What is "time resolution"? Options

[Axon]

So I've become involved in a rather colorful argument (I'm Publius in the thread) with somebody on stevehoffman.tv. The original thread revolved around shooting down an old audiophile canard, about how subsample delays cannot be represented in PCM. In the course of that debate, I've begun to question a couple things.

• Is it ever accurate to use the term "time resolution" in any sort of technical context? To the best of my knowledge, it has no universally agreed upon technical definition. Most of the times I've seen it used are either for SACD/DVD-A marketing fluff, or to describe FFT window lengths. I'm tempted to just go quasi-logical-positivist on everybody and say that it is a completely meaningless phrase.
• Is there any meaningful time-domain constraint on audio quality that is directly related to the sampling period? Subsample delays (as I've shown above) are not meaningfully related. Bandwidth is a frequency-domain attribute. Pre-echo potentially gets more audible at lower sampling rates, but this is not a concern with sigma-delta ADCs, and it is of debatable audibility at 44.1 to begin with. Some DSP operations may be harder to implement at lower sample rates, but most of the issues involve seem implementation-related. I'm suspecting that there are no clear general limits as to what can and cannot be accomplished in PCM, except with respect to very domain-specific or system-specific situations; and so any claims of 44khz always being limited in ways different from bandwidth may be regarded with skepticism.

[2Bdecided]

ChiGung,

Your experiment wouldn't work. By knocking the sample rate down to 11kHz (and implicitly limiting the bandwidth to 5.5kHz) you would change the waveform dramatically. For simple synthetic waveforms, we could say correctly whether none, some, or all peaks would stay in the same place depending on the content of the original signal. However, for complex waveforms, we can't say anything sensible about what would happen to individual waveform peaks.

For example, if you have a bass drum and a high hat playing at the same time, most of the waveform excursion will be due to the bass drum (which will survive the 5.5kHz low pass filter), but the exact peak location will also depend on the "wiggles" in the waveform due to the high hat itself. These high frequency "wiggles" will be butchered by a 5.5kHz low pass filter, so the peak will move!

The only way you can be absolutely sure that it's a fair experiment, and that the low pass filter isn't significantly moving the peak by removing part of the signal that forms the peak itself, is to ensure that the low pass filter doesn't remove anything - i.e. that the original doesn't contain any frequencies above 5.5kHz, or, to put it another way, that the downsampled version still satisfies Nyquist with reference to the original content.

Nyquist is right yet again - what a surprise!



The basic problem is here:

Fundamentaly the 11kHz pcm's potential to resolve detail 'should' seem to be 1/4 of the CDs 44kHz record.

That would go,
44kHz pcm 'time resolution' = 4* 11kHz pcm 'time resolution'

You are implying that these two things are directly proportional in a real and limiting sense, whereas, until you get to the absolute limit, many orders of magnitude better than the limits of human hearing, and many orders of magnitude better than anything we expect the system to achieve, the two things are completely independent.


Rather than talking about sample rate and "time resolution", let's talk about the number of hours I'm awake in a day, and the number of bananas I eat that day.

Fundamentally, the number of bananas I eat if I'm awake for 8 hours "should" seem to be 1/2 of that if I'm awake for 16 hours.

What's wrong with this statement? On the face of it, it seems like intuition. However, the reality of the situation is that the number of bananas I eat in a day has nothing to do with how long I'm awake. I might not have any bananas in the house, and I might not go shopping. I might go to the market I buy a big bunch of bananas at a bargain price and eat several of them. I'll probably just have one a day for my lunch (I'm so boring and predictable) no matter how long I'm awake. The simple truth is that, whatever blind intuition may try to tell you, the practical real world truth is that the number of bananas I eat in a day is completely independent of how many hours I'm awake.


Similarly, the time resolution of a PCM system is independent of the sample rate!

oddly, here and elsewhere this formula is eclipsed with not insubstantial excursions into test pattern replications and counter intuitive textbook quotations.

"oddly"?! What's odd about reality not conforming to blinkered misguided uninformed intuition?!


As for "excursions into test pattern replications" - if you look carefully at my previous post, and the waveforms, I've performed your latest thought experiment already - but with the only type of waveform where it will work - a carefully controlled one!


Finally...

A plot could be made of their distribution of correlation, perhaps it would tend to be a bell curve? for pink noise only? What would the limits of correlation be?

In a suitably controlled version of your experiment (like mine) it wouldn't be a bell curve, it would be a single point! That would certainly be the case with the parameters you propose (44.1>192kHz).

If you try it with real music, it might be a bell curve, or it might be some other distribution (with a cut off corresponding to the point you decide the peaks don't match) - but that's got nothing to do with the temporal resolution of PCM, and everything to do with an experiment where you change something intentionally at random, and then measure how much you've changed it!

Cheers,
David.