[Mojo ideas]

Can you two please take this private and stop polluting the thread...

Only some people believe in Pixels I do
as J M Barry once said

 

[bhobba]

Boy this business really annoys me sometimes....

It gets on my nerves as well. This is just Shannon's sampling theorem - nothing mystical.

If you want to see something actually 'mystical' check this out (watch out like Shannon's Sampling Theorem math required).

There is this mathematical function, the so called zeta function which is ζ(s) = 1/1^s + 1/2^s + 1/3^s ......... Now put in k = 0, -1, - 2 etc and you get for k=0 1+1+1+1......, or k=-1 1 +2 + 3 etc. These look like infinity. But not so fast attendant reader - lets actually calculate it for all values -k.

∑(-1)^k*ζ(-k)*x^k/k! = ∑∑ n^k*(-x)^k/k! = ∑ ∑(-nx)^k/k! = ∑e^(-nx). Let S = ∑e^(-nx). e^xS = 1 + S so S = 1/e^x - 1 = 1/x*x/e^x - 1. But one of the definitions of the so called Bernoulli numbers Bk, is x/(e^x - 1) = ∑Bk*x^k/k! or taking the1/x into the sum S = ∑ B(k+1)*x^k/(k+1)! after changing the summation index so you still have powers of x^k. Thus you have ∑(-1)^k*ζ(-k)*x^k/k! = ∑ B(k+1)*x^k/(k+1)!. Equating the coefficients of the power of x^k you have ζ(-k) = (-1)^k*B(k+1)/k+1.

This result implies the bizarre identities ζ(0) = 1+1+1+1....... = -1/2 and ζ(1) = 1+2+3+4....... = -1/12.

Now that's magic - not like Shannon's sampling theorem that is simple basic engineering math. The why of the above is much more complicated - but unsuitable for here. You can do a bit of internet research on that one or go to a science forum and ask someone - I personally use Physics Forums. I will have to tell you though only some people actually understand it (technically it's got to do with complex analysis and avoiding the pole at s = 1 that causes the infinity - but that is likely meaningless unless you know complex analysis). You will hear all sorts of views like such sums are just definitions etc. In fact its used in calculations of real physical problems like the so called Casmir Force so is not just a definition - it has real physical consequences.

This is just to point out you cant really discuss this stuff, including what Rob does, without REALLY knowing the math behind it. English is a very poor medium for doing that. Rob has explained it many times - its simple math and the consequences irrefutable, but some just do not get it.

Thanks
Bill

 

[514077]

It gets on my nerves as well. This is just Shannon's sampling theorem - nothing mystical.

If you want to see something actually 'mystical' check this out (watch out like Shannon's Sampling Theorem math required).

There is this mathematical function, the so called zeta function which is ζ(s) = 1/1^s + 1/2^s + 1/3^s ......... Now put in k = 0, -1, - 2 etc and you get for k=0 1+1+1+1......, or k=-1 1 +2 + 3 etc. These look like infinity. But not so fast attendant reader - lets actually calculate it for all values -k.

∑(-1)^k*ζ(-k)*x^k/k! = ∑∑ n^k*(-x)^k/k! = ∑ ∑(-nx)^k/k! = ∑e^(-nx). Let S = ∑e^(-nx). e^xS = 1 + S so S = 1/e^x - 1 = 1/x*x/e^x - 1. But one of the definitions of the so called Bernoulli numbers Bk, is x/(e^x - 1) = ∑Bk*x^k/k! or taking the1/x into the sum S = ∑ B(k+1)*x^k/(k+1)! after changing the summation index so you still have powers of x^k. Thus you have ∑(-1)^k*ζ(-k)*x^k/k! = ∑ B(k+1)*x^k/(k+1)!. Equating the coefficients of the power of x^k you have ζ(-k) = (-1)^k*B(k+1)/k+1.

This result implies the bizarre identities ζ(0) = 1+1+1+1....... = -1/2 and ζ(1) = 1+2+3+4....... = -1/12.

Now that's magic - not like Shannon's sampling theorem that is simple basic engineering math. The why of the above is much more complicated - but unsuitable for here. You can do a bit of internet research on that one or go to a science forum and ask someone - I personally use Physics Forums. I will have to tell you though only some people actually understand it (technically it's got to do with complex analysis and avoiding the pole at s = 1 that causes the infinity - but that is likely meaningless unless you know complex analysis). You will hear all sorts of views like such sums are just definitions etc. In fact its used in calculations of real physical problems like the so called Casmir Force so is not just a definition - it has real physical consequences.

This is just to point out you cant really discuss this stuff, including what Rob does, without REALLY knowing the math behind it. English is a very poor medium for doing that. Rob has explained it many times - its simple math and the consequences irrefutable, but some just do not get it.

Thanks
Bill

My dad was a civil engineer, with an almost savont-like understanding of higher maths. He couldn't understand why other people couldn't deal with simple math in the same way that I find it interesting that some people can't hear and arrange harmonies in their heads; let alone identify notes and chord permutations. I think we're lucky Rob takes the time to dumb-down his explanations so we can begin to try to understand how his DACs work.

 

[bhobba]

My dad was a civil engineer, with an almost savont-like understanding of higher maths. He couldn't understand why other people couldn't deal with simple math in the same way that I find it interesting that some people can't hear and arrange harmonies in their heads; let alone identify notes and chord permutations. I think we're lucky Rob takes the time to dumb-down his explanations so we can begin to try to understand how his DACs work.

Too true. Too true.

There is so much in this hobby people argue about that is obvious to those that actually listen - but still its argued. The difference here is what Rob talks about is beyond objective question, its mathematical certainty. It's not arguable, even if, like the zeta function I gave the info on, it can look weird. What you can argue about, and all people can have an opinion, is how it sounds. Its a personal thing. I have quite a few audiophile friends and we argue all the time about preferences for one piece of gear or another. I am very much looking forward to getting the M-sampler when released and either a Dave or TT2 to directly run some speakers I want to get - probably the TT2 because I am not sure the 2W of the Dave is enough for the 92db speakers I have in mind - they will be using the seas exotic drivers and the speaker will be similar to this:
http://www.salksound.com/model.php?model=Exotica+Monitors.

I know a speaker maker who can build it, but he does stuff stuff like lining the cabinet with steel and he is Duelund capacitor mad, so they are likely to be used as well - even if just for bypass. Audio is a strange thing, he normally is a very open guy to audio stuff but refused to believe me when people were raving about the Duelund bypass capacitors, he thought it all hooey.:
https://www.hificollective.co.uk/co...-silver-foil-precision-bypass-capacitors.html

How can a capacitor that small make any difference? Well I got some in and he gave it to him to try with a mock up crossover on a new speaker he was designing. He was shocked. Bought I think 50 of them, he has to buy them in lots of 50 as a wholesaler, and uses them in all his speakers now. Strange but true. They work particularly well with the Jantzen Superior which is his standard Capacitor now (he used straight Duelund VSF before but they are not made any more except on special order) - he uses mostly Janzen bypassed by the Duelund. It works even better with the Jupiter Copper but for large values they can be really pricey - same with Duelund Cast which is even more pricey. I personally am not that enamoured with the Duelund RS - they sound slightly dry to me - I like the Jupiter Copper better. But Duelund Cast is king - however not at that price thank you very much - the difference above the Jupiter is simply not worth such a huge price difference IMHO.

Thanks
Bill

 

[Currawong]

Now I understand how digital works better, I'm rather annoyed having found all the fundamental errors in the discussion of impulse responses.

Living in a country where we regularly have earthquakes, while reading the discussion on CA over MQA and digital filtering, I was thinking that even though a quake is the result of a sudden shift inside the Earth, I have always recalled that what we feel at the surface doesn't start with a sudden shock and then fade, but starts with gentle shaking that builds to a crescendo then fades away slowly. A quick search for seismographs revealed that I was correct.

Nothing like nature to show us the truth of the matter.

 

[bhobba]

Now I understand how digital works better, I'm rather annoyed having found all the fundamental errors in the discussion of impulse responses.

Sigh. I will go through it carefully.

Shannon's sampling theorem guarantees a band-limited signal at the frequency f/2 transmitted digitally at f can be reconstructed EXACTLY - no if's or buts. What Rob does is he uses a very accurate reconstruction method that involves a sync function. He does the digital math involved using technical lingo with a million taps which his calculations have shown is necessary to achieve 16bit accuracy in the reconstruction. Other methods have errors - small errors to be sure but the interesting thing is Rob found them quite audible. He demonstrated this to other engineers who were shocked - they did not believe such small errors were audible.

Now for the transient thing. Suppose you have a signal with transients that have content above f/2. When you band limit it the filter that does that gets rid of some of that detail - in fact it can create whats called 'ringing' when those frequencies bang into the filter. It is claimed by some this is the cause of the so called digital sound - but other explanations are possible such as the accuracy of the filter. This is something Rob wants to check out with his highly accurate filter and I am sure he will report what he finds.

Now for MQA what you do is a very gentle filter above 24k. What they have found is when you do that for nearly all material (something like over 99.9%) if you just take the first 16 bits then no other filter is required - all frequencies above that are below 16 bits. There is whats called aliasing by using such a gentle filter - but that is below 16 bits so is not a worry - well hopefully anyway. This is explained here:
http://www.aes.org/e-lib/browse.cfm?elib=17501

Have a look at figure 13. At 16 bit accuracy there is nothing above 48k so that's all you need to transmit. The filter is technically called a spline filter - the MQA guys analyse the music to determine what filter to use to ensure everything above 48k is below 16 bits. In fact on much materiel even if you do no filtering (see figure 10) there is virtually no material above 48k so no spline filtering at all is needed. Now Rob is not a fan of spline filtering - he can explain why - its quite gentle and supposedly does not cause any audible effects. That's the theory anyway - if its audibly true or not is another matter. I do not think Rob has any issue with material that has nothing above 16 bit and just stripping the lower bits - it's that spline filter that is sometimes used I think he worries about.

Even though in MQA there is no material above 48k you still need to reduce its sample rate to 48k - Rob would probably contend his M-Scaler down sampler will do a better job of that than what the MQA guys use. In fact I suspect since much material requires no spline filtering Rob would say don't bother - just use his filtering technology that is a brick-wall at 48k.

There is so much we do not know about the audible effects of this stuff only time will tell about MQA.

Thanks
Bill

 

[csglinux]

Now for the transient thing. Suppose you have a signal with transients that have content above f/2. When you band limit it the filter that does that gets rid of some of that detail - in fact it can create whats called 'ringing' when those frequencies bang into the filter.

I would replace the word "can" with "has to" The only way that wouldn't happen is if you were to add additional filtering below Nyquist. That's the fascinating part of all this to me. The sync filter seems to sort out all bandwidth-limited signals, but what is the correct thing to do when trying to recreate a recording of a fast cymbal hit, gunshot, shock-wave, blast-wave etc? Anybody have an answer (or opinion) on this?

P.S. @Currawong might have a point re. the earthquake seismograph plot. I'd expect any medium that the impulse travels through (i.e., the earth) is going to preferentially damp the highest frequencies. How would the earth be smart enough to cut off the highest frequencies and still avoid the Gibb's phenomena?

 

[Arpiben]

In order to perfectly reconstruct a bandwidth-limited sampled signal there is no other option than applying a sinus cardinal (sinc) convolution in time domain.
Applying a sinc in time domain is equivalent to apply a perfect Low Pass Filter in frequency domain.
Cf. Nyquist, Shannon, Whittaker, Kotelnikov...
A bandwidth-limited sampled signal will be perfectly (100% accuracy) reconstructed with infinite time ( Cf J.Fourier)

In real life time is finite, sampled signals are not all properly bandwidth limited, sampled signals may suffer from ADC steps (decimation,aliasing,timing accuracy,etc).
Reconstructing the signal with sinc is computationally heavy. Nevertheless,despite real life, different methods allow to reach a desired accuracy with acceptable latency.

Mathematically, in order to perfectly recreate a sound the prerequisite is to sampled it at least twice its maximum frequency.
In instrumentation (digital scopes,spectrum analyzers),the analog signal is sampled around 5x times the allowed bandwidth in order to be able to catch aperiodic/random transients

In terms of sound perception, and how one may perceive the differences among reconstruction filters types or accuracy is not my domain. I simply believe what I can measure.

@csglinux and @Currawong waves length involved in earthquake are under 25 Hz (periods are from seconds to hours).




.

 

[csglinux]

Mathematically, in order to perfectly recreate a sound the prerequisite is to sampled it at least twice its maximum frequency.

...which we can never attain if we're trying to record a non-linear, self-steepening wavefront.

So I think there's a question of whether we want to intentionally add further filtering, and if so, how. This part seems more art than science, but we should probably respect certain mathematical properties of the signal, such as being total variation bounded in time.

waves length involved in earthquake are under 25 Hz (periods are from seconds to hours).

Bats and mosquitoes probably think of 22 kHz as sub-bass It's all a question of scale.

P.S. Interesting paper here on gunshot SPL measurements: http://www.sandv.com/downloads/0908rasm.pdf

Superficially, it looks like the impulse has pre- and post-ringing, but the descriptions around fig. 6 indicate there are physical explanations for all of these (sounds of the trigger pull, combustion and movement of the bullet before leaving the barrel). It seems that air, at least, respects the total variation of the impulse. There's no pre-ringing. I'm guessing tectonic plate shifts aren't perfect impulses and that transmission is through a pretty inhomogeneous mix of different types of soil, rock, etc.

 

[Currawong]

I would replace the word "can" with "has to" The only way that wouldn't happen is if you were to add additional filtering below Nyquist. That's the fascinating part of all this to me. The sync filter seems to sort out all bandwidth-limited signals, but what is the correct thing to do when trying to recreate a recording of a fast cymbal hit, gunshot, shock-wave, blast-wave etc? Anybody have an answer (or opinion) on this?

P.S. @Currawong might have a point re. the earthquake seismograph plot. I'd expect any medium that the impulse travels through (i.e., the earth) is going to preferentially damp the highest frequencies. How would the earth be smart enough to cut off the highest frequencies and still avoid the Gibb's phenomena?

The thing is, there aren't any instant transients in nature, including music. Even if you hit a drum, there is a ramp-up, even if it is quite short. Nothing accelerates to full speed instantly. Here's a cymbal hit, shamelessly nicked from a post at CA. I believe sampled at 96k:

I have a 96k track of drums and cymbals only which I've been looking at as well. I might edit this post with a sample of that instead.

Now for the transient thing. Suppose you have a signal with transients that have content above f/2. When you band limit it the filter that does that gets rid of some of that detail - in fact it can create whats called 'ringing' when those frequencies bang into the filter. It is claimed by some this is the cause of the so called digital sound - but other explanations are possible such as the accuracy of the filter. This is something Rob wants to check out with his highly accurate filter and I am sure he will report what he finds.

This is very helpful. This explanation only has to do with the ADC or re-sampler though doesn't it?

 

[csglinux]

The thing is, there aren't any instant transients in nature, including music.

Of course, but there doesn't need to be. The ramp-up only needs to be fast enough w.r.t. to our sample rate and then we have to do something (or choose to do nothing).

With symbol crashes and gunshots, etc., there are enough reflections going on that it's difficult to separate out and identify the physical cause of all the little peaks and troughs, but I'm not sure molecular dissipation in air (or compressible solids, like rock) can itself generate pre- or post-ringing. (For it to generate pre-ringing, it would seem part of the signal would have to travel faster than the speed of sound in the medium.) As elegant as the infinite sinc function might be for bandwidth limited signals, I'm not sure that's enough for fast transients, because of the risk of generating Gibbs oscillations which wouldn't occur in nature. Sure, the amplitudes may be tiny and the frequencies very high, but it's interesting from a mathematical point of view (at least it is to me) to understand what would be needed to reproduce nature without these artefacts.

Rob's previous posts indicate that Davina needs to address this via ADC as well as the subsequent DAC process.

 

[Currawong]

As elegant as the infinite sinc function might be for bandwidth limited signals, I'm not sure that's enough for fast transients, because of the risk of generating Gibbs oscillations which wouldn't occur in nature.

A transient any faster than what is bandwidth limited to audible frequencies would be of a higher frequency than is audible. That's obvious, isn't it?

 

[csglinux]

A transient any faster than what is bandwidth limited to audible frequencies would be of a higher frequency than is audible. That's obvious, isn't it?

I don't agree. The Fourier reconstruction of that transient contains all frequencies - and you're only throwing away those that are inaudible. I have some test tracks of two impulse signals varying deltas apart (down to 5 microseconds). I'll try to dig these out and pm them to you. You'll be surprised how you can hear above 20 kHz (based on your arguments above). I don't think we understand enough of how the ear/brain process sound to conclude exactly what's audible and what's not.

Maybe it's of low priority, but I've never liked the argument that we don't need to worry about it because "sharp transients don't occur in music". There's no law preventing anybody from recording an album of dirac delta functions - and of course, the classic follow-up, the square-wave album. You might not think it sounds very good, but then I don't think rap sounds very good An ADC -> DAC should be able to work perfectly (or as well as we can possibly make it) in all cases, even extreme limiting cases.

 

[csglinux]

Well that's weird. @Currawong - I just tried to send you a pm, but headfi says I'm not allowed to. I hope that's just a headfi bug and that none of my prior rantings have caused you to block me?

Anyway, here are those impulse test tracks: https://www.dropbox.com/sh/qmkr6429gqpxfhe/AABxRdfmfgkOxoyTdYBsqZmba?dl=0

There's a readme file in the folder.

 

[rayl]

I don't agree. The Fourier reconstruction of that transient contains all frequencies - and you're only throwing away those that are inaudible. I have some test tracks of two impulse signals varying deltas apart (down to 5 microseconds). I'll try to dig these out and pm them to you. You'll be surprised how you can hear above 20 kHz (based on your arguments above). I don't think we understand enough of how the ear/brain process sound to conclude exactly what's audible and what's not.

Maybe it's of low priority, but I've never liked the argument that we don't need to worry about it because "sharp transients don't occur in music". There's no law preventing anybody from recording an album of dirac delta functions - and of course, the classic follow-up, the square-wave album. You might not think it sounds very good, but then I don't think rap sounds very good An ADC -> DAC should be able to work perfectly (or as well as we can possibly make it) in all cases, even extreme limiting cases.

While I agree with the general comment that transients outside the hearing frequency range do matter (else I wouldn’t be a Chord fan!)... bec “hearing range frequencies” are defined relative to tones, and do not rule out transients from being heard per se though of course, there is some limit to the ability to hear transients for the same reason as my next comment:

I will point out that I find ‘recording a square wave” objectionable as a concept for a square wave is a mathematical construct that cannot occur in a physical analog world. Aside from quantum phenomenon, nothing responds “instantaneously” in the physical world. Everything has “some” bandwidth limit.

Hearing is no different... how fast the ear drum can move, how fast electrical signals can transition within the brain, etc.... but such limits are much higher than for detection of a steady tone.