Time resolution of digital sampling

[Don Hills]

In another thread, the following was said:

 

5 hours ago, manueljenkin said:

... science studies do show humans can discern a time precision way above what a 48khz sampling rate can reliably capture (5micro seconds is what I remember, check some MIT studies and stuff).

 

This is incorrect. 48 KHz sample rate has a "time precision" in the picosecond range. Here's something I wrote for 44.1 KHz:

The time resolution of a 16 bit, 44.1khz PCM channel is not limited to the 22.7µs time difference between samples. The actual minimum time resolution is equivalent to 1/(2pi * quantization levels * sample rate). For 16/44.1, that is 1/(2pi * 65536 * 44100), which is about 55 picoseconds. To put that in perspective, light travels less than an inch in that time.

 

Shannon and Nyquist showed that as long as you keep all components of the input signal below half the sampling frequency, you can reconstruct the original signal perfectly - not just in terms of amplitude, but in terms of temporal relationships too. They only addressed sampling, and assumed infinite resolution in amplitude. With a digital signal the precision is limited by the number of amplitude steps, leading to the above formula.

 

If anyone has difficulty in understanding why the above is true, please post. I'll try and explain it in other terms. It's non-intuitive that an event that occurs between samples can be accurately captured.

 

If you want to see a real world demonstration of a single event (the edge of a square wave) being accurately sampled between sample points, check out Monty's show and tell at the 20:55 mark. If anyone following this thread hasn't seen the video before, I strongly suggest you take the time to watch it all.

 

 

[Don Hills]

I'll try.

In the following, assume that the input to be digitised is valid, that is, contains no frequencies equal to or greater than half of the sampling frequency.

If a peak that would sample to a value greater than "digital full scale" occurs between samples, the value of that peak is accurately captured in the samples before and after that peak. To reach that peak and also be valid, the input signal before and after the peak must have a finite slope. The reconstruction filter performs the calculations to recreate that slope and there's only one correct curve that joins the slopes before and after the peak. There's a very good Benchmark paper on the subject:

https://benchmarkmedia.com/blogs/application_notes/intersample-overs-in-cd-recordings

[Don Hills]

Another way of looking at it is to consider an "intersample over" as being no different than a lower level signal. Provided that the signal amplitude at the moment of sampling is no greater than can be represented by the digital value (is accurately captured), the values of the signal in between the samples will be accurately reproduced after reconstruction. (Provided, of course, the hardware is designed to handle intersample overs.) The Benchmark article makes this point with the Steely Dan track - no clipping, all of the samples accurately captured the signal amplitude, but many DACs clip the intersample overs.

Note that filterless NOS DACs don't accurately reproduce values in between the samples. Far from being a barrier to accurate reproduction, the reconstruction filter is mathematically essential for accurate reproduction.

[Don Hills]

8 hours ago, PeterSt said:

... Not sure where the math in the OP (from Don) originates, but 71ns is what I would say. Just because my DAC shows that, and which is the result of a transient which originally requires infinite frequency to do its job (but which transient I flattened 16x).

It is electrical behavior which I think can be calculated with math (Voltage jump is +2V, but the one sample turned into 33 samples). ...

 

It came from JJ. It is the math describing the effect.

A transient might occur between sample times. When you (correctly) low pass filter the transient before sampling, the filtering spreads the transient energy over a period of time. The sample times before and after the time of the transient then capture the energy. If you move the transient occurrence to a different time between samples, the values sampled will change. To detect that the transient has "moved", at least one digital sample value has to change. The formula describes how far the transient has to "move" to change at least one bit.

[Don Hills]

7 hours ago, manueljenkin said:

... The time precision quoted in the first post here is an extremely optimistic scenario, taken at steady state, which is not a real world occurence. Unless you're trying to validate oscillators in their steady state, it doesn't work out that well. And things don't stop here. During Reconstruction, you're doing another sinc convolution over the samples = an additional transient smearing. In steady state, you can put a sinc filter with 20khz over a 5khz sine of amplitude X how many times you want and you'll still get an output of X, because there's an infinite train of 5khz behind and ahead to support the math. But in reality we don't have such signals sampled for infinite time. In the transient states you will actually get a ringing (and gets worser each iteration).

...

The time precision quoted does not depend on a "steady state". As I pointed out in my previous post, it is just as valid for a transient.

 

You'll need to provide some proof of the "ringing" getting worse for each iteration. In Monty's video, as well as showing the effect of moving a transient event between sample times, he showed that once the signal has been filtered during the ADC - DAC process, it can be run through the chain again and the "ringing" does not change. For there to be "ringing" at the output of the filter there has to be signal presented to the filter that is outside its passband. Once that signal has been removed by the filter, there is nothing to increase the "ringing" on subsequent passes.

 

* I dislike the term "ringing" to describe Gibbs effect, but its (mis)use is sadly far too common.

 

[Don Hills]

9 hours ago, pkane2001 said:

...

Hearing down to -200dB means (to me, at least) being able to detect, with ears, a signal (or its effects/modulation of another signal) at -200dB. No? Maybe it's that language barrier thing that Peter was talking about.

...

 

I interpreted his meaning as being what you wrote in parentheses. Of course, for the effect to be audible it has to result in an audible change of the "main" signal, and therefore have caused a much higher level of distortion than -200dB. 

 

[Don Hills]

18 hours ago, JoshM said:

Don, your posts here are wonderful.
 

How much do you think we should worry about intersample overs? Are they audible? Should we apply digital volume attenuation in Audirvana, HQ Player, etc. to avoid overs? Or does the extra processing involved in using digital volume attenuation outweigh the benefits of avoiding overs?

 

I’ve been using a true peak plugin with Audirvana to see which albums I have clip, then applying digital volume attenuation. However, I still debate whether I’m doing the right thing, as then I’m no longer sending “bit perfect” data to the DAC. 

 

Thank you.

 

I don't worry too much about intersample overs. I feel that music with aspirations to high fidelity won't have been pushed to the limit in mastering, and conversely music that has been pushed hard will likely already have more audible damage.

 

As to audibility, it of course depends on the specific case. I could probably generate a test signal where it was quite audible, but I don't think it's a big problem in everyday music. It will also depend on your DAC's behaviour when processing intersample overs - some will handle them better than others.

 

The simplest prevention method is to attenuate the digital signal before the DAC by exactly 1 bit (6.02dB). This requires minimum processing and should handle all except pathological cases. Personally, I feel that maintaining "bit perfectness" is overhyped. A competently implemented digital volume control will be effectively as transparent as an equivalent analogue control.

 

For those albums where you find overs, have you tried comparing the before and after attenuation signals? Note that you need to apply attenuation post-DAC to make both signals the same volume for comparison. My feeling is that you'll be hard put to hear a difference.

 

[Don Hills]

19 hours ago, Miska said:

...

To what I've seen, over 90% of RedBook material has overs. And for over 90% of it, -3 dBFS is enough to avoid it. And there are clear mathematical reasons why this is the case.

...

 

 

Personally, I don't spend much time worrying about them. By definition, an intersample over occurs between 2 samples. Therefore, its frequency content is above 22 KHz. Provided that the DAC clips cleanly and the ultrasonic content doesn't upset the following equipment, it should be inaudible. (Of course, in real life things are rarely that ideal.) But I'd still rather see them avoided at source than having to allow for them in playback.

[Don Hills]

1 hour ago, fas42 said:

 

What??!!  An intersample over is above 22kHz? ...

 

Yes. In your example with an 11 KHz sine wave, clipping of the intersample peak will result in harmonic distortion. And those harmonics begin at 22 KHz...

 

Edit: I think I can come up with special cases where there may be distortion products below 22 KHz. I'll see what I can do with Audacity.

[Don Hills]

31 minutes ago, danadam said:

https://forum.cockos.com/showthread.php?t=201868

 

Thanks. I'll try it.

[Don Hills]

If I recall correctly, the red in Audacity indicates where at least 3 consecutive samples are at the digital limit ("0dB"). This is usually real clipping, where the level of the signal being sampled at the sample moment is greater than that represented by the sample value. It is possible to have inter-sample overs where all of the sample values are valid (the sample value accurately represents the value of the input voltage at that moment). 

 

Try generating and filtering the white noise at a lower amplitude than 99%. Now normalise it. There should be no red, but there may be intersample overs. I'll try it myself hortly.

[Don Hills]

18 hours ago, opus101 said:

 

I've done this and its possible to get no red, yes. But then how to see when there are intersample overs?

 

There's another way to 'see red' with this - don't do filtering just upsample the white noise. When there's no clipping (red) at 44.1k but the red appears at higher rates would you say that's an instance of intersample overs?

 

Yes, that makes sense. "True peak" meters do this to indicate intersample overs.

[Don Hills]

3 hours ago, yamamoto2002 said:

When bit depth becomes ₀א, time resolution becomes ₁א ?🤔

 

 

Yes. See the equation in the OP... ☺️