Understanding Sample Rate

[mansr]

32 minutes ago, Don Hills said:

What JJ actually said was "1/ ( 2 pi bandwidth number_of_levels)"

Yes, that is correct.

[mansr]

6 minutes ago, adamdea said:

Don, I too share your concern becasue I have assumed JJ got it right. But he definitely said it was

"1/ (2 pi fs nlevels)

for CD that is 1/( 2 pi 44100 65536)."

https://hydrogenaud.io/index.php/topic,91126.0.html

Fs isn't bandwidth. Are we reading the same thing. Sorry if I am being obtuse.

That's definitely wrong.

[mansr]

Just now, adamdea said:

Just to be clear I assuming that you are saying that what he said was wrong not that I have misquoted him.

He's wrong, the quote is accurate. Maybe he just misspoke, but what he said there is wrong.

[mansr]

5 minutes ago, adamdea said:

Great. Can you just talk me through your correction from 220 ps to 110ps

Scroll down a little on that page. As esldude points out, a level above ½ LSB will be rounded up to 1.

[mansr]

2 minutes ago, adamdea said:

Ah so the minimum time difference is equal to the difference which will cause a 1/2 lsb amplitude change in a 22.05Khz wave at the sampling instant.

Right. The 110 ps figure is for a full amplitude signal at the highest frequency, 22.05 kHz. This represents the best case. Now look at some other limit cases. For example, a full amplitude sine wave with a very long period, say a year, changes so slowly that it takes more than a minute to go from zero 1 LSB. At another extreme, a sine wave with an amplitude of 1 LSB needs a phase shift of 30° in order for a sample point initially at the zero-crossing to register a 1.

[mansr]

9 minutes ago, adamdea said:

Thanks 

I'm still a bit troubled. A part of me feel that if a 22.05khz wave is sampled forever and then phase shifted slightly, the minimum time difference to register one sample with a different value (out of all samples) will be different from the figure required to get a difference in the value of one sample per cycle.

Is your calculation not the time difference required to get a 22.05 khz wave perfectly zero crossing at a sample instant to read the lsb at the sampling instant. What trouble me is that any lsb could always tick over to the next value for any change in phase if the value was previously just under the threshold where it rounded up. 

Such is the nature of quantisation. A small phase shift will cause a change in some samples but not in others. The reconstructed signal thus gets distorted a little differently in addition to the phase shift.

 

9 minutes ago, adamdea said:

Something nags at my mind that this is  not the question people want to answer when they ask what is the time resolution of the system. In a dithered system the sample values will change anyway, My instinct is that for an infinitely long sampling period we can detect any change in phase eventually.

Disregard dither to begin with. Now consider a sine wave sampled at some multiple of its frequency. In each period of this this wave, the samples values will be exactly the same. If we now shift it slightly, but not enough to trigger an LSB change at any of the sample points, every period will have exactly those same sample values in perpetuity. If the sample rate isn't a multiple of the signal frequency, the sample values will still be periodic, just with a longer period and the same logic applies.

 

9 minutes ago, adamdea said:

Is the better question what is the minimum change in the location of the peak of a sine wave which can be detected from the sample values? Is that the same as the question you answered?

The derivative of the sine is zero at the peaks. To cause an LSB flip there, the phase shift has to be much greater than at the zero-crossing.

[mansr]

1 hour ago, Ralf11 said:

do we hit that limit in the real world?

 

I thought Johnson noise, etc. were the practical limits

24-bit exceeds the practical analogue noise limits. Cooling your DAC in liquid helium just isn't practical.

[mansr]

2 minutes ago, adamdea said:

I take your other points. On this one- it's not the point I was thinking about. That wasn't what would it take to make the lsb flip at peak it was more- for a sine wave being sampled at twice its frequency what is the minimum change in peak location (which might be  between the sample values) which can be detected from the samples. (perhaps this yields the same answer as your question what is the minimum time shift to generate a change in the lsb at the zero crossing). 

Sampling at exactly twice the frequency isn't allowed.

[mansr]

1 minute ago, adamdea said:

OK I know not to satisfy the sampling theorem , but you were using the example of a 22.05Khz  sine in a 44.1khz sampled system....

An infinitesimally lower frequency is allowed, so to calculate the limit of the time resolution, using the limit of the frequency is fine.

[mansr]

Just now, adamdea said:

I think the points I was making still stand with a frequency very lightly under half the sample rate. It is noted that the sampling intervals will change for each cycle of the wave.

If you're sampling just above twice the frequency, the phase accuracy for a small number of samples depends on the relative phase of the samples to the signal. If the samples are near zero-crossings, you get better accuracy. Over a longer period, you'll get some samples closer to the zero-crossing than others. I'm not sure what the worse case is.

[mansr]

4 minutes ago, adamdea said:

I think I make it either 4 x10^-8 seconds for sampling at peak .

cos-1[1-1/(2^16)])/2pi *22050                     since cos starts at peak, taking the minimum time diference to get the 1 to drop by 1 lsb .  

Does that look right?

Should be 1 - 1/2^15 since values are both positive and negative.

[mansr]

7 minutes ago, adamdea said:

But don't we still have the fact that it only has to tick under half way between full scale and next one down?

Right. I keep forgetting that.

[mansr]

9 minutes ago, buonassi said:

Let me ask you guys.  Is it possible I prefer the minimum phase reconstruction filter because its own phase distortion during playback is negating the phase distortion introduced during analogue to digital capture/encoding? 

That's unlikely. Audio equipment tends to use either linear phase (no distortion) or minimum phase (delayed high frequencies). What you suggest would mean some step in the processing used a filter skewed towards maximum phase. I've never heard of anything doing this, and I can see no reason why anyone would want it.