Which is more accurate: digital or vinyl?

Hello Al.

Thanks for the note, but I find the arguments unconvincing. While it is easy to speak of step functions being "smoothed out", it is imprecise. To make the statement precise, the smoothed function must be measured from real devices rather than theoretical, if for no other reason that every RC filter introduces its own distortion. Once an empirical function is obtained with adequate precision, it may be possible to fit the curve analytically, or, at worst, as an approximation using some technique such as cubic splines. Then, when an expression for the smoothed function is obtained, the analysis can be re-run, and an amended error figure derived. In the absence of such a Herculean effort, which should, of course, be borne by those who market the technology, I think that we are entitled to simplify the problem as I have done (see below).

Furthermore, I hold little hope that this effort will much reduce the distortion figure. Perhaps this is why we have not seen it reported. I alluded to the problem in my previous post - the smoothed curve will lag the sine except at the peak and trough. Hence the smoothed curve will closely approximate another smooth function, albeit one with two higher frequency distortion components, both of which will be some function of frequency. That other smooth function will not be a sine, having a (relative) hollow on the left edge and a bulge on the right. The RMS error, being referenced to a true sine function, will remain high.

As for your riposte, that a 176 Hz tone would be 5% distorted, that is not implausible to me. I find even the mid-range on CD's to be unclear compared to analogue (Linn Unidisk source into electrostatics). You are absolutely right to make the calculation and challenge me on it, but I have already made that calculation and found it plausible, so I suppose we must agree to disagree on that point.

If you would like to proceed as I suggest in the first paragraph, and achieve a better approximation, I applaud your devotion to science. And I will modify my opinions with a dose of humble pie if you prove me wrong.

Thanks for engaging.

Terry

all digital recordings are made using analog mikes - so unless there is A/D converter that can do it 100% identical as mikes picks it up - then all analag recordings will be better (as long as there is no digital involved in the process) - which was true in vinyl days....
kind of same as photography - kodachrome was always better than digital....

Hi Terry,

Rather than getting into a lot of esoteric mathematics that would be necessary to provide a quantitative perspective on all of this, I’ll just make a couple of additional qualitative points. I suspect that following your rebuttal we'll then, as you say, have to agree to disagree.

I agree that the low pass filtering/analog reconstruction process cannot be done to absolute perfection. However, consider the spectral components that distinguish an audio frequency sine wave from that sine wave as sampled at 44.1 kHz. The spectral components that distinguish those two waveforms are all at ultrasonic and RF frequencies, and as such are essentially inaudible to us. (The reason I say “essentially” is that, as you may be aware, some seemingly credible studies have suggested that we may be somehow able to sense the presence of frequencies up to perhaps as high as 100 kHz if they are accompanied by frequencies that are within the nominal 20 kHz range of our hearing). Consider especially the spectral components corresponding to the transition times between steps. Those are at radio frequencies!

Yet in referring to them as “distortion,” and citing that “distortion” as the basis for defining the threshold of sample rate acceptability, your analysis implicitly assigns audible significance to ALL of these ultrasonic and RF spectral components, little or no differently than if they were some low order distortion components lying well below 20 kHz. It also implicitly assigns audible significance to these ultrasonic and RF spectral components that is no different than if during the analog reconstruction process no filtering were applied to them at all.

Second, consider the hypothetical situation where an infinitely long sample record is available, with each sample having infinite resolution (i.e., zero quantization noise). The rationale behind your contention that 250 samples per cycle are necessary to achieve 5% distortion would seem to be no less applicable to that situation than it is to real world digital scenarios, despite the fact that (as I think you would agree) only a little more than 2 samples per cycle are necessary in that hypothetical situation.

The bottom line, IMO and with respect , is that I doubt your contention that a sample rate of more than 100x the Nyquist rate is necessary to achieve reasonable (although still high!) levels of distortion would be likely to receive widespread support even among the most ardent vinyl advocates, or at least those among them who have sufficient technical background to comprehend the issues.

Regards,
-- Al

Hello Al.

Thank you for your expert and thoughtful response. I find myself agreeing with your premises while disagreeing with your conclusions.

I agree with your aside concerning filtering, but, would you not agree that every capacitor introduces distortion? And that therefore we should be concerned with physical measurements rather than idealizations? I hope that this does not misrepresent your point.

I also agree that the spectral components are all above 20KHz. Would you not agree that this creates a very rich ultrasonic environment? And further, that this is mainly generated from harmonies in a fairly narrow 4 octave range, suggesting that the ultrasonics are also clustered? I note that different frequencies "beat" against each other; e.g. 33KHz and 34KHz signals beat to form their difference, or 1 KHz. Further, these beats will be related to the fundamentals in no simple respect, producing distortions which have not been characterized. If they were especially irritating, only a small audio component would be required to render digitally processed signals unpleasant. Which is what some of us observe.

Were it true that ultrasonic distortion was inaudible, SACD would be no improvement on CD, which is not observed. Therefore, I stand by the assertion that total distortion is what is important, until it is proved otherwise.

Having said that, I agree with your (implicit) point that another useful simulation would use linear interpolation between subsequent sample points. Then it would be an empirical question of which method better approximated the physical effects, and whether the ear responded as the approximation would lead us to expect. A Ph.D. dissertation there.

Your point about a periodic waveform of infinite duration is absolutely correct. I was restricting myself to waveforms which are physically possible. Since physical possibility precludes the use of the Shannon Sampling Theorem to justify reasoning, I stand by my assertion.

I also suspect that many will disagree with me, for whatever reason. I respect your reasons, but nevertheless must disagree.

Thank you for an enjoyable and enlightening discussion. Respectfully,

Terry

Hi Terry,

I agree with your aside concerning filtering, but, would you not agree that every capacitor introduces distortion? And that therefore we should be concerned with physical measurements rather than idealizations?

Absolutely. The various non-idealities of low pass filters, in both the recording and playback parts of the chain (anti-aliasing and reconstruction filters, respectively) are a major issue in digital audio.

I also agree that the spectral components are all above 20KHz. Would you not agree that this creates a very rich ultrasonic environment? And further, that this is mainly generated from harmonies in a fairly narrow 4 octave range, suggesting that the ultrasonics are also clustered? I note that different frequencies "beat" against each other; e.g. 33KHz and 34KHz signals beat to form their difference, or 1 KHz. Further, these beats will be related to the fundamentals in no simple respect, producing distortions which have not been characterized.

Agreed. In fact, arguably the most important reason for low pass filtering the d/a output is to eliminate (or at least greatly attenuate) beat frequencies that would otherwise arise as a result of non-linearities downstream in the system (and perhaps to some extent in our hearing mechanisms as well).

Were it true that ultrasonic distortion was inaudible, SACD would be no improvement on CD, which is not observed. Therefore, I stand by the assertion that total distortion is what is important, until it is proved otherwise.

As indicated in this Wikipedia writeup:

Because of the nature of sigma-delta converters, one cannot make a direct technical comparison between DSD and PCM. DSD's frequency response can be as high as 100 kHz, but frequencies that high compete with high levels of ultrasonic quantization noise.[36] With appropriate low-pass filtering, a frequency response of 50 kHz can be achieved along with a dynamic range of 120 dB.[2] This is about the same resolution as PCM audio with a bit depth of 20 bits and a sampling frequency of 96 kHz.

So although comparison between the parameters of the two formats is not straightforward or precise, it would seem clear that the performance of DSD is, at least potentially, superior to that of redbook cd in terms of dynamic range, and also in terms of providing greater margin relative to the Nyquist rate. That increased margin can be expected, at least potentially, to lessen the side-effects of anti-aliasing and reconstruction filters that may occur at audible frequencies, just as it can for hi rez PCM, relative to redbook PCM.

In summary, I think that our positions are similar in a lot of respects, but we agree to disagree on the need for a sample rate that approaches the one you have advocated. My thanks to you, also, for a stimulating and mutually respectful discussion.

Regards,
-- Al