[gregorio]

[1] Dude, sorry but I think It is just impossible for any kind of finite file will be carriying infinite information.

[2] I need proof ...

1. What you think is impossible is completely irrelevant! What's relevant is the established proven facts, which are completely unaffected by your personal suppositions and beliefs!

2. "Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rare) of the original samples. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are bandlimited to a given bandwidth, such that no actual information is lost in the sampling process. ... The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples:
If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.
A sufficient sample-rate is therefore 2B samples/second, or anything larger. Equivalently, for a given sample rate fs, perfect reconstruction is guaranteed possible for a bandlimit B < fs/2." - Taken from the Nyquist-Shannon Sampling Theorem page of Wikipedia.

The bold emphasis is mine, the use of "perfect fidelity", "no actual information lost", "completely determined" and "perfect reconstruction" equals infinite information. If you require the actual proof, here's a link to the original 1948 paper by Claude Shannon (http://math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf) where the mathematical proof was published. The practical invention of digital audio was based on this proven theorem (which again was mentioned in the OP)!

You're free to believe whatever you like of course but if you're going to dispute the Nyquist-Shannon Theorem you're going to need a whole lot more than your personal belief, A Fields Medal and a Nobel Prize or two would be a good starting point!!

G

 

[HAWX]

@gregorio Wow, OMG!. Is this message from a professional audio engineer who's job is to know and utilise the digital audio. You have quoted a theorem which can be found on Wikipedia again and said digital audio is based on this theorem again. Didn't know that, thanks.

Enough talking, Implement your claim as an expereinced professional audio engineer, or give way.

Please post here the proof of a sine wave which's amplitute is -50.5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 dBFS and frequency is 10,000.55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 Hz which has encoded by 44.1 kHz and 16 bit bit depth audio file.

Oh, and there's no "PERFECT" or "lossless" word, EVEN ONCE in the paper you sent me. You have made fun of me, I will make fun of you as well.

And I beleive you can encode a tone which has a frequency with a presicion to 100 million digits, take pi for instance, which plays for 1 second wav file and still takes 176.4 KB of storage. You will win a perfectly reconstructed Nobel Prize for that, with infinite fidelety.

 

[castleofargh]

Thanks @gregorio for answering no 5, what you say is logical. My English is not native thank you for clairty.

@castleofargh ..... I think I have already said "My further understandings are only about the digital audio format, not about If "High Res" is necessary for music reproduction. You can think about the signal is produced by a computer, and decoded by other computer, NO PLAYBACK IN REAL LIFE." regarding my understandings. You're still saying "spesify where you are looking at the playback chain" and limitations of dacs amps reconstruction and bla bla, I'm not even talking about the playback of the file.

I will repeat my understandings with asking questions and examples about 1 and 3.
1) For a given computer generated tone, (not captured with electronic devices) how presice the tone will be assigned to approximate dBFS without any dither comparing, 24 vs 16 bits? Let's say normally tone should be assigning about -50.5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 dBFS. (100 digits after the decimal)
I think the 24 bit assignation will have more digit after the decimal point thus leading to more precise assignation where the precision of a number is the total number of significant decimal (or other) digits.

3) For a given computer generated continuous tone which has the frequency of 10,000.55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 Hz (200 digits after the decimal), how presice this tone frequency value assignation will be, comparing sample rates of 44.1kHz and 192 kHz, without any dither. (Although I don't think the dithering will make a diffrence)
Again, I think the higher the sampling rate will be more presice to the original computer generated waveform.

I'm not asking this to get infromation about the playback fidelty of sampled file to human ears through dac/amp/headphones. My question was about math.

if we look only at digital values which is missing the point of recording and playing back a sound, then sure you can use more bits to record a bigger number without any limit as to the size of those numbers. let's record Pi ^_^.

Dude, sorry but I think It is just impossible for any kind of finite file will be carriying infinite information. I have already looked the original post and many of the external links given in this thread. I need proof, or It would be great to make test in some kind of program wich showes me what I'm talking about.

You are right about 44.1 kHz and 16 bits are more than enough for digital audio playback, which I also agree. But that's not I'm asking for. Thanks.

how about y=X² ? did we forget about math along the way?
digital reconstruction isn't based only on the precision and number of samples, we also use the fact that an audio signal is composed of sine waves. which have a behavior we know perfectly. with 2 samples we have crap as is, but if we add perfect band limiting where it counts, then Nyquist proved that we could reconstruct the signal.(we don't have perfect band limiting but as you want to stick to digital numbers without doing anything with them, I guess that's fine).
I imagine that's where you reply that with 64bit samples we could reconstruct an even more precise signal than with 24bit. did I guess right?^_^
and I'd say yes in theory but no in practice. because how do you record the value at such a resolution in the first place? from the microphone? the preamp? the ADC? no real life situation allows for 24bit resolution of a recorded band. we only make containers for that resolution.

 

[HAWX]

@castleofargh Yeah, that's why I said computer generated. Not real life sound captured by mic.. and now it's 3rd time saying it again btw..
Look I know math in basics. You migh contruct the fuction for the first 100 million digits of the pi number, though It won't be easy as y=X², and solving would not be instant by consumer computers today. We did not forget the math along the way. If Nyquist theorem applies fully as you say, read my post above, accomplishing what I have requested will solve my problem.

 

[VNandor]

@castleofargh Yeah, that's why I said computer generated. Not real life sound captured by mic.. and now it's 3rd time saying it again btw..
Look I know math in basics. You migh contruct the fuction for the first 100 million digits of the pi number, though It won't be easy as y=X², and solving would not be instant by consumer computers today. We did not forget the math along the way. If Nyquist theorem applies fully as you say, read my post above, accomplishing what I have requested will solve my problem.

Real life sound captured by a mic contains a sum of sine waves only, just as a computer generated tone. In fact, any kind of sound can be described as a sum of sine waves. If you think that is not true, I suggest you look up fourier-transform.
Once a signal is band limited, it can be captured and replayed perfectly with the correct sampling rate.

 

[HAWX]

Real life sound captured by a mic contains a sum of sine waves only, just as a computer generated tone. In fact, any kind of sound can be described as a sum of sine waves. If you think that is not true, I suggest you look up fourier-transform.
Once a signal is band limited, it can be captured and replayed perfectly with the correct sampling rate.

Thanks, I'm saying computer generated because I want to discard recording chain limitations. I know the mathematical theorem, I just want to see an real life example in digital audio format for the conditions I provided above..

 

[castleofargh]

ok I think I get it, you're not talking about sound at all. your point is that a discrete value is discrete. well congrats on that, you're right.

what Greg has been trying to say to you is that a 2khz sine wave with noise is a perfect 2khz sine wave, plus some noise. by increasing the bit depth what you effectively do is changing the noise content, not the 2khz sine. and when we use discrete values, the approximation of each sample creates extra noise.
as I've explained not long ago, you're free to look at the resulting amplitude and say the 2khz wave is wrong by the noise value. but it's also true that the 2khz is perfectly reconstructed and then we have some noise. the same way a piano and a singer are still a piano + a singer despite the signal being a single amplitude value.

 

[bigshot]

You won't get a "real life example" that discards recording chain limitations because in real life, recording chain limitations are always there.

But it doesn't mean that those limitations have an effect on human ears. Theory is always great in theory. But the ultimate goal is to make our sound systems sound good to us. That's the best thing to keep focused on.

 

[gregorio]

Oh, and there's no "PERFECT" or "lossless" word, EVEN ONCE in the paper you sent me. You have made fun of me, I will make fun of you as well.

There's no need for me to make fun of you, you're doing a perfectly good job of that all by yourself! ...
Section 19, Page 34 of the Shannon paper states: "If a function of time f(t) is limited to the band from 0 to W cycles per second it is completely determined by giving its ordinates at a series of discrete points spaced [1/2W] seconds apart in the manner indicated by the following result." I can't paste the actual equations but you can read them for yourself. Hopefully you understand that "completely determined" effectively means infinite precision (unlimited decimal places)? This is the undisputed proof YOU requested!

Secondly, as a professional audio engineer I own and use industry standard professional audio tools. The signal generator I own is capable of setting a sine wave frequency in 0.1Hz increments and my DAW is capable of amplitude output adjustments in increments of 0.1dB. Scientific laboratory tools are needed for higher precision signal generation, not audio industry tools. Therefore, I cannot even generate the signal you are suggesting in the first place!

Lastly, you really should read what you're replying to and understand what is being said before jumping to the conclusion that you're being made fun of. If you don't understand then ask but threatening personal attacks is a childish response!

G

 

[HAWX]

I have asked you several times already. IF you don't know the value you can always say I DON'T KNOW. Really. You are talking too much but the work done is 0. You cost me too much time.

For the dBFS part, I have found it myself.
"For example, if the amplitude of the electrical signal being sampled ranges from -10 volts to +10 volts and we use one byte for each sample, each number does not represent a precise voltage but rather a 0.078125 V portion of the total range. Any sample that falls within that portion will be ascribed the same number. This means each numerical description of a sample's value could be off from its actual value by as much as 0.078125V -- 1/256 of the total amplitude range. In practice each sample will be off by some random amount from 0 to 1/256 of the total amplitude range. The mean error will be 1/512 of the total range.
This is called quantization error. It is unavoidable, but it can be reduced to an acceptable level by using more bits to represent each number. If we use two bytes per sample, the quantization error will never be greater than 1/65,536 of the total amplitude range, and the mean error will be 1/131,072." https://docs.cycling74.com/max5/tutorials/msp-tut/mspdigitalaudio.html

IF you give the " discrete points " perfectly (which I wonder),you will obtain the waveform perfectly. But I want to see that wheather the exact same case in digital audio from first hand. That's why I'm here.

@gregorio "Scientific laboratory tools are needed for higher precision signal generation, not audio industry tools. Therefore, I cannot even generate the signal you are suggesting in the first place! " Even I, far from audio engineering, can generate more presice tone than that! Here you go: http://onlinetonegenerator.com/

I am quite sure that just as that site does, a computer program can produce artificial waves for given loudness and frequency, and measure/inspect the samples for their Hz and dBFS. Does anybody know a program can do that?

 

[gregorio]

[1] For the dBFS part, I have found it myself....
This is called quantization error. It is unavoidable, but it can be reduced to an acceptable level by using more bits to represent each number.
[2] Even I, far from audio engineering,can generate more presice tone than that!

1. What do you mean you found it for yourself, that's all in the ORIGINAL POST!!!! How does quantization error stop you from getting infinite resolution and how does it disprove the Nyquist-Shannon Theorem? For the last time, read the damn original post and ask questions if there's something you don't understand rather than trying to use your ignorance as a battering ram!

2. Please, tell me why I, as a professional audio engineer, ever need a signal generator with more precision than a 1/10th of a Hz and more precision than a 1/10th of a dB!

G

 

[HAWX]

1. What do you mean you found it for yourself, that's all in the ORIGINAL POST!!!! How does quantization error stop you from getting infinite resolution and how does it disprove the Nyquist-Shannon Theorem? For the last time, read the damn original post and ask questions if there's something you don't understand rather than trying to use your ignorance as a battering ram!

2. Please, tell me why I, as a professional audio engineer, ever need a signal generator with more precision than a 1/10th of a Hz and more precision than a 1/10th of a dB!

G

1st)I don't need to discuss it anymore. And I didn't said my findings compromises the Nyquist-Shannon Theorem.

2nd) It is quite sad that YOU DON'T KNOW OR YOU CAN'T EVEN THINK that is possible to create an audio file with more presicion than a tone generator, by using just a plain computer. I have never said you need to have. I expected you already knew or you could understood it by JUST THINKING.. You were the engineer who has extensive amount of experience and information about digital audio, remember?

You have made your point and said what you think, which is infinite presicion. OK. No need to repeat or argue it again and again. Cool?


Now for the other people, I need to see a sample in an audio file which can be assigned by infinite presicion, from first hand, or try it myself. For loudness part, a sample can be assigned to 2^16 value (16 bits), or 2^24 (24 bits) That's it. I am NOT talking about playback capability or noise floor or dither and all that.

Now for demonstration I need to know "For a given computer generated continuous tone let's say a tone that has to be encoded with the frequency of 10,000.55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 Hz (200 digits after the decimal), how presice this tone frequency value assignation will be, comparing sample rates of 44.1kHz and 192 kHz, without any dither. " Please provide me the results of tone which has assigned to this value in an audio file, in real life. Or recommend be some program where I can create this tone in various sample rates.
I know this amount of presicion is completely unnecessary in digital audio world. Thanks.

 

[Squeaky Duck]

C'mon guys, this is a FRIENDLY forum to share ideas and knowledge, not bash on each other.

 

[old tech]

1.

1. Science knows! Sure, some audiophiles (or more typically, those who sell equipment to audiophiles), come up with ridiculous ideas all the time and even sometimes present those ridiculous ideas as "different facts" but they're not really facts, they're just marketing bulls***. The basic facts of digital audio were invented 90 years ago, proven mathematically 70 years ago and no-one since has dis-proven them. In fact, doing so would invalidate the basis of all digital information theory and therefore demonstrate that no computer based technology works. This, along with most of the other facts in this post, were discussed in the OP, are you sure you've read it?
G

The fundamentals were mathematically proved long before that. The discrete properties of sound were shown to be equivalent to a continuous wave form over a hundred years before Nyquist et al. Indeed, the mathematical properties of sound were sort of understood back in the days of Euclid.

 

[old tech]

@gregorio
Oh, and there's no "PERFECT" or "lossless" word, EVEN ONCE in the paper you sent me. You have made fun of me, I will make fun of you as well.

On that part you are correct, in a strict sense. There is no perfect or lossless audio even in the natural world of acoustic physics. Acoustic sound has quantization properties if you drill down far enough into air molecules. Whether that is relevant to human hearing is another matter. One thing is certain, sound can never be lossless once it passes through a transducer (ie a change of energy state) converting air pressure to an analog electrical signal and then again, converting the analog signal to acoustic energy. However, digital processing of the signal between these two necessary transducers is going to be more lossless than any analog processing, particularly processing that involves transmitting that signal over a distance or passing it through more transducers, eg tape heads, T/T cartridges etc.

https://www.st-andrews.ac.uk/~jcgl/Scots_Guide/iandm/part12/page2.html