular one augmented with higher or lower frequency wave to varying amplitude.

Each of the additional waves only needs a couple of points to represent their frequency. As per usual it appears that CA article really doesn’t get what the Nyquist theorem is saying…

However it is right when it says that it is possible for a perfect 8kHz signal sampled at 96kHz to sound better than sampled at 192kHz if the DAC has trouble with 192kHz (ie is a cheaper DAC). The whole point is that a perfect DAC will produce a perfect 8kHz signal when sampled at 16kHz. But the quality of the output has little to do with the conversion and more to do with the analogue output. One of my earliest posts on HA was asking about DACs in AVRs as I am keen to understand the practicalities on consumer grade stuff.

What determines a good DAC?
I assume the main issues are post the conversion to analogue and that these issues are the usual ones that impact an amp (pre or power), such as clean power supply, suitable analogue filters, etc.
However are there any differences in the digital part of the amp (such as up/over sampling circuits, or the conversion itself)?

[DonP]

Each of the additional waves only needs a couple of points to represent their frequency. As per usual it appears that CA article really doesn’t get what the Nyquist theorem is saying…
....
The whole point is that a perfect DAC will produce a perfect 8kHz signal when sampled at 16kHz.

This key mistake in citing the Nyquist theorem leads to no end of potential trouble. The sampling frequency has to be greater than 2x the maximum signal frequency you want to reproduce.

[pdq]

If you know that the waveform consists of a single unvarying sine wave whose frequency is less than half the sampling rate then a fairly small number of data points are required to determine the waveform's frequency and amplitude.

On the other hand, if there are multiple frequencies or the amplitude is not constant then you will need a longer observation period.

[2Bdecided]

If you know that the waveform consists of a single unvarying sine wave whose frequency is less than half the sampling rate then a fairly small number of data points are required to determine the waveform's frequency and amplitude.

On the other hand, if there are multiple frequencies or the amplitude is not constant then you will need a longer observation period.

For any arbitrary waveform, correctly sampled, you can reconstruct all frequency components under fs/2 perfectly - but you need an infinite number of sample points.

If we consider quantisation it's far worse.

If however we consider that we don't care about anything beyond ~120dB down, it becomes easily realisable in 1990s-style DSP.

I know everyone here knows this. That computeraudiophile thread is just a parallel universe which I don't want to enter

Cheers,
David.

[Arnold B. Kruege...]

If you know that the waveform consists of a single unvarying sine wave whose frequency is less than half the sampling rate then a fairly small number of data points are required to determine the waveform's frequency and amplitude.

On the other hand, if there are multiple frequencies or the amplitude is not constant then you will need a longer observation period.

For any arbitrary waveform, correctly sampled, you can reconstruct all frequency components under fs/2 perfectly - but you need an infinite number of sample points.

If we consider quantisation it's far worse.

If however we consider that we don't care about anything beyond ~120dB down, it becomes easily realisable in 1990s-style DSP.

I know everyone here knows this. That computeraudiophile thread is just a parallel universe which I don't want to enter

Whenever infinity is mentioned, the temptation to become pedantic can be overpowering. Take this for exactly a pedantic question, because that is what it it is.

Isn't infinity an indefinite number? That's what I was taught in first semester calculus 48 years ago, if memory serves. I'm under the impression from my work in calculus through grad school that equations involving infinity only make sense if you talk about infinity as the limit. IOW, a sampled wave approaches perfection as the number of sampled points approaches infinity.

In the real world nothing is perfect, and significance and relevance should be our greatest interest. As I think about it, in audio the number of sample points and the numerical precision of those samples are not serious limiting issues in our best or even mediocre currently implemented systems.

[Porcus]

Isn't infinity an indefinite number? That's what I was taught in first semester calculus 48 years ago, if memory serves. I'm under the impression from my work in calculus through grad school that equations involving infinity only make sense if you talk about infinity as the limit. IOW, a sampled wave approaches perfection as the number of sampled points approaches infinity.

Well, in first-semester calculus, infinity is not a number at all. In higher courses, it might even be treated just as a number. Now you are talking about 'the' limit, and that 'the' is not necessarily a unique concept (first-semester calculus may or may not involve the distinction between pointwise and uniform limits).

For each given frequency < half the sampling frequency, a sine wave on time interval [-T, T] will be sampled better as T grows, and -- in the appropriate sense -- tend to perfection.
But:
Fix the time interval [-T,T] and let the frequency increase to half the sampling frequency. Then the sample quality tends to pretty bad. (And no, you don't have do do a sweep.)

So transients near the Nyquist may be badly sampled, in principle. But even if you can hear 22.05, it does not mean that you will hear that this is bad -- this physiology issue beats me, but: do you really hear a tone before the hair cells in the inner ear have reached a steady resonance with the sound? That takes some oscillations, sampler takes some oscillations to pick up the frequency, wouldn't it?