Which is more accurate: digital or vinyl?

Hello Almarg.

Reciprocating your respect, truly, I wouldn't expect you to have encountered it before.

I bought a UNIX box in 1999 to do simulation research with Maple (the pro math package). Then I found that, sadly, the journals don't like results which don't parrot the mainstream "wisdom". So I did recreational things like investigating this. In any case, it's unpublished, so you'll have to do it yourself.

The algorithm is quite simple: set the number N of samples per waveform, calculate the step functions appropriately, and calculate the difference squared between that and a sine wave. Divide by the area under the sine. That gives you RMS distortion.

Let N increase. At about N=250 you will see the distortion falling towards 5%.

Oversampling does not help much. Unless the original signal is also processed this way, you merely end up with a curve that more closely approximates a distorted sine.

Regards,
Terry

Hi Terry,

It seems to me that the flaw in that analysis, as my previous post intimated might be the case, is that it does not take into account low pass filtering that is applied in the d/a process to smooth out the stepped character of the sampled waveform.

Essentially, your distortion percentage is incorporating ultrasonic spectral components that represent sampling artifacts (as opposed to distorted musical information), which ultimately get filtered out.

Another way to look at it is that were your claim true, then for redbook cd an audio frequency of 44100/250 = 176 Hz would be distorted by 5% when it is played back, and higher frequencies would be distorted by a far greater percentage than that. Clearly the cd medium, while far from perfect, does better than that!

Regards,
-- Al

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