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Topic: AES 2009 Audio Myths Workshop (Read 166183 times) previous topic - next topic
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AES 2009 Audio Myths Workshop

Reply #100 –
Quote from: John Eppstein on
Quote from: Notat on
With a couple reasonable assumptions around signal level and dither, a digital console or workstation absolutely operates as an ideal linear system.
sez you. the reality is different unless you start piling on the qualifiers such as "within the stated pass band".
You are correct, "within the stated pass band" is an important assumption that I did not include in my original post. All are very compatible with the real world. I don't think there are any others.

AES 2009 Audio Myths Workshop

Reply #101 –
Quote from: dwoz on
Quote from: Notat on
Quote from: dwoz on
Actually, you apparently follow quite well.  I agree completely with your characterization, EXCEPT that you're not describing a real-world system, but a hypothetical one, that you'd be hard pressed to discover in actual use.

With a couple reasonable assumptions around signal level and dither, a digital console or workstation absolutely operates as an ideal linear system.


Can you prove this?  It sounds like you're making a conjecture.  I know how fun that is, I am always accused of making conjecture. 

If one were to have to prove this, how would one even go about accomplishing that proof?

By the way, are you rebutting Ethan?  He said that dither is basically inaudible, so in mentioning it, do you refute him?

As a practical matter with today's 24-bit converters, Ethan may be right. As a matter of mathematics, I think he goes too far.

The linearity proof goes like this:
  • Niquist-Shannon tells you that all of the information within the frequency range of interest is encoded in the digital sampled version of the signal.
  • Quantization theory tells you that limited bit resolution adds noise to your signal.
  • Dither ensures that quantization noise is not correlated with the signal. An uncorrelated signal can be treated as a separate signal added to the original - signal and noise come in, signal and noise go out, there is no interaction or intermodulation between the two.
  • The operations performed in the signal path of a digital mixer are addition and multiplication. Addition and multiplication are linear operations. Any operation built from a linear combination of linear operations is itself a linear operation.

AES 2009 Audio Myths Workshop

Reply #102 –
Quote from: Notat on
Quote from: dwoz on
Quote from: Notat on
Quote from: dwoz on
Actually, you apparently follow quite well.  I agree completely with your characterization, EXCEPT that you're not describing a real-world system, but a hypothetical one, that you'd be hard pressed to discover in actual use.

With a couple reasonable assumptions around signal level and dither, a digital console or workstation absolutely operates as an ideal linear system.


Can you prove this?  It sounds like you're making a conjecture.  I know how fun that is, I am always accused of making conjecture. 

If one were to have to prove this, how would one even go about accomplishing that proof?

By the way, are you rebutting Ethan?  He said that dither is basically inaudible, so in mentioning it, do you refute him?

As a practical matter with today's 24-bit converters, Ethan may be right. As a matter of mathematics, I think he goes too far.

The linearity proof goes like this:
  • Niquist-Shannon tells you that all of the information within the frequency range of interest is encoded in the digital sampled version of the signal.
  • Quantization theory tells you that limited bit resolution adds noise to your signal.
  • Dither ensures that quantization noise is not correlated with the signal. An uncorrelated signal can be treated as a separate signal added to the original - signal and noise come in, signal and noise go out, there is no interaction or intermodulation between the two.
  • The operations performed in the signal path of a digital mixer are addition and multiplication. Addition and multiplication are linear operations. Any operation built from a linear combination of linear operations is itself a linear operation.



So, basically "a digital console or workstation" is not "ALL digital consoles or workstations", but only the specific ones that only do summing, gain changes, and...panning.  Every OTHER digital console is not included in your term?

AES 2009 Audio Myths Workshop

Reply #103 –
Quote from: googlebot on
Quote from: dwoz on
Can you prove this?  It sounds like you're making a conjecture.  I know how fun that is, I am always accused of making conjecture. 

If one were to have to prove this, how would one even go about accomplishing that proof?


Induction is always problematic. But it is to be going easy for you to find out, just as it was easy for me to find out: Load up an original and a looped back sample into an ABX program and see if you can differentiate it. The last time I tried I stopped after 20 loop backs on a DG Archiv Produktion of Beethoven. It was just too hard. And that was just looped through the commodity sound chip of an Apple Macbook Pro.

If you're rather out for more general statements. The limits of human auditory perception are well researched. Just measure the looped back version and compare it to those thresholds. You'll see, that your concerns aren't justified by reality.


Ahhh....I see.  so basically if something is equally LINEAR to something else (like, source to output in a console), then it will be sonically indistinguishable?  I would LIKE to agree, but that seems to have already been thoroughly debunked up higher in the thread.  You're basically calling Arnie Krueger and Notat a liar.  Are you correct, or are they? 

Notat said:
Quote
QUOTE (dwoz @ Mar 21 2010, 11:49) *
Implication you're making is that analog console sonics are non-differentiable. (for modern kit). I think that's a bold statement to make that will get you in trouble later.

QUOTE (Notat @ Mar 21 2010, 15:52) *
Arnold said that analog consoles are linear. Linearity doesn't say a whole lot about about how a console sounds just that it doesn't generate certain types of distortion.

QUOTE (Arnold B. Krueger @ Mar 21 2010, 16:28) *
Exactly.

<snip>

I only mentioned nonlinear distortion, so it is hard to understand how one might logically progress from my statement to a statement that system (in this case analog console) sonics are non-differentiable. Straw man, anyone? ;-)

AES 2009 Audio Myths Workshop

Reply #104 –
Quote from: googlebot on
Induction is always problematic. But it is to be going easy for you to find out, just as it was easy for me to find out: Load up an original and a looped back sample into an ABX program and see if you can differentiate it. The last time I tried I stopped after 20 loop backs on a DG Archiv Produktion of Beethoven. It was just too hard. And that was just looped through the commodity sound chip of an Apple Macbook Pro.

If you're rather out for more general statements. The limits of human auditory perception are well researched. Just measure the looped back version and compare it to those thresholds. You'll see, that your concerns aren't justified by reality.


Let's be rigorous about this, okay?  I am not exactly sure what you mean when you say "looped back sample", and "original".  We're talking about whether we can test that a console or workstation is ideally linear.  Could you define those terms more specifically?

Also, if we're talking about linearity, it's either ideally linear or it's not, right?  we can just hook up an oscilloscope to the output and learn everything we need to know, right?  or simply invert the polarity of the output and sum it with the input and look for whatever didn't null, right?

Why is this about listening?

AES 2009 Audio Myths Workshop

Reply #105 –
Quote from: Notat on
Quote from: dwoz on
I must confess...I've never heard of "linear distortion".  What is it?  Isn't ALL distortion by definition non-linear?

Quote from: Arnold B. Krueger on
Systems can have only 4 general kinds of signal response faults: linear distortion (frequency and/or phase response errors) , nonlinear distortion, random noise, and coherent interfering signals.

Maybe you wouldn't classify frequency response errors as "distortion". Not everyone does. It's a just a terminology thing. Nothing to get hung up on.



So, basically, when Arnold says "linear distortion" then says "non-linear distortion", he means the same thing?

AES 2009 Audio Myths Workshop

Reply #106 –
Quote from: Notat on
Quote from: John Eppstein on
Quote from: Notat on
With a couple reasonable assumptions around signal level and dither, a digital console or workstation absolutely operates as an ideal linear system.
sez you. the reality is different unless you start piling on the qualifiers such as "within the stated pass band".
You are correct, "within the stated pass band" is an important assumption that I did not include in my original post. All are very compatible with the real world. I don't think there are any others.


It depends on who you are conversing with and what the purpose of the discussion is. For instance, what you say isn't true if the device is operated in the centre of the sun, for instance. I'll bet its behaviour will not be linear then. So yeah, see, reality is more complicated than your simple models.

Now whether this is a reasonable thing to say would depend on what my purpose is, doesn't it?

AES 2009 Audio Myths Workshop

Reply #107 –
Quote from: dwoz on
So, basically, when Arnold says "linear distortion" then says "non-linear distortion", he means the same thing?

Are you being purposely obtuse or are you really not that smart?

This isn't exactly rocket science you know!

Linear distortion is any change in the signal that is not level dependent.

Non-linear distortion IS level dependent.

AES 2009 Audio Myths Workshop

Reply #108 –
Quote from: dwoz on
So, basically "a digital console or workstation" is not "ALL digital consoles or workstations", but only the specific ones that only do summing, gain changes, and...panning.  Every OTHER digital console is not included in your term?
Equalization is also done with multiplication and addition. My point is that there's not any unintentional non-linearities in any properly-implemented digital console. There are no compromises designers need to make. There are no minute imperfections or any non-ideal behavior in the way DSPs do the math.

Consoles can do non-linear processing (e.g. dynamic range compression, distortion plugins...). You turn that on and you're telling the console you don't want linear behavior - it does what you ask.

AES 2009 Audio Myths Workshop

Reply #109 –
Quote from: pdq on
Quote from: dwoz on
So, basically, when Arnold says "linear distortion" then says "non-linear distortion", he means the same thing?

Are you being purposely obtuse or are you really not that smart?

This isn't exactly rocket science you know!

Linear distortion is any change in the signal that is not level dependent.

Non-linear distortion IS level dependent.


My dear friend, your reply is interesting on many levels.  First off, it is a vicious personal attack.  Next it is an ad-hominem attack on the premise of the question.  Next, we can talk about rudeness.  But all that aside, I'm afraid your answer didn't help me much.  What do you mean by "level-dependent"?  Do you mean, depending on the input level?  So, "linear" distortion is distortion that is there no matter what voltage or frequency I apply to my input, but "non-linear" distortion is not?

Also, I have found, FINALLY, a couple references that actually use the words "linear distortion".  The references seem to be specific to loudspeakers, and seem to anecdotally mention level-dependency as an factor, but not as a definitional term.  Can you clarify?  Can you pop over some references where someone may have laid this out in a discussion of signal processing circuits?  In other words, the references I saw seem to suggest that MOST non-linear distortion is level dependent, (still not quite sure what that means), but that you could certainly have non-linear distortion that wasn't level dependent.  Can you clarify?

Maybe it's just an old-fashioned term that's gone out of use lately...

Thank you for your forthcoming well-reasoned reply!

AES 2009 Audio Myths Workshop

Reply #110 –
Wikipedia covers the topic in depth: http://en.wikipedia.org/wiki/Distortion

This is a very fundamental audio/electrical engineering topic...

AES 2009 Audio Myths Workshop

Reply #111 –
Quote from: Notat on
Quote from: dwoz on
So, basically "a digital console or workstation" is not "ALL digital consoles or workstations", but only the specific ones that only do summing, gain changes, and...panning.  Every OTHER digital console is not included in your term?
Equalization is also done with multiplication and addition. My point is that there's not any unintentional non-linearities in any properly-implemented digital console. There are no compromises designers need to make. There are no minute imperfections or any non-ideal behavior in the way DSPs do the math.

Consoles can do non-linear processing (e.g. dynamic range compression, distortion plugins...). You turn that on and you're telling the console you don't want linear behavior - it does what you ask.


Ok, that's a good answer.

But I'm wondering.  You seem to be making a very sweeping and absolute statement here.  The way I'm reading this, you aren't saying anything about audibility or "as far as is reasonably necessary", you're saying flat-out that they are linear, period.  Is that a correct reading of your intent?

AES 2009 Audio Myths Workshop

Reply #112 –
The inherent non-linearities of any given electrical system can be quantified as "total harmonic distortion" and "intermodulation distortion", as well as others. I'm not sure where the audibility threshold of THD starts, but it's a measurable characteristic of any electrical system, whether digital or analog.

AES 2009 Audio Myths Workshop

Reply #113 –
Quote from: John Eppstein on
Quote from: Notat on
Quote from: dwoz on
Actually, you apparently follow quite well.  I agree completely with your characterization, EXCEPT that you're not describing a real-world system, but a hypothetical one, that you'd be hard pressed to discover in actual use.

With a couple reasonable assumptions around signal level and dither, a digital console or workstation absolutely operates as an ideal linear system.

sez you. the reality is different unless you start piling on the qualifiers such as "within the stated pass band".


A defined pass band is a natural property of *any* real world system. Therefore presuming one does not necessarily require any specific qualifiers at all.

Since we (hopefully) all know that we are talking about audio, we also know that the traditional audio passband is 20Hz - 20KKz.  As a rule, digital systems lack inherent, practically significant LF roll-offs. That leaves only 20 KHz as a possible area of discussion. We just went through a period of weirdness a fw years back where a few people people tried to profiteer by extending the HF passband well above 20 KHz and making false claims about audible benefits. That madness seems to have pretty much evaporated since then, at least in the mainstream. Furthermore, people who actually invested big money in that weirdness, received negligable rewards, millions were lost by a few large corprations, and a few execuitives simply lost their jobs.

So, the reality is not actually any different from what Notat said,  unless I somehow missed something.

AES 2009 Audio Myths Workshop

Reply #114 –
Quote from: Canar on
The inherent non-linearities of any given electrical system can be quantified as "total harmonic distortion" and "intermodulation distortion", as well as others. I'm not sure where the audibility threshold of THD starts, but it's a measurable characteristic of any electrical system, whether digital or analog.


Thank you for that.  Not exactly sure how it is relevant, but it's certainly interesting.

I have found a bit of discussion that implies that "linear distortion" is gain and phase anomalies as a function of frequency...i.e. that it is a "systemic" problem independent of program, where "non-linear distortion" is gain and phase anomalies as a function of amplitude...i.e. that is a "systemic" problem that is dependent on program.

I like this definition, because it seems to solve a pervasive and awful problem that I see all over the internet audio forum world...the tendency to bundle, mix, and match various different CLASSES of problems together, ad hoc, to mix-and-match cause and effect, to slur the difference between domain and the value of products.

I think this was the basis for my objections to the Ethan Winer "four measurements".  It wasn't so much that they are wrong, per se, but that each category is a jumbled mis-match of strange-bedfellows.

...and everyone always seems to have their own way of grouping them.

For example, harmonic distortion is not a CAUSE of distortion, it's a product of distortion.

your thoughts?

AES 2009 Audio Myths Workshop

Reply #115 –
Quote from: dwoz on
For example, harmonic distortion is not a CAUSE of distortion, it's a product of distortion.
It is neither a cause nor a product of distortion. It is a type of distortion. It is a measurable characteristic of an electrical system.

AES 2009 Audio Myths Workshop

Reply #116 –
Quote from: John Eppstein on
First, you're assuming that all components will always be operated strictly within their linear region which in pro audio is not always true.


Not really. The actual assumption is that if someone wants to operate all components in a useful signal chain in their linear region, that is generally practical and feasible.

Everybody who understands gain staging understands that merely operating all components in a useful signal chain can take a little planning and skill.  The ability of unskilled or careless people to make big messes can't be understated.

Quote
Second you're assuming that all production examples of a specific part will always adhere strictly to their published spec.


Not really.

1. Many important properties of much are gear are simply not fully specified.
2. Specified performance is often far better than the minimum required to be effective.
3. Much equipment performs far better than specified, in many ways.
4. The performance of much digital equipment is unbelievbly consistent. For example, noise floors are often created by digital means and are therefore identical for every piece of equipment that works at all.

Quote
In reality this is NEVER the case - there is always a tolerance range.

NO GIVEN PART EVER EXACTLY MATCHES THE SPEC SHEET.


So what? Please see items 1-4 above.

AES 2009 Audio Myths Workshop

Reply #117 –
Quote from: Arnold B. Krueger on
A defined pass band is a natural property of *any* real world system. Therefore presuming one does not necessarily require any specific qualifiers at all.

Since we (hopefully) all know that we are talking about audio, we also know that the traditional audio passband is 20Hz - 20KKz.  As a rule, digital systems lack inherent, practically significant LF roll-offs. That leaves only 20 KHz as a possible area of discussion. We just went through a period of weirdness a fw years back where a few people people tried to profiteer by extending the HF passband well above 20 KHz and making false claims about audible benefits. That madness seems to have pretty much evaporated since then, at least in the mainstream. Furthermore, people who actually invested big money in that weirdness, received negligable rewards, millions were lost by a few large corprations, and a few execuitives simply lost their jobs.

So, the reality is not actually any different from what Notat said,  unless I somehow missed something.



My understanding of the engineering design reason for extended bandwidth systems is really quite different than yours.  Instead of it being some kind of hyperbolic "madness" as you call it, it was really a lot more about improving the linearity of the hardware within the defined passband.

It's no secret that transient response is directly related to bandwidth, after all.  That's been part of signal processing curriculum since about the time they started sending telegraph signals across the country, putting the pony express out of business!

The information and anecdotal stories I've heard point to extended bandwidth having very little to do with the idea of audibility of signals above 20k, but more that having that bandwidth moves a lot of intractable electronic component design problems up out of the pass band...such as filter ripple, slew rate problems, pre-echo of IIR, phase anomalies at the crossover points of filters, etc.

This is the thinking and modus operandi of manufacturers who make gear for serious audio production.  What it translated to in the marketing materials aimed at audiphile home theater listeners, is a world I spend very little time in. 

As I've said before, I think it is a fundamental mistake to conflate the professional audio production market with the audiophile market...the needs and goals are entirely orthogonal to each other.

Your thoughts?

AES 2009 Audio Myths Workshop

Reply #118 –
Quote from: dwoz on
Quote from: Notat on
Quote from: dwoz on
I must confess...I've never heard of "linear distortion".  What is it?  Isn't ALL distortion by definition non-linear?

Quote from: Arnold B. Krueger on
Systems can have only 4 general kinds of signal response faults: linear distortion (frequency and/or phase response errors) , nonlinear distortion, random noise, and coherent interfering signals.

Maybe you wouldn't classify frequency response errors as "distortion". Not everyone does. It's a just a terminology thing. Nothing to get hung up on.



So, basically, when Arnold says "linear distortion" then says "non-linear distortion", he means the same thing?


Not at all.

I believe that the following reference has just lately been cited, but the question above shows that people must not be taking the citation very seriously:

link to Wikpedia article about distoriton

So, to repeat something that is pretty fundamental in audio:

1. Linear distoriton - signal processing errors that are commonly quantified by frequency response and phase response estimates. 

2. Nonlinear distoriton - processing errors that are commonly quantified by THD, IM, jitter, and flutter and wow.

3  Noise - random signal source that in nature usually has a thermal origin. Pseudorandom noise is usually the reult of trying to approximate random noise by digital means.

4. Interferring signals - things like power line hum, communication signals, harmonics from switchmode power supplies, etc.

AES 2009 Audio Myths Workshop

Reply #119 –
Quote from: Arnold B. Krueger on
Quote
Second you're assuming that all production examples of a specific part will always adhere strictly to their published spec.


Not really.

1. Many important properties of much are gear are simply not fully specified.
2. Specified performance is often far better than the minimum required to be effective.
3. Much equipment performs far better than specified, in many ways.
4. The performance of much digital equipment is unbelievbly consistent. For example, noise floors are often created by digital means and are therefore identical for every piece of equipment that works at all.



That's a pretty bold statement, pardner!  If I was to post this, you would jump on me and ride me like a circus pony, for offering a bunch of unsubstantiated conjecture.

Would you like to offer a bit of supporting documentation or evidence for this rather bold assertion?

I will offer a bit of my own, to start the ball rolling:  The Creative Labs soundblaster card has recently-published specs that state that the card's A/D and D/A produce THD + IMD + Noise of 0.002%.  By anyone's standards, that's pretty spankin' good!  TWO THOUSANDTHS of ONE PERCENT. 

Wow.

HOWEVER....you get down into the "mice type" at the bottom of the spec sheet, buried in the legal stuff and sales contact information, is a little "qualifier":  the "reference signal" for the specification, is a 1kHz sine wave at full input range level.

In other words, if you want to reproduce 1kHz sines, then BUY THIS CARD.  It will excel.  But if you attempt to put anything remotely resembling a music program through it, well, it's caveat emptor.

Your thoughts?  I'd like to see your examples too!

AES 2009 Audio Myths Workshop

Reply #120 –
Quote from: dwoz on
As I've said before, I think it is a fundamental mistake to conflate the professional audio production market with the audiophile market...the needs and goals are entirely orthogonal to each other.

Your thoughts?

If by orthogonal you mean that professional audio equipment must work well and audiophile equipment is used to pick people's pockets then I quite agree. Other than that there should NOT be any great difference between them.

AES 2009 Audio Myths Workshop

Reply #121 –
Quote from: dwoz on
But I'm wondering.  You seem to be making a very sweeping and absolute statement here.  The way I'm reading this, you aren't saying anything about audibility or "as far as is reasonably necessary", you're saying flat-out that they are linear, period.  Is that a correct reading of your intent?

Yes. Mathematicians are in it for the sweeping and absolute statements. The math doesn't lie (I've shown you the proof) and the processor does the math correctly. The only place audibility is involved is in choosing sample rate (bandwidth) and bit resolution (S/N ratio).

AES 2009 Audio Myths Workshop

Reply #122 –
Quote from: pdq on
Quote from: dwoz on
As I've said before, I think it is a fundamental mistake to conflate the professional audio production market with the audiophile market...the needs and goals are entirely orthogonal to each other.

Your thoughts?

If by orthogonal you mean that professional audio equipment must work well and audiophile equipment is used to pick people's pockets then I quite agree. Other than that there should NOT be any great difference between them.


I mean the word "orthogonal" in the taxonometric sense.  And, I specifically said "market".  A particular piece of equipment is just a particular piece of equipment, and it's suitability of purpose in any given market is defined by the market, not the equipment.  In an absolute sense, it either works, or it doesn't.

AES 2009 Audio Myths Workshop

Reply #123 –
Quote from: dwoz on
Quote from: Arnold B. Krueger on
A defined pass band is a natural property of *any* real world system. Therefore presuming one does not necessarily require any specific qualifiers at all.

Since we (hopefully) all know that we are talking about audio, we also know that the traditional audio passband is 20Hz - 20KKz.  As a rule, digital systems lack inherent, practically significant LF roll-offs. That leaves only 20 KHz as a possible area of discussion. We just went through a period of weirdness a fw years back where a few people people tried to profiteer by extending the HF passband well above 20 KHz and making false claims about audible benefits. That madness seems to have pretty much evaporated since then, at least in the mainstream. Furthermore, people who actually invested big money in that weirdness, received negligable rewards, millions were lost by a few large corprations, and a few execuitives simply lost their jobs.

So, the reality is not actually any different from what Notat said,  unless I somehow missed something.



My understanding of the engineering design reason for extended bandwidth systems is really quite different than yours.  Instead of it being some kind of hyperbolic "madness" as you call it, it was really a lot more about improving the linearity of the hardware within the defined passband.


Yes and no.

If it is 1962 and you are Stuart Hegeman and you are designing a Citation II Tubed Power Amp for Dr. S Harmon MD, then you may find that you need to control response up to 50 or 100 KHz in order to have good performance at 20 KHz.
If it is 2010 and you are looking at DSP-based digital  audio hardware to buy, it is entirely possible that equipment with a brick wall filter 100 dB deep at 22 KHz that will have response within 0.1 dB at 20 KHz and its phase response will be "linear phase". IOW, it will behave like a short delay.

Quote
It's no secret that transient response is directly related to bandwidth, after all.


What seems to be a secret is what sort of transient response actually matters for audio. Some people show off the sort of 10 KHz square waves that come out of CD players and want us all to say Yecch! They are misleading people.

Quote
That's been part of signal processing curriculum since about the time they started sending telegraph signals across the country, putting the pony express out of business!


That would relate more to people's thinking before the era of modern psychoacoustics. Read Zwicker and Fastl yet?

Quote
This is the thinking and modus operandi of manufacturers who make gear for serious audio production.


It is the boutique segment of the pro audio world that wants us to look at 10 KHz square waves from CD players and lust after their new "high resolution" toys.  Unfortunately this includes people who should know better.

Quote
What it translated to in the marketing materials aimed at audiphile home theater listeners, is a world I spend very little time in.


To me audio is audio from musican to listener.

Quote
As I've said before, I think it is a fundamental mistake to conflate the professional audio production market with the audiophile market...the needs and goals are entirely orthogonal to each other.


I was very happy to deconstruct Your last thoughts on that, but I don't think you ever responded.

Did you read it before the grass was cut?

AES 2009 Audio Myths Workshop

Reply #124 –
Quote from: Notat on
Quote from: dwoz on
But I'm wondering.  You seem to be making a very sweeping and absolute statement here.  The way I'm reading this, you aren't saying anything about audibility or "as far as is reasonably necessary", you're saying flat-out that they are linear, period.  Is that a correct reading of your intent?

Yes. Mathematicians are in it for the sweeping and absolute statements. The math doesn't lie (I've shown you the proof) and the processor does the math correctly. The only place audibility is involved is in choosing sample rate (bandwidth) and bit resolution (S/N ratio).



very good.  Now, we've determined that as idealized theoretical systems, the digital console or DAW should be perfectly linear.

Now, this was all in the context of whether a hypothetical system is ever attainable in the real world.  Since no DAW can ever be useful if you can't get audio into the darned thing, then we have to include the A/D converters in the model, no?

You've asserted that the systems "do the math correctly".  If I take that as a given, then I ask you to explain about the anti-aliasing filters on those "pesky" converters.  Specifically, most modern designs don't really use analog brick wall filters anymore...too much ripple and phase non-linearity down into the passband.  The analog filter has been replaced or complemented by a digital filter.  But some digital filters introduce their own problems....pre-echo is one.  IIR filters (implemented in the digital side) have minimized pre-echo, but introduce phase anomalies!

Thus, while the idealized console/DAW itself is "perfect", there is NO perfect idealized model of one of the primary, key components, the A/D converter. 

So, I ask...is this important to consider when you make your statement?