Technically meaningless AI word salad

[ampetrosillo]

I'll amend my previous statement.

There aren't necessarily infinite zero crossings in any time interval. There *can* be, though.

[Andreya_Autumn]

As I said, due to the nature of sampled digital audio (a train of impulses multiplied by the analog signal) there are actually infinite zero crossings for any time interval.

Sorry, I agree with the rest of your take but this just isn't true afaict. The sample train itself is completely valueless between samples. There is no signal there that can cross zero. There is also not zero. There is nothing.

If there were "nothing" resampling wouldn't be possible. It is rather that there is nothing in the file, in the CPU, between two samples. But at a certain point, those integers or floating point values will have to be converted into a signal. If there were nothing, the reconstruction process would be basically invention. Instead the reconstruction filter (the ideal one at least) is simply a brickwall lowpass.

So yes, we agree there is nothing between the samples in the sample data itself. And we also agree that you need to run the reconstruction process for the signal to have a continuous value. So far so good.

So... where could these infinitely many zero-crossings of which you speak exist, exactly? EDIT: now I see your post above. Ok good.

[ampetrosillo]

Well, technically, a sampled signal is a Dirac impulse (infinite amplitude for an infinitely small time, defined as having an area equalling one) times the value of the signal at the time of the impulse, and there are as many impulses as samples, spaced at a specified interval we will call tau. So technically between each value, we could think of the digital representation of the signal as having infinite zeroes. Or even not zeroes. It could actually be undefined. We could have just noise. EDIT: But the most useful hypothesis, to humans, and one that we actually strive to approximate, is that what's in between is an endless string of zeroes. In fact we employ what we commonly know as antialiasing filters. If there weren't zeroes, what we are representing digitally could be an infinity of aliased signals that happen to sound like what's in between DC and f_Nyquist.

[havran]

It's not zero, it's nothing. It's not a crossing, it's a pause for reflection. But when an infinite cluster of zeroes finally meets some ones, that's where the magic can happen.

[havran]

I'll amend my previous statement.

There aren't necessarily infinite zero crossings in any time interval. There *can* be, though.

I can't really follow this discussion, but in the digital domain isn't there some timewise quantum in play, which would limit the number of zero crossings to a finite number within any given time interval?

[ampetrosillo]

You know how N is a subset of Z, which is a subset of Q, which is again a subset of R (and finally C?). Discrete mathematics can exist in isolation, yes. But it is possibly more useful to us (not always; but in certain cases, it is) if we consider it as a special limitation of the continuous realm. Digital discrete-time sampling and the DFT are useful to us because digital technology is easier to make than analog technology. But the idea (when we humans decide to use it) is that it is, basically, a functional compromise of what we generally decide to view as a continuous universe (to put it briefly - the universe is not necessarily continuous, and actually there are hints that it may actually be... discrete).

[ampetrosillo]

I'll amend my previous statement.

There aren't necessarily infinite zero crossings in any time interval. There *can* be, though.

I can't really follow this discussion, but in the digital domain isn't there some timewise quantum in play, which would limit the number of zero crossings to a finite number within any given time interval?

If we could actually put an infinite number of zero crossings in between samples we'd essentially be infinitely oversampling a signal. It turns out that we can add any number of zero crossings to a digital signal A and get another signal B that is identical to A from DC to A's Nyquist frequency, and then complete "silence" afterwards up until B's Nyquist. Then you'd get the same silence, reversed (to be more precise, the imaginary counterpart of that silence) up until A's Nyquist, then A reversed (or actually its imaginary counterpart) up until A's DC, then everything all over again times infinity.

EDIT: I was tired, I had had a couple of beers. If you just put zeroes in, you are merely including the spectral aliases within the signal. (Which actually makes my point even better). You need a lowpass afterwards to have only your signal, and you are essentially interpolating.

[havran]

I'll amend my previous statement.

There aren't necessarily infinite zero crossings in any time interval. There *can* be, though.

I can't really follow this discussion, but in the digital domain isn't there some timewise quantum in play, which would limit the number of zero crossings to a finite number within any given time interval?

If we could actually put an infinite number of zero crossings in between samples we'd essentially be infinitely oversampling a signal. It turns out that we can add any number of zero crossings to a digital signal A and get another signal B that is identical to A from DC to A's Nyquist frequency, and then complete "silence" afterwards up until B's Nyquist. Then you'd get the same silence, reversed (to be more precise, the imaginary counterpart of that silence) up until A's Nyquist, then A reversed (or actually its imaginary counterpart) up until A's DC, then everything all over again times infinity.

For the sake of understanding, is there some universally agreed-upon or cosmically determined zero-point that is used to reckon where and when the zero-point crossings occur, or is it more relativistic than that -- in which case, why isn't everything just one continuous flatline?

[ampetrosillo]

Well. Everything is relativistic really. Or rather, our chosen¹ human representation is relativistic. For instance, there is no "real" 0V and 100V. When we're talking about pressure waves, or AC signals, it is all relativistic, again. Why should everything be a continuous flat line? Why not just noise? Which we might as well be, in the context of the universe.

¹ I say chosen because, for instance, the idea that time is a line does not conform to certain African cultures' view of time.

[havran]

Probably just a matter of time before someone comes along to trot out, "One man's noise is another man's signal."

And I'll admit that it was incorrect or at least inaccurate of me to land on "continuous flatline", given that sound occurs or is perceived to occur in at least a 4D world, or some might say an 11-dimensional world (not sure if that would include time), or maybe even an imaginary-number-dimensional world!

(Hoping I'm the first to propose that last one, in case I turn out to be right for a while.)

[Andreya_Autumn]

EDIT: But the most useful hypothesis, to humans, and one that we actually strive to approximate, is that what's in between is an endless string of zeroes.

Who says that? I have never encountered that idea before. And my initial gut feeling is to disagree that this is a useful hypothesis. Seems like a mental model with much potential to cause confusion, somewhat in the same way that visually representing sample data with a stair-step can.

To those having trouble following the conversation I recommend this:

I've timestamped the most relevant part but really the whole thing is worth a watch.

[Andreya_Autumn]

For the sake of understanding, is there some universally agreed-upon or cosmically determined zero-point that is used to reckon where and when the zero-point crossings occur, or is it more relativistic than that -- in which case, why isn't everything just one continuous flatline?

Yes. That point is zero. What that zero means depends on context.

Acoustically, 0 means the normal air pressure at the location of interest.
In analog electronics 0 means the voltage the circuit returns to when there's no sound. Sometimes that's literally 0 volts, sometimes (say in vacuum tubes) it's a fixed bias voltage.
In digital audio zero means the literal number 0.

Zero crossings happen when the signal crosses that point. Air pressure/voltage/numbers greater than the respective zero state, followed by pressure/voltage/numbers lesser than it. Simple as that.

The reason the digital case is trickier to reason about is you don't know the value of your signal at any point in time other than the sample period. Only in the exceedingly rare case that a zero-crossing happens to fall precisely on the sample interval (resulting in an exact 0 value) do you know precisely when the zero crossing happens. Most of the time, what you have is a value >0 followed by one <0, and thus the knowledge that sometime in between those two points, a zero-crossing will occur in the analog signal a converter will make from your samples.

Exactly one zero-crossing in fact. In order for there to be more than one, the signal would have to wiggle up and down more than once inside the sample period. Which is literally the simplest image I can think of for what it means to contain harmonic content above nyquist (and thus not be a valid reconstruction).

Indeed as you point out Ampesotrillo, once you oversample, you will get more opportunities to correctly represent multiple zero-crossings between each samples of the *original* signal. And if you theoretically oversample infinitely, well yeah you have infinite such opportunities. But what I think you might be missing is that's not the same as the signal *actually having* infinite zero crossings.

Consider a signal which is two sine waves, a loud one at 1Hz and a much quiter one at 10000000Hz.
If you record it at a standard samplerate (correctly band-limiting it first of course), you get two zero crossings per second as the slower wave crosses up and down.
If you record it with an infinite sample rate, do you get infinite zero crossings? Of course not. Twice per second, around the times the louder 1Hz wave crosses, you get a bunch of them. Then none, as the louder one moves far enough off 0.

[ampetrosillo]

EDIT: But the most useful hypothesis, to humans, and one that we actually strive to approximate, is that what's in between is an endless string of zeroes.

Who says that? I have never encountered that idea before. And my initial gut feeling is to disagree that this is a useful hypothesis. Seems like a mental model with much potential to cause confusion, somewhat in the same way that visually representing sample data with a stair-step can.

To those having trouble following the conversation I recommend this:

I've timestamped the most relevant part but really the whole thing is worth a watch.

If you limit analysis to the digital samples you have, a digitally sampled signal only has content from DC to Nyquist. It's really the whole idea behind the Nyquist-Shannon theorem (or somebody may say the Whittaker theorem, or the Kotelnikov theorem). It doesn't even have aliases, it's really just the single spectrum. But if you generalise it and expand the discrete to continuous, the closest equivalent is samples interleaved with infinite zeroes. And there you do get the aliases, as many as the infinite zeroes in between samples. When I studied Fourier analysis, I started from the continuous and only later did we restrict analysis to the discrete. I'm not expecting to be right, hey? But in practice, the only kind of sample addition we do when we upsample is zero padding.

[Andreya_Autumn]

Oh were you taught to think of it that way? By whom if so, like where can I read more? The idea of a *continuous* signal which is somehow zero everywhere except at the points you happened to sample doesn't make any sense to me personally. I can imagine it conceptually, but it would have all these infinitely steep slopes etc etc... I don't see how it has much to do with the *actual* reconstructed signal you get when you run the converters. But if you were taught this model in school maybe there's something I'm missing.

Anyway, as a mix engineer and not a mathematician, seems to me the most intuitive/useful way to conceptualize a continuous signal made from discrete samples is, well, the reconstructed signal. That's what we care about in the end.

And yeah, oversampling by adding zeroes in between the existing samples, then lowpassing does work, because (with the right filter) it approximates well what you would've got if you'd sampled the same continuous signal twice as often. But there are also other ways to compute the missing sample values, and with Sinc interpolation the approximation can get arbitrarily good.

[Zeisner]

That text is ridiculous. I'm getting "stochastic erosion" vibes again.