Neil Young Announcement - Blu-Ray is the way

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D. Lundberg

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We begin studying a sine wave that for convenience 1) has a frequency equal to half the sampling frequency;
You need to be aware that reconstructing a waveform at exactly half the sample rate is a special case that only works in theory.

ArnoldLayne said:
Then we move on to another case using the same wave as before but altering point 3) so that the peaks don't coincide with the exact moments of sampling. Let's say that the samples measure the wave when it reaches 80% of its amplitude. At t1, t2, t3 etc we obtain samples 0.8, -0.8, 0.8 etc. As far as I can see it is no longer possible to reconstruct the wave. Frequency is still correct but amplitude will be underestimated.

I guess (and hope) this is somehow taken into account in the signal theory used when developing our consumer electronics. For example, would 44.1, 48 or 96KHz PCM gives us sufficient sample points to estimate the true amplitude of a 20KHz wave even if its maximum value is not present among the samples? If so, what are the maths behind?
The above is based on a common misunderstanding of how waveforms are reconstructed.

It doesn't matter if there are any points at the peaks of the waveform or not, because it's not reconstructed in a "connect-the-dots" fashion.
The parts of the waveform that are between the sample points do not have to be guessed or estimated. They are completely known from the points you do have.

The only things that matter is that you have sufficient sample points and that the signal is band limited, because then there is only one "legal" way of reconstructing the waveform!
Any other way you can think of to reconstruct it will have frequency content above the "legal" range, and thus not conform to the Nyquist theorem.

So the sampling theorem basically says that there is only one possible way to draw a waveform through points along an axis if it's band limited to half the sample rate.

If you're interested in the math behind the theorem you can read Claude E. Shannon's proof here: http://www.stanford.edu/class/ee104/shannonpaper.pdf
 

ArnoldLayne

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Hmm, I think I have to explain myself further still. Please bear with my "connect-the-dots" explanations. It's been decades since I studied maths and I'm too rusty to get into formula dribbling. The overall purpose is to study which information can be stored in the samples.

I would like to qoute the following from the Shannon paper:

Theorem 1: If a function f(t) contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1/2 W seconds apart.

In the special case I mentioned above (frequency W, peaks not coinciding with sample points) I think it's clear that theorem 1 holds if and only if we are dealing with normalized waves. In other words, if we are only interested in identifying the frequency and do not care about reconstructing phase nor amplitude. Being a special case at the border of the theorem's scope, we should of course not haste into conclusions. But it would be interesting to know if theorem 1 assumes normalized waves in general, e.g. for all frequencies below W too. I am still digesting the referenced documents but I think I can see that:

- The Shannon paper is dealing with phase as something to be filtered down to an acceptable level, not something being identified in samples and reproduced.

- The Lavry "paper" does not state if it is dealing with normalized waves or not. In the latter part the plotted waves definitely seem normalized.

So I'm afraid that despite your most appreciated efforts I am still not convinced. Using the Nyqvist theorem to prove that "all information can be sampled" or that "all information about frequency can be sampled" are two very different things in the context of audio. So I think it would be very interesting to know if the theorem is useful beyond normalized waves. It should have at least some importance for the discussion of digital formats.

If my layman hunches happened to lead in the right direction, how come that a CD still sounds fairly good to most people? How can it represent dynamics and timbres in music if phase and amplitude information gets lost? One possible explanation would be that most of the frequencies are much lower than the sampling rate. For all those waves, even a worst case phase shift equivalent to half the sampling frequency would result in a relatively small error. But I should leave that for later, until I've figured out if signal theory based on the Nyqvist theorem actually discards phase information or not. Or if there is a "second part" that deal with phase issues. Any help with that would be most welcome, of course.

Cheers
AL
 
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D. Lundberg

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In the special case I mentioned above (frequency W, peaks not coinciding with sample points) I think it's clear that theorem 1 holds if and only if we are dealing with normalized waves.
Peaks not coinciding with sample points is not a special case, in fact it is very common as you approach the Nyquist limit. And it's not a problem because it doesn't matter where the sample points are, just that they are evenly spaced and that you have sufficient points to represent the original waveform.

But there is actually a real practical problem caused by this. There is plenty of software and hardware used in recording and processing digital audio where the visual representation just shows the sample points and not the actual waveform. And if you have sample points close to the maximum level (0dBFS), then the actual reconstructed waveform may have peaks exceeding it (and thus cut off).

So when working with digital audio you should not consider the highest sample point the peak of the waveform.

So I'm afraid that despite your most appreciated efforts I am still not convinced. Using the Nyqvist theorem to prove that "all information can be sampled" or that "all information about frequency can be sampled" are two very different things in the context of audio. So I think it would be very interesting to know if the theorem is useful beyond normalized waves. It should have at least some importance for the discussion of digital formats.
I'm not quite sure what you mean by "normalized waves". I know what normalization is (changing the amplitude of a signal), but I'm not sure how it fits in this context.
Do you mean sine waves? Because it's important to know that all waveforms (no matter how complex) are basically sine waves, or sums of sine waves.

But let's go to the basics.

Any waveform can be completely described by four characteristics: frequency, amplitude, phase (relative to other waveforms) and dynamic range.

And by recording only the amplitude (at sufficient intervals) the entire waveform can be accurately reconstructed, including its amplitude, frequency, phase and dynamic range!
That is because there is only one waveform that can be reconstructed from those specific amplitude values (as long as the waveform is band limited).

Did that clear things up?

Otherwise I recommend checking out "Digital Audio Explained" by Nika Alrdich:
http://www.amazon.com/Digital-Audio...=sr_1_1?ie=UTF8&s=books&qid=1250348132&sr=8-1
 
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Almen

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Maybe what is refered to as "normalized" is the common practice to filter the signal through a "sinc function" before sampling, when dealing with digitalization in practice? I am sorry if this has been stated and I am misunderstanding something.

The sinc filtering means that all information in the original (bandwidth limited) analog signal indeed is kept when sampling has been performed. So it works both in theory and practice! :)
 

ssully

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Maybe what is refered to as "normalized" is the common practice to filter the signal through a "sinc function" before sampling, when dealing with digitalization in practice? I am sorry if this has been stated and I am misunderstanding something.

'Normalizing' typically just refers to bringing the file peak amplitude value to some target value, typically 0dBFs. I think that's what ArnoldLayne is referring to.

Btw, there is a 'danger' in normalizing to values near 0dBFS, but it's not a flaw in Nyquist, it's a flaw in monitoring and modern mastering and CDPs. If your track is compressed/limited in the modern fashion such that many peaks lie near 0dBFS, and you normalize the peak *sample* to 0dB, there will very likely be some waveform peaks that lie *between* two sample..they shoot 'over' the 0dB limit. These are called intersample peaks or intersample overs. Not a big issue with 'old school' mastering, where only one or a few peaks ever approached 0dB during the course of a track. But potentially more of a problem now.

Proper digital level monitoring during recording and production avoids them. Staying away from heavy compression and limiting avoids them. Good CDP design avoids introducing distortion due to them.


The sinc filtering means that all information in the original (bandwidth limited) analog signal indeed is kept when sampling has been performed. So it works both in theory and practice! :)
Indeed, Nyquist works, and moreover it and digital don't have to work 'perfectly' to still be vastly more measurably accurate than *any* analog recording process, and I hope ArnoldLayne has gotten over it.
 

Almen

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'Normalizing' typically just refers to bringing the file peak amplitude value to some target value, typically 0dBFs. I think that's what ArnoldLayne is referring to.

Yeah, I just remembered the term "normalized sinc function", so I thought there might be a mixup, there.

Here is some more info about using the sinc in A/D conversion: Sinc Filter
 

Almen

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Theorem 1: If a function f(t) contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1/2 W seconds apart.
In the special case I mentioned above (frequency W, peaks not coinciding with sample points) I think it's clear that theorem 1 holds if and only if we are dealing with normalized waves. In other words, if we are only interested in identifying the frequency and do not care about reconstructing phase nor amplitude.

Methinks here lies the rub. f(t) defines a function in the time plane, and thus phase and amplitude is fully determined.

One possible explanation would be that most of the frequencies are much lower than the sampling rate. For all those waves, even a worst case phase shift equivalent to half the sampling frequency would result in a relatively small error.
I think you are a little too much preoccupied with the frequency. The signal is sampled in the time plane, after the sinc filtering, and thus all information is stored.
 

ArnoldLayne

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With "normalized" I meant unitary amplitude and/or neutral phase. I saw this menioned in webpage related to Fourier transforms but later on I could not recall the link. It was actually this that led me to try to understand how (if) digital sampling allows for perfectly reconstructing waves, phase information included. I have skimmed through the recommended litterature and a low of Wiki articles. But I'm afraid I'm still not convinced that the Nyquist theorem is fully applicable to music. Firstly, I find it difficult to grasp if authors are making appropriate distiction between continuos and discrete Fourier transforms before drawing their conclusions. Secondly, I have understood that standard signal theory runs into problems when in comes to transients but not found any study on to what extent it also constrains the representation of lesser randomness. Just to prove that it's not only me that seeks "life beyond Nyquist", I'll put this link: http://en.wikipedia.org/wiki/Time-frequency_representation .
 

Almen

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ArnoldLayne said:
With "normalized" I meant unitary amplitude and/or neutral phase.

What do you mean by "neutral" phase? I would imagine it difficult to establish a phase of a music signal in that sense. You are of course right that the amplitude must be normalized (or matched) to the A/D-converter, so that dynamics are used fully without limiting the signal.

But I'm afraid I'm still not convinced that the Nyquist theorem is fully applicable to music.

There are some (very ambitious, double-blind) listening tests performed where the listeners could not differ between a signal going through an A/D-D/A conversion and the original signal. This would imply that it at least is possible to apply Nyquist to music, would it not? Why would music be a special case?

An eye opener for me was an article about this, unfortunately so far only in Swedish, published on the site for the Swedish Audio Engineering Society: The Sinc. If you don't know Swedish, maybe you can look at the pictures? :)
 

Almen

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Yeah, but since there is inherently a lot of talk about specs and performance, there will be a lot of talk about technicalities like this. The discussion was prompted by the rather innocent question "The higher the sampling rate and bit rate used for a transfer, the closer one comes to the original wave form, correct?", which was explained by ssully to be not quite correct, and then... well.

I do think that technical discussions have a place here, but I can understand the frustration when it seems to take over a thread. What is the general opinion? On the other hand, this thread would be quite dead sans Nyqvist, since there seems to be nothing else to discuss right now...
 

Odys

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I just noticed that Amazon is selling the blu-ray for $209.99, which is the cheapest I remember seeing it and is currently selling for even less than the dvd set.
 
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