Which is more accurate: digital or vinyl?

Analogue. Do the mathematics:

By the Fourier Theorem, we must only consider sine waves. A sine wave must be sampled 250 times to achieve 5% RMS distortion or less (the bear is when they cross zero, if I remember my simulations correctly). Undamaged adults can hear to 20KHz. Therefore a signal must be sampled at 250 x 20,000 = 5MHz to achieve less than 5% distortion throughout the accepted bandwidth.

And I will shriek if I hear Shannon's Information Theorem (mis) quoted again. That theorem requires a continuous Fourier Transform - i.e. has been infinitely repeating since minus infinity, through the present, and on into plus infinity, whereupon the samples may be reassembled to give good results. But the universe is only 13,000,000,000 years old - a long way from infinity (infinitely long way, actually).

So digital will rival a Revox A77 when sampling frequency exceeds 5MHz. As for rivalling a Studer ... no way.

01-24-12: Terry9
A sine wave must be sampled 250 times to achieve 5% RMS distortion or less ...

With all due respect, as someone who has taken several advanced courses dealing with digital sampling theory, and has designed digital circuits implementing FFT's and other digital signal processing functions, I have never before encountered such a statement.

Are you sure you are not confusing sampling with quantization? Are you sure you are taking into account the low pass filtering or other techniques that are used to reconstruct the analog waveform during the d/a conversion process?

In any event, can you provide some supporting documentation or rationale for that claim?

Regards,
-- Al

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 In any case, it's unpublished, so you'll have to do it yourself.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