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Re: Noise is Noise Or is It?

Posted by Peter Qvortrup (M) on February 27, 2007 at 09:18:13

In Reply to: Re: Noise is Noise Or is It? posted by Dave Kingsland on February 23, 2007 at 15:06:52:

Dear Dave,

I did think that someone might come back with an answer like that, even though I hoped we had built in sufficient information to illustrate that there is a distinction between the digital and analogue cases when MUSIC or another “random” pattern is the signal, because signal processing theories are generally blind to the application, as they were most likely derived by university lecturers and telephone engineers, without music as a specific application.

Let me quote from Terry Pratchett's Discworld, Moving Pictures,

"Reality is not digital, an on and off state, but analogue, something gradual. In other words, reality is a quality that things possess in the same way that they possess say, weight."

Ignoring the amplitude domain/dimension, because for this argument, time and frequency are sufficient.

Digital: The time domain is quantised, and therefore the set of samples, from time zero to time infinity can be counted according to integers, 1,2,3, no matter what the sampling rate. The set of samples has a one to one correspondence to Aleph 0, the set of countable numbers.

Analogue: Analogue is not a discrete process, therefore cannot be counted by integers, and would correspond to the set of real numbers Aleph 1 (or c) depending on the acceptance of the continuum hypothesis, but that makes no difference to this argument.

Cantor proved that Aleph 1 (or c) is greater than Aleph 0.

Therefore digital can never be analogue, no matter what the sampling rate, EVEN IF IT WERE INFINITE.

Cantor also showed that any subdivision of the real number line also has a correspondence to the entire line. That may have some interesting implications, and might conflict with certain sampling theorems.

Maybe one could enter an argument that at some level the analogue time domain would be subdivided by Planck time, but one would have to examine the derivation of Planck time, and how it relates to this set theory argument.

Looking at it this way points to the analogue noise floor having “no grain”, it is continuous, the digital noise floor would be composed of discrete frequency bands. Dither may/would modulate noise out of bands, but again I’m not sure whether it’s possible to translate from a correspondence with Aleph 0 to c, I don’t think it is at first thought.

This is essentially the issue, so whilst much of what you say is correct according to conventional theory it does not actually cover the full subject, but to comprehensive prove this one would have to spend a lifetime in research and there is unfortunately no money in that and commercial pressures against it are comparable to what was wielded against hemp in the 1930s, so even if one came up with an answer it is likely to be suppressed by powerful commercial interests.

My argument is essentially based on: “How do you average music to detect it?” Maybe there are periodic elements in there which can be averaged. With digital you get one sample, every 22 odd microseconds, with analogue the time domain is continuous, so the question is, what period do you have to average over for it to make sense?

There is much further theory that can be used, in number theory for example, but we would have to study it further, and to transfer, for example, transfinite numbers into information theory or signal theory or whatever is the kind of thing which takes 20 years and you win a Nobel prize for, provided you live long enough!

Going back to the textbooks and digging out the equations and then quoting them assumes that whoever did the original assumptions and subsequent calculations actually did them both including the maths with a view to achieve the best possible end result i.e. sound, and even if they did, then by the time products were designed using these theories and their calculations, I think that commercial expedience (meaning achieving low cost primarily) would have been higher on the agenda, as is mostly the case with this type of technology (question to solve is/was, "How do we get round these problems quickly and cheaply?" not, "How do we do this properly?"), so whilst the text books provide a "solution" and arguments for why, they do not necessarily hold anything close to the final answer to the real requirements of highly variable and complex waveforms contained in music, because if they did then what we have from digital reproduction media would be closer to or even better than the best analogue reproduction solutions they were intended to replace and live up to the original slogan "Perfect Sound Forever".

So take you pick, either the theory with its associated assumptions and calculations are at worst wrong or at best incomplete, or the products that are designed based around them have be designed cutting corners.

Just in passing allow me to give you a simple question that current theory has problems answering,

"Did you ever try a silver cable?”

Standard answer to that is,

“I don’t need to it will make no difference.”

However when you try it between transport and DAC it makes a significant difference, in ways which cannot be explained with any current available "tools"

This discussion has got us thinking though, about number/group/set theory and Georg Cantor’s work, so we shall dig deeper when time allows.

Sincerely,
Peter Qvortrup
Andy Grove

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