Time resolution of digital sampling

[pkane2001]

Can someone please provide a clear mathematical definition of "static" vs "dynamic" signals and how this invalidates the sampling theorem or the ability of the Fourier transform to convert data between time and frequency domains? 

 

[pkane2001]

1 hour ago, manueljenkin said:

 

@pkane2001heres your answer.

 

Tl;dr,  Multiply the fft of heaviside function with fft of a sinc filter and see how it behaves in the transition point of the heaviside function. You can also do it by convolving them in time domain. See if it is really "Bandlimited".

 

Also another correction. I never said "static" and "dynamic". I was only talking about "transients", vs "steady state".

 

You did in our earlier conversation, so I assumed you are still talking about the same thing. Forgive me, I didn't understand it then, and still don't understand it now.

 

 

I'm still not sure what a transient vs. steady state means. Is transient a Heaviside function or a Dirac pulse? A mathematical function doesn't need to be bandlimited. But how is this related to audio, where signal is always bandlimited, where infinite slope functions do not exist?

[pkane2001]

2 hours ago, manueljenkin said:

Well I'm saying the very act of bandlimiting the signal to 22khz is cutting off transients. The transients may not be as abrupt as heaviside function, but there is no guarantee it is naturally Bandlimited to 22khz, I certainly haven't seen any studies that prove it. 

 

That's correct, and I wouldn't think you need a study to show that. There are plenty of natural and artificial sounds that have frequencies well above 22kHz. But there are also plenty of studies that show humans can't hear much above 20kHz, and this also gets significantly worse with age. So, even if we are missing some components of the Heaviside function in the recording above 22kHz, is this really significant for audio reproduction (for humans)?

 

2 hours ago, manueljenkin said:

Transient response is just the property exhibited by the system during the time from inertia to steady state. If you have worked with Laplace transforms, steady state is when the e^(-st)'ish terms decay to zero, or in practical situations, we take it when it could be approximated to zero. It generally is taken at 5 times the time constant when the transient components are less than 1% as compared to initial stage (e^-5 nearly equals 0.0673).

 

OK, but what is the significance of this? Nothing in the real world has a 0 time transition from one state to another, except maybe at the quantum entanglement level :) A transient response, a transition from state A to state B over time t is just part of a naturally band-limited signal that is just as much subject to the sampling theorem and Fourier analysis as any other. 

[pkane2001]

28 minutes ago, PeterSt said:

Hi Paul, it looks like you just don't want to read it. And you don't want to see (through) the real-life plots either ?

 

Peter

 

Hi Peter,

 

Quote

Not correct. Synthesizers can and they do (and don't let an ADC loose on them or else you are correct 🤪).

Hmm? Digital isn't real, Peter. Digital isn't a signal. It's a mathematical representation of a signal. So yes, inside the computer, one can encode a 0-time transition. Show me any real signal that transitions between two states in 0 time.

 

Quote

 Why ? just because you think so ? Can you prove it ? I won't repeat all the posts and plots again about the opposite (I already did it twice). Btw, I don't think you did not read them (you were here all the time). You just seem to ignore what I show.

(I know, my English 🤫)

No, i didn't ignore you. I've been extremely busy, and have not kept up with this thread or what's been posted (and there's been a lot!) Just surfacing for air for a few hours. I'll go look for your examples of 0 time transition, but they better be good, or else you're wasting my time! ;) See below as to why I don't need to prove it.

 

Quote

Why do you call that a state ? There is no such thing as a state or changing state etc. There's a steep rise (or drop) of level. I am afraid you miss the point somewhere. Maybe investigate "Steady State" better ? I am not sure. But it is the opposite of a state of a signal as what you seem to refer to (again, I am guessing).

 

All physical phenomena have a state. A state is just a point in multiple dimensions, including time. A transition between two states is a vector.  A Heaviside function is a step transition between two points with infinite slope. There's no physical (not mathematical!) signal that transitions between two states in zero time, that would violate known laws of Physics. Infinities are not easy to find in the natural world, so if you've found one, publish a paper and quickly!

 

Quote

The only thing which definitely would be true is that after we applied the reconstruction in the (sinc) fashion we know it, Shannon / Nyquist has been applied. And then it is too late ...

So you must think prior to that situation.

 

It's too late for what? Why do I care about discarded frequencies that are far out of my ability to hear them? Regardless of how hard you try to reproduce an infinite slope of a signal, you'll never succeed, so there is always something missing compared to the idealized, digital or mathematical encoding of a pulse or a step.

 

 

[pkane2001]

7 minutes ago, PeterSt said:

 

I'd really like to see your explanation of why you have such a response. To me it does not make a but of sense. Please read carefully what Manuel says, and please know (or acknowledge) that a transient is not a frequency. It just is not.

 

Peter, you really should trust mathematical theorems a little more. After all, they are proven beyond any doubt, unlike audiophile claims. All signals can be expressed in the frequency or time domains, interchangeably, without any loss of information.  That's the Fourier theorem. A transient is just a signal, and therefore can be analyzed as a set of frequencies. An idealized transient (0 time) will contain an infinite number of frequencies. A real-world transient will always be band-limited. It just is ;)

 

 

[pkane2001]

8 minutes ago, PeterSt said:

So if I set that (as the artist), would I like that to be a sinus at playback ? Or do we think it remains to have that shape after reconstruction ?

 

Sorry, but as an artist you don't get a choice, Peter. Fourier decided this for you over 200 years ago: any signal can be decomposed into a sum of sines. Even the sawtooth one that you like so much ;)

 

[pkane2001]

Just now, manueljenkin said:

Only for any "truly periodic" signal. If he has a sawtooth running from -infinity to +infinity, yeah. Otherwise, no!

 

A finite-length signal is easy to handle as if it's infinite by windowing in the time domain. And before you bring up tradeoffs of window functions as an issue, there are some very good windows available that have tiny effects on time domain at the limits of floating point calculations and well below any audibility levels (-200dB or less). So yes, nothing is perfect in the real world, but it can be as perfect as you'd like it to be, as long as you still keep it finite.

[pkane2001]

5 minutes ago, manueljenkin said:

Not if I have transients existing in my signal. It can modulate anything to any large extent! This 200db dip may exist that way for a normal infinite period sinusoid whose components won't leak out much, but not so for transients. The transients could modulate the sampled signals heavily even on a perfect low pass, let alone an imperfect one.

Are you still talking about infinite slope transients?

[pkane2001]

4 hours ago, PeterSt said:

 

Please notice that we should not make the mistake in thinking that this will be put through to the output. So even if any stupid proposes a 1 sample Dirac pulse to the input, *any* filtering means already diminishes that to a more gradually peak. Just like I showed it.

So it is not necessary to be funny. Haha. Oops.

 

One could also attest that any Dirac (made by stupids) can only exist in digital and in the bandlimited system (domain); would the bandwidth be infinite, then the pulse very theoretically would be allowed to be infinite, but it would be useless because no hardware would be able to deal with it (just because in physics infinity does not exist).

 

In a next post I will have an attempt to make this clear better. Or to let you (guys) possibly think differently. Throw out your textbook math and such. Create new math if it is deemed necessary (that would be comfortable !).

 

New math is not necessary, since the old math is perfectly adequate. Anything new you come up with must not contradict a proven mathematical theorem. It doesn't matter how many other ways you try to approach it, if your result goes against a proven theorem, your way is wrong and not the other way around.

 

[pkane2001]

1 minute ago, manueljenkin said:

Please "prove" Nyquist Shannon's sampling to work without artefacts for the transients I mentioned. I give you option for two scenarios - 5 and 6.

 

I've shown you the proof it won't work. Look at the ft of the functions. If you actually manage to prove it, you would have invented "new" math proof.

 

YOU WILL NEVER REPRODUCE A PERFECT HEAVISIDE FUNCTION OR A DIRAC PULSE. PERIOD.

 

No matter how much new math you invent, this is not possible. The best thing you can do is approximate it within some bandlimit. That creates "artifacts", such as Gibbs, that are much, much larger than anything a proper windowed FFT would cause. Accept that real world cannot be perfect, and an ideal pulse is not possible. 

[pkane2001]

Just now, manueljenkin said:

You have another option. The 6th one. In time domain it's the piecewise multiplication of time delayed heaviside with same time delayed sine. It is no longer a heaviside. This is not infinite slew. But this is also transients. You can certainly make this happen in real life. Show me how you'll go about perfectly bandlimiting, sampling and reproducing this without artefacts using sinc filters.

 

No such option. Time domain and frequency domains are equivalent and interchangeable. One cannot exist without the other. That's the Fourier theorem. If you manipulate time domain, you are affecting the frequency domain and vice versa. You don't get a choice to change just one.

[pkane2001]

Just now, manueljenkin said:

You're distorting math to your own opinions. This scenario is perfectly valid.

 

Nope. I'm stating exactly what the Fourier theorem states. I'm not re-interpreting it or coming up with new math - you are.

 

[pkane2001]

25 minutes ago, manueljenkin said:

But the signal is perfectly valid, just that it cannot be handled properly using Fourier transform. And can't be handled by Nyquist Shannon sampling. Before you twist my words again, this is VALID REAL WORLD SCENARIO, but cannot be handled reliably by Nyquist shannon sampling because a sinc filter cannot perfectly bandlimit this signal.

 

So, you are concerned that Shannon-Nyquist cannot be used to perfectly reconstruct an idealized, infinite-slope pulse? Are you just ignoring all my posts that infinities do not exist in the natural world, or do you disagree with this statement? 

 

[pkane2001]

5 minutes ago, manueljenkin said:

Here is the signal. I've marked it in yellow.

 

Sorry, but you'll have to explain this to me in more detail. A delayed sine function cannot be represented by Nyquist-Shannon? Are you serious? That will be big news to every single CD ever recorded that starts with zero level signal and then plays music (sine waves).

[pkane2001]

repeat

[pkane2001]

1 hour ago, manueljenkin said:

Do the Fourier transform of this signal (in time domain you represent it as Piecewise multiplication of a time delayed sine and a time delayed heaviside) then multiply it with Fourier transform of sinc. Now tell me what you get, whether it is band limited or not. (This sine frequency is less than fc). Passing this function through a sinc will for sure create out of band components. You cannot bandlimit this using sinc.

 

Only a delay in steady state sine (if it began from time =- infinity and runs through to time +infinity) can be bandlimited properly without having any out of band components during low passing. The above is a transient scenario and it will have out of band components. Especially high right at the point where it stops being dc 0 and starts becoming a sinusoid.

 

A DC transitioning to a sine wave and back is just another idealized signal. It's a multiplication of a square window with a sine wave. As we discussed, a perfect square edge requires infinite bandwidth to reproduce, and therefore is not possible in the real world. Therefore a transition from DC to sine cannot be reproduced perfectly. Where's the problem?

[pkane2001]

2 hours ago, manueljenkin said:

You keep shifting goalposts. Next thing I'll hear from you is that signals above 20khz don't exist in real world, or that real world is always steady state. This is a valid signal, take it or leave! I'm out of this discussion, not wasting my time here anymore. I got what I wanted to proceed further in my journey. Have your trophy.

 

I've been fairly consistent in what frequencies humans can and cannot hear:

So, yes, if you are designing equipment to reproduce music for dolphins, then you may want to worry about ultrasonic frequencies. For humans, not so much.

 

[pkane2001]

57 minutes ago, manueljenkin said:

Hang on.. where did I mention I was multiplying it with a "square window". When on universe did a Heaviside function become a square window. It doesn't transistion back to dc after that, it stays a sine wave. Another classic case of you skewing the information that was conveyed to you, to suit your agenda.

 

 

I suspect you know that most finite-length recordings (ie, all of them) have a beginning and an end. So yes, a square window is a step function at the beginning, going from 0 to 1, and then another one, at the end, going from 1 to zero. Sorry, should've said "rectangular", aka, Dirichlet window.

[pkane2001]

1 hour ago, jabbr said:

 

Your assumption is that humans cannot respond to frequencies beyond what sine waves humans can perceive. Stated another way: you are assuming the human auditory system is linear.

 

Not much about the human brain nor its sensory organs is linear.

 

This discussion has been about applicability of Fourier and Nyquist to artificially constructed signals, and not about audibility. Go ahead and pick a higher sampling rate if you're worried about missing ultrasonics. And if you want a perfectly reproduced Heaviside function, then pick infinite sampling. But I warn you, you may have to pay a lot for storage for an infinite length recording.

 

I've tested myself and found no perception or detection of any ultrasonics in test signals or musical content. So, yes, I assume that I don't respond. Maybe my brain is just too linear ;)

 

[pkane2001]

2 hours ago, jabbr said:

This is where the test that you select is relevant. Have you tried to reproduce Kunchur's methodology?

 

If we are going to be objective, let's do it correctly.

 

If I were to try Kunchur's methodology, I'd change at least one thing that he got wrong. He didn't level match the filtered and unfiltered signals. Level differences at or just below 0.2dB can still be detected in the audible band, so the results of Kunchur's test are suspect. 

 

He had assumed that level differences below 0.7dB are inaudible. The detection results he reported were more likely caused by the level differences at 7kHz than any ultrasonic content. His test subjects detected differences just at the 0.23dB level. Where the detection was even more consistent and reliable is also where the level differences were greater, so that does cast some doubt on the study conclusion.

[pkane2001]

1 minute ago, manueljenkin said:

I'd recommend you to publish this as a paper/journal. As of now this is just your speculation.

No need, I’m not the one claiming anything about audibility of ultrasonics. Just reporting what’s been published in many studies and the results of my own tests.
 

Kunchur’s results stand out as very different compared to most everyone else, so some skepticism is not out of place here. 

[pkane2001]

58 minutes ago, jabbr said:

If you have a counter hypothesis, yes you could repeat the experiment with tighter leveling.

 

Just because other tests haven’t demonstrated anything is irrelevant because this is a specific hypothesis. Has anyone refuted this by repeating the experiment?

Why ask me? You brought up his paper. I’m not interested in defending or refuting it, but there are obvious problems with methodology that need to be addressed.

[pkane2001]

36 minutes ago, danadam said:

Agreed.

Some further reading :-)

https://hydrogenaud.io/index.php?topic=73598.0

and about the level differences specifically at the end of this post and in the next one: https://hydrogenaud.io/index.php?topic=73598.msg701379#msg701379

That FAQ is particularly telling. Kunchur states that the incorrect JND estimate he had used is not something he can go back and redo, as that’s too much work.

 

He then, incorrectly, states that despite the audible level differences, this somehow does not invalidate the conclusions. This is curious, as it clearly can provide a completely different explanation for the results that is wholly unrelated to timing differences or frequency contents of the signal. Hmm..

 

PS: by the way, @sandyk is a beautiful human being. I know he’ll disagree, but he’s just too modest ;)

[pkane2001]

8 hours ago, manueljenkin said:

Let them run with this thread with their "opinions". Skepticism is still an opinion, only when you truly analyse verify and set bounds, it becomes something worthy of consideration. Not worth looking into unless they provide it as a verified paper. As of now, there is no official verified publication that these guys have provided to refute kunchur's study.

 

Look, Kunchur himself said he knows his JND value was incorrect, four times larger than the level detected in newer experiments. This is one thing I picked up on first reading of his study. That's all I stated. It's not just my opinion, it's the author's (emphasis is mine):

 

Quote

It was found that the level changes in the experiments (~0.2 dB) were subliminal (four times smaller than the published level JND) making it likely that the discrimination depended on more than just level changes.

 

My papers also propose quantitative neurophysiological models to explain what might be happening in the timing/phase domain. One forum poster asked why I did not re-measure the level JNDs. In scientific research we have to start with what has already been published and cannot go back to the beginning of time and re-measure and reprove everything ever published (unless there is a special reason to doubt the previous results) otherwise it will be impossible to move forward. The present work took about five years. To redo the level JND thresholds properly will take at least two years.

 

Remember, in the paper he had assumed that anything below of JND of 0.7dB was not detectable.

[pkane2001]

3 hours ago, Audiophile Neuroscience said:

 

Exactly so... Déjà vu

 

 

Yep

 

 

1+

 

 

A plea falling on deaf ears

 

 

He does not

 

 

Yep

 

 

Yes but he has a "self- validating" App.....the kind they like at ASR ! 🤣🙄

 

 

 

You are always a breath of fresh air, David, with your sarcasm and personal attacks you add value to every conversation.