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.

[Woodinville]

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.

Having read that thread, or some of it, it's clear that there are several issues being confuted by people who do not want to let go of their pet belief system.

The first issue is that of pure delay. You've killed that. You can point out that if you store the data in 16 bit signed, that yes, there is a limit, it's directly related to the sampling rate times the number of levels available for quantization... i.e. directly related to the SNR and the sampling rate.

The second is that of jitter. You need to beat on these people to point out that jitter is different for each sample, and your time delay is not.

Time delay has been reported as audible down to 5 to 10 microseconds in binaural settings with a great deal of care and signal prep involved. No lower.

The cited nanosecods, etc, are just stuff and nonsense.

What you need to do with the jitter crowd is to point out that the spectrum of the jitter is the big deal.

You can have 1 part in 1000 (relative to sample period) if the jitter has only frequencies below 1 Hz.

You can hear much more than that if you have high frequency jitter, and a high frequency signal.

Looking onward, we see the twirp who insists that things have to be periodic in order for subsample resolution to work.

Perhaps he needs to be acquinted with both the Nyquist theorem and the fact that for all real signals, Fourier synthesis works.

I'm not going to get an ID to go argue with those guys. Blah.

This post has been edited by Woodinville: Oct 5 2006, 23:21

[ChiGung]

The first issue is that of pure delay. You've killed that. You can point out that if you store the data in 16 bit signed, that yes, there is a limit, it's directly related to the sampling rate times the number of levels available for quantization... i.e. directly related to the SNR and the sampling rate.

Just me doing the arguing in that thread really. On the fly (though it was never a prime point) I hedged that what you refer to as 'delay' is dependant on the measure size like you state.
However you seem also unwilling to acknowledge too, that such 'delay' is no definite attribute of the undownsampled source, it is an attribute summed circumspectly from the phases of frequencies surviving the downsample. The frequencies which didnt survive the downsample, contained the information required to resolve the true subsample detail of 'time localised energy spikes'.

The second is that of jitter. You need to beat on these people to point out that jitter is different for each sample, and your time delay is not.

Comments about jitter came right at end... or maybe its grown...eek, anyway there enough on the plate already.

Time delay has been reported as audible down to 5 to 10 microseconds in binaural settings with a great deal of care and signal prep involved. No lower.

Informative, although I have never argued the against the subsample time resolution of frequencies surviving the implicit lowpass of the sample rate. The arguement was about whether such precision can be fairly refered to as 'time-resolution'

Looking onward, we see the twirp who insists that things have to be periodic in order for subsample resolution to work.

Ahem* that would be me, but an unkindly misrepresented myself.
Again, my point is -"subsample resolution" of what exactly? the subsample levels implied in PCM have to assume all energy above the nyquist frequency is zero - so yes 'things' do have to be periodic, specificaly the component periods of the frequencies which inform the spaces between samples all have to be less than 2 samples in length - because shorter frequencies are not informed by the record. Subsample detail of the waveform is for standardisation purposes assumed to be consistent with the remaining information, because the origional information is lacked. So what can 'time resolution of PCM' refer to? any synthetic measurement we can take? Is there not already a fair candidate for this term?

Perhaps he needs to be acquinted with both the Nyquist theorem and the fact that for all real signals, Fourier synthesis works.

I am acquainted with it thanks, and understand some limitations. Just trying to pass that on.

I'm not going to get an ID to go argue with those guys. Blah.

You didnt need to

This post has been edited by ChiGung: Oct 6 2006, 00:49

[saratoga]

Looking onward, we see the twirp who insists that things have to be periodic in order for subsample resolution to work.

Ahem* that would be me, but an unkindly misrepresented myself.
Again, my point is -"subsample resolution" of what exactly? the subsample levels implied in PCM have to assume all energy above the nyquist frequency is zero - so yes 'things' do have to be periodic, specificaly the component periods of the frequencies which inform the spaces between samples all have to be less than 2 samples in length - because shorter frequencies are not informed by the record. Subsample detail of the waveform is for standardisation purposes assumed to be consistent with the remaining information, because the origional information is lacked. So what can 'time resolution of PCM' refer to? any synthetic measurement we can take? Is there not already a fair candidate for this term?

Perhaps he needs to be acquinted with both the Nyquist theorem and the fact that for all real signals, Fourier synthesis works.

I am acquainted with it thanks, and understand some limitations. Just trying to pass that on.

You mention periods "less than 2 samples in length". This is a band limited signal, there are no such periods. I think you are trying to say that since you can't have frequencies higher then the nyquist frequency, then it must be true that you cannot have frequencies closer together then 2/fs. However, this is not true. The Nyquist theory allows exact reconstruction for band limited signals, which means you can resolve infinately many points between any two points within the reconstructed waveform just by upsampling and filtering.

Let me give an example. If you sample at 10Hz and are bandlimited to 5Hz, you can resolve the amplitudes at 1Hz, 1.0000000000000000000001Hz, etc perfectly. Its just that since you're bandlimited, you have no new information at these interpolated points (I.E. they're exactly determined by the value of the adjecent points). But you can still retrieve their values using resampling, sinc interpolation or whatever.

Regarding your periodic remark, I'm afraid I don't follow.

[ChiGung]

The subsample levels implied in PCM have to assume all energy above the nyquist frequency is zero - so yes 'things' do have to be periodic, specificaly the component periods of the frequencies which inform the spaces between samples all have to be less than 2 samples in length - because shorter frequencies are not informed by the record. Subsample detail of the waveform is for standardisation purposes assumed to be consistent with the remaining information....

You mention periods "less than 2 samples in length". This is a band limited signal, there are no such periods.

Exactly, frequencies of that period are required to render time localised energy between the samples, which we obviously dont have until we have access to non-downsampled records.

I think you are trying to say that since you can't have frequencies higher then the nyquist frequency, then it must be true that you cannot have frequencies closer together then 2/fs.

I would never knowingly say anything to that effect'

Regarding your periodic remark, I'm afraid I don't follow.

I didnt exactly make the periodic remark, I think woodinville was refering to explainations in the other thread like this:

It was shown that the phase of a sinusoidal pattern which is assumed as perfect and constant can be resolved to a fraction of the sampling interval.
This subsample accuracy was possible because the pattern recorded is not a discrete event, it's impression is recorded throughout many consecutive samples and its exact formation is inferable (idealy).
For all discrete or unassumable events, PCM records can only specify time of occurence to within a whole length of the sampling interval. Time resolution can only be improved when a known pattern can be observed throughout multiple samples -which is the case for computing the phase of synthetic frequency components, but not at all when trying to refine the temporal location of unassumed events.

For conversion and processing purposes etc PCM is interprated as a composite of exclusively periodic entities (frequencies) But localised energy need not fit into any periodic cycle, so we cant locate it precisely with the periodic tools (frequencies) even though we can locate all the individual tools precisely.

techie: "captain we have located a 'spike' event on the PCM sensor"
captain:"what is the position of the spikes peak teki?"
techie:"324.37643 sampling intervals exactly captain"
captain:"how can you be so precise?"
techie:"because time delays are quite precisely encodable in PCM"

But a natural spike, will have an unknown frequency spectrum, the tools to locate the true peak with certainty had to be removed before the downsample, so we can make the best guess by assuming the 'subsample deviators' were all flat anyway but thats just a guess, the true peak could have been anywhere in the sample interval. If it was actualy somewhere other than the record suggests most likely, that information was contained in the lowpassed higher frequencies which now manifest as the unrecorded gaps between samples.

waffle, waffle, waffle.

'It just refers to how precisely you can localize energy in the time domain. Generally this is just the sample period, but not always.

That is how I seesaw it. Ive just been clumsily trying to explain this really, from a few different angles.

[saratoga]

The subsample levels implied in PCM have to assume all energy above the nyquist frequency is zero - so yes 'things' do have to be periodic, specificaly the component periods of the frequencies which inform the spaces between samples all have to be less than 2 samples in length - because shorter frequencies are not informed by the record. Subsample detail of the waveform is for standardisation purposes assumed to be consistent with the remaining information....

You mention periods "less than 2 samples in length". This is a band limited signal, there are no such periods.

Exactly, frequencies of that period are required to render time localised energy between the samples, which we obviously dont have until we have access to non-downsampled records.

No. I'm saying that no such information ever existed. Not just in the sampled signal. Ever. The signal is band limited to no more then half the sampling rate, therefore there is NO information between between samples, either in the information we have or the original analog signal.

I think you are trying to say that since you can't have frequencies higher then the nyquist frequency, then it must be true that you cannot have frequencies closer together then 2/fs.

I would never knowingly say anything to that effect'

You just did. "Frequencies of that period are required to render time localised energy between the samples" Surely you realize how these statements are equivilent?

Regarding your periodic remark, I'm afraid I don't follow.

I didnt exactly make the periodic remark, I think woodinville was refering to explainations in the other thread like this:

It was shown that the phase of a sinusoidal pattern which is assumed as perfect and constant can be resolved to a fraction of the sampling interval.
This subsample accuracy was possible because the pattern recorded is not a discrete event, it's impression is recorded throughout many consecutive samples and its exact formation is inferable (idealy).
For all discrete or unassumable events, PCM records can only specify time of occurence to within a whole length of the sampling interval. Time resolution can only be improved when a known pattern can be observed throughout multiple samples -which is the case for computing the phase of synthetic frequency components, but not at all when trying to refine the temporal location of unassumed events.

I was refering to your statement that "things have to be periodic". That doesn't seem related to what you posted above because you make no mention of periodicity, or if it is I cannot follow your logic.

For conversion and processing purposes etc PCM is interprated as a composite of exclusively periodic entities (frequencies) But localised energy need not fit into any periodic cycle, so we cant locate it precisely with the periodic tools (frequencies) even though we can locate all the individual tools precisely.

PCM does not assume anything is periodic. No frequency treatment is required to dervive it either. The frequency/time trade off you're refering to is a property of the fourier transforms, but not of PCM which could be implemented with nothing but time domain tools.

techie: "captain we have located a 'spike' event on the PCM sensor"
captain:"what is the position of the spikes peak teki?"
techie:"324.37643 sampling intervals exactly captain"
captain:"how can you be so precise?"
techie:"because time delays are quite precisely encodable in PCM"

But a natural spike, will have an unknown frequency spectrum, the tools to locate the true peak with certainty had to be removed before the downsample, so we can make the best guess by assuming the 'subsample deviators' were all flat anyway but thats just a guess, the true peak could have been anywhere in the sample interval. If it was actualy somewhere other than the record suggests most likely, that information was contained in the lowpassed higher frequencies which now manifest as the unrecorded gaps between samples.

We're assuming the signal is band limited, so we absolutely can exactly locate your spike. Think about how an ideal sinc interpolator works. You superimpose sinc functions to get the exact value of the function at that point. There is no approximation like you seem to be thinking. It really is exact.

What you're saying would be true of a signal that was not band limited, but if it was not bandlimited, we would not be able to reconstruct the signal anyway. Alternatively, it would be true if you assumed the signal was not bandlimited, but that you had an antialiasing filter in place to band limit it. In that case, you could in fact say that the antialiasing filter delocalizes your spike by discarding the extra frequency content needed to exactly localize it. However this is NOT what everyone else is discussing which is why they keep saying PCM and not antialiasing filter. We're assuming that input signal is already bandlimited, and thus it can be perfectly reconstructed.

This post has been edited by Mike Giacomelli: Oct 6 2006, 06:31