[71 dB]

Can you please dial down the drama.

Already feeling better so yes, I can. I'm done with µ-law.

 

[gregorio]

[1] In perceptual coding, audio data is selectively removed (compressed) based on how unlikely it is that a listener will notice the removal. ...That [definition] is true for modern codecs, but it is not a general rule/definition.
[2] In your exchange with ILoveMusic, there are several ideas that are unclear, misleading, garbled or incorrect. Not all of that from you(!) in the above quote(!!), but rather in the entire exchange.
If you’re so inclined, perhaps you can comment on the following facts:

1. That definition doesn't work though. For example, reducing a 24bit signal down to 16bit with the application of noise-shaped dither would also qualify as "perceptual coding" and therefore most CDs produced in the last 20 years or so are "perceptual coding". In modern usage, the term "perceptual coding" is used specifically as I stated and does not include redbook CDs.

2. I made it clear a number of times that I was not attempting to be absolutely accurate, instead, I was looking for a view or "visualization" which would help him understand some of the fundamental basics. For example, in that same post I stated "BTW, all the above is not exactly correct or incorrect either! It's just another way of looking at the issue, a way which avoids some incorrect conclusions/assumptions." with this in mind, I'll address your points:

1. ... The analog signal from a microphone, the signal on an analog interconnect, the output of a DAC or the signal on the speaker wire from an analog amplifier are all voltage amplitude vs. time (time domain). Frequency information is not available unless one transforms the signal, using a spectrum analyzer for analog data or a Fourier transform for digital data.
2. 16 bit in not perfect. [2a] If I have an original signal, convert it to 16 bit resolution, and use that to create a reproduced signal, the original and reproduced will not be identical. That is, subtracting the two does not give all zeroes.
3. 1 bit delta-sigma coding (used in DSD and SACD) is not the same as 1 bit LPCM coding. Usually, talking about 16 bits implies linear pulse code modulation.
4. Shannon-Nyquist tells us we need to sample at greater than twice the highest frequency of interest, not greater than or equal to twice. Twice the highest frequency is inadequate.

1. That was exactly my point. All that is actually measured/stored in the digital domain is amplitude vs. time. Frequency is derived from that "amplitude vs time" information, so effectively there is only one domain "amplitude vs time".
2. 16bit is perfect, any bit depth it's perfect and to suggest otherwise requires disproving the Nyquist/Shannon Theorem. However ..
2a. Two points: Firstly, if you convert your original signal to 16bits and then reproduce that signal you are going to need an Analogue to Digital Converter and a Digital to Analogue Converter, both of which, by definition, are not just digital devices but analogue devices as well. If you compare an input signal to an output signal you are therefore not only evaluating 16bit digital but 16bit plus two analogue stages, neither of which can be absolutely perfect. Secondly and in addition, while 16bit is perfect and Nyquist/Shannon and the Sampling theory are true/correct, that does NOT mean that the theory is perfectly implemented in ADCs and DACs, there is of course variations of implementations between different makes and models. In fact, some audiophile DACs not only didn't implement the theory perfectly, they deliberately broke the theory; Filter-less NOS Dacs for example. If we're going to be precise about it, no DAC implements the theory absolutely perfectly, a perfect Sinc function is impractical for example. In practise, even very cheap implementations can get surprisingly close to perfect.
3. Both (DSD and PCM) follow the same basic rules of digital audio theory. If we're going to get into the fine detail though, then the situation is rather complex and the lines between delta-sigma and PCM become blurred. For example, for 25 years or more all professional PCM ADCs (as far as I'm aware) actually digitize the input signal using a form of delta-sigma encoding, typically a handful of bits with sample rates in the many megahertz range and then decimate down to the user defined PCM sample rate and bit depth.
4. Again, yes if we're going to be precise about it but for the sake of discussion and a simple "view", 2x sample rate is acceptable. In effect, the difference we're talking about is a Nyquist Point for CD (44.1kS/s) of 22.04kHz instead of 22.05kHz.

G

 

[bigshot]

We were discussing Nyquist and reproducing upper harmonic frequencies. That's sampling rate, not bit rate.

Assuming the normal room tone of a room being somewhere above 30dB, in order for a 96dB dynamic range to be audible above the room tone all the way down to -96dB, you would need to be into the volume range range pushing past the threshold of pain. I sure don't listen to music that way,

Redbook is already overkill.

 

[gregorio]

[1] That is why, since the early 1990s I guess, production has moved from 16/44.1/-96 to 24, 32-float/44.1, 88.2, 96, 192/-144 to -infinity dynamic range, instead of remaining at the same specs as Redbook, as it did through most of the eighties.
[2] Every time processing is done, in the digital domain, quantization occurs, and thus dither must be applied.

1. Not really. In the '80's the vast majority of production (mixing) was still analogue, even though the recording was often digital. In the 1990's recording progressed from 16bit to 24bit, not because 24bit had more accuracy/sound quality than 16bit but simply because it provided far more headroom, making multi-channel recordings much easier.

2. No, that's not correct. Yes, every time processing is applied in the digital domain quantisation error occurs but dither does NOT need to be applied. In the '90's, commercial digital mix environments were usually 48bit fixed or 32 float. Many hundreds of processing steps would be required for the cumulative quantisation error to get even near audibility and therefore dither is not typically required. Today (and for quite a few years), commercial mix environments are virtually all 64bit float and the quantisation error is so tiny, you'd need to sum together thousands of processing steps to even get within the theoretical limits of 24bit!

[1] But bit-depth(8, 16, 24, etc) does determine the amount of quantization error/noise. So even if conditions(within the bandwidth of the sampling rate) are met, there will always be 'noise'. 48dB down from full-scale for 8bit, 96dB down from full-scale for 16bit, and so on.
[2] 96dB dynamic range? Under any conditions? You're simply wrong. Upon what do you base your statement?

1. Mmmm, that's not exactly correct. If it were correct, 1bit (SACD) would be un-listenable with only 6dB of dynamic range. Of course, SACD is very listenable, the noise is effectively a great deal further down than -6dB, due to noise-shaped dither. Therefore:

2. Even in most top class commercial recording studios a 96dB dynamic range is typically more than uncomfortable. With a studio noise floor of say 20dB, a 96dB range above that would put the peaks at 116dB which is not far off the threshold of pain, let alone "comfortable". However "96dB dynamic range" is rather arbitrary, referring to the point above, the dynamic range of CD/16bit is not 96dB, it's effectively about 120dB, due to noise-shaped dither. Can you think of any conditions where a dynamic range of 120dB would be comfortable? In practice, virtually all commercial audio is kept within a 60dB dynamic range, as more than that would be uncomfortable for the majority of consumers.

G

 

[TheSonicTruth]

1. Not really. In the '80's the vast majority of production (mixing) was still analogue, even though the recording was often digital. In the 1990's recording progressed from 16bit to 24bit, not because 24bit had more accuracy/sound quality than 16bit but simply because it provided far more headroom, making multi-channel recordings much easier.

2. No, that's not correct. Yes, every time processing is applied in the digital domain quantisation error occurs but dither does NOT need to be applied. In the '90's, commercial digital mix environments were usually 48bit fixed or 32 float. Many hundreds of processing steps would be required for the cumulative quantisation error to get even near audibility and therefore dither is not typically required. Today (and for quite a few years), commercial mix environments are virtually all 64bit float and the quantisation error is so tiny, you'd need to sum together thousands of processing steps to even get within the theoretical limits of 24bit!



1. Mmmm, that's not exactly correct. If it were correct, 1bit (SACD) would be un-listenable with only 6dB of dynamic range. Of course, SACD is very listenable, the noise is effectively a great deal further down than -6dB, due to noise-shaped dither. Therefore:

2. Even in most top class commercial recording studios a 96dB dynamic range is typically more than uncomfortable. With a studio noise floor of say 20dB, a 96dB range above that would put the peaks at 116dB which is not far off the threshold of pain, let alone "comfortable". However "96dB dynamic range" is rather arbitrary, referring to the point above, the dynamic range of CD/16bit is not 96dB, it's effectively about 120dB, due to noise-shaped dither. Can you think of any conditions where a dynamic range of 120dB would be comfortable? In practice, virtually all commercial audio is kept within a 60dB dynamic range, as more than that would be uncomfortable for the majority of consumers.

G

Man you just love saying "No", "Not correct". Maybe I just got the timelines a little wrong??

Well, the information I posted, that you claim is "not correct", is being retransmitted all over the web, on my Facebook pages and blogs. And guess what Gregorio: I'm getting people to change both how they buy music and what they listen to it on!

They, like I, are tired of having our chains jerked.

 

[castleofargh]

most of us understand that there isn't much of anything perfect going on anywhere from recording down to playback output. I understand how some find weird that @gregorio would insist on some perfect aspects of digital audio so often while it's obvious that he himself recognizes the limits of a practical application as he just did. but here is why I believe he does that and is right to do so(maybe I'm wrong, he'll tell me^_^). to me it's about an important aspect of digital audio that fails to get across when we stop at visualizing the signal as sampled dots on a graph over time. digital audio is more than that, it's supported by wave properties.
for example, we know that we don't have the ability to create missing data out of nothing. we can add stuff but it won't be the previously lost data. if I delete half of my post(real delete, so no CTRL+Z), there is no simple math trick to reconstruct it or part of it. with that in mind, the intuitive idea of a quantification to N bits, could be that all the data below the least significant bit is lost. be it the quieter signals, or the extra amplitude accuracy for all the music(same thing). I know how easy it is to think that way. but that intuitive idea is false, and it's easy to demonstrate. based on what we know about data, we shouldn't be able to retrieve information below the LSB if that information was truly lost/cut out/discarded. and yet we do just that all the time with noise shaping. we have defined a lower limit with the bit depth encoding, and then we go retrieve information the least significant bit of that code shouldn't be able to quantify.
that's why even though it can seem like a small distinction in most conversations, or even as zeal in favor of digital audio, the mental model where the signal is perfectly captured, plus some noise, is the better simplified model. that way, when we move the noise around and the original signal appears in the audible frequency range, it actually makes sense instead of feeling like it's witchcraft. and again DSD is the living extreme example of that fact. we encode a 1bit signal, yet we can use that to reconstruct stuff with better than 100dB of dynamic/SNR in the audible frequency range. and adding even more samples allows, after some fooling around, to retrieve even lower levels. showing that the bit depth wasn't in the first place putting any strict limit of how accurate the amplitude of the captured signal can be. the signal was really there all along plus some noise.


a counter example to that would be to filter out a 60khz signal and encode at 44.1khz. no matter how many bits we'd use to record, we wouldn't reconstruct that 60khz tone. the data this time has effectively been discarded. which agrees with the sampling theorem. 44.1khz sampling isn't able to completely capture a 60khz signal. and the band limiting makes very sure of that. like my half deleted post, this is a concrete examples of what happens with effectively lost data. it's pretty much lost. ^_^

so IMO all this is making a strong case in favor of looking at quantization noise, as noise over a fully captured signal within the allowed frequency range. we can't really capture anything perfectly or reconstruct perfectly in the real world, but there is meaning in avoiding the assumption that we're just dealing with quantization errors below which the signal is forever lost/discarded.



does this make sense, or did I just enter the twilight zone in my own head?

 

[bigshot]

Even without dithering, there is more than enough oomph in redbook to reproduce recorded music perfectly for the purposes of listening to music in the home. I think there is a test of that in one of the videos in my sig. How many times does the lily need to be gilded?

 

[jgazal]

(...) Anyone...even you!....could write an article. (...)

most of us understand that there isn't much of anything perfect going on anywhere from recording down to playback output.
(...)
does this make sense, or did I just enter the twilight zone in my own head?

I identify with "writing articles" from the "twilight zone in my own head"!

 

[SoundAndMotion]

That definition doesn't work though. For example, reducing a 24bit signal down to 16bit with the application of noise-shaped dither would also qualify as "perceptual coding" and therefore most CDs produced in the last 20 years or so are "perceptual coding". In modern usage, the term "perceptual coding" is used specifically as I stated and does not include redbook CDs.

Well, when I found it here, I thought it caught the gist of the idea, but you have indeed found a loophole. But your loophole does not only use the "perceptual" (noise-shaped dither) part of the definition, your argument includes the idea that RBCDs use "lossy compression". It it were 2 in the morning in the dorms, the pizza just arrived, and we just reloaded the bong or grabbed a six-pack, I'd be up for arguing for hours whether down-sampling and truncation to create CDs constitutes "lossy compression". But I moved out of the dorms 40 years ago and haven't "smoked" in nearly as long, so I'm not interested in this idea.

I made it clear a number of times that I was not attempting to be absolutely accurate, instead, I was looking for a view or "visualization" which would help him understand some of the fundamental basics. For example, in that same post I stated "BTW, all the above is not exactly correct or incorrect either! It's just another way of looking at the issue, a way which avoids some incorrect conclusions/assumptions."

Cool, that's a neat trick. Indeed, I didn't read that part. If you put that in your sig, you never have to worry about being wrong again. You just say "Dude, read my sig". Sweet.
As for helping "him understand some of the fundamental basics", how can you explain the need for anti-aliasing filters, the idea of frequency folding, the need for anti-imaging (reconstruction) filters, even really fundamental stuff like bandwidth and frequency response without the frequency domain?

1. That was exactly my point. All that is actually measured/stored in the digital domain is amplitude vs. time.
1a.Frequency is derived from that "amplitude vs time" information, so effectively there is only one domain "amplitude vs time".

1a. I don't even get how you can make this mistake. Freq. and time domains are 2 domains. There are complete textbooks on freq. domain analysis, that don't work in the time domain, and vice versa. I mentioned convolution vs. point-by-point multiply... they are different.
1. Yay! I'm so glad you have this brand new perspective that corrects so very many of your previous posts, e.g.

Since the dawn of audio recording right up to the present day we measure/convert and store just two properties, frequency and amplitude.

among so many dozens more over the years. We don't measure/convert frequency. We measure/convert amplitude as a function of the independent variable, time.

2. 16bit is perfect, any bit depth it's perfect and to suggest otherwise requires disproving the Nyquist/Shannon Theorem. However ..
2a. Two points: Firstly, if you convert your original signal to 16bits and then reproduce that signal you are going to need an Analogue to Digital Converter and a Digital to Analogue Converter, both of which, by definition, are not just digital devices but analogue devices as well. If you compare an input signal to an output signal you are therefore not only evaluating 16bit digital but 16bit plus two analogue stages, neither of which can be absolutely perfect.

2. No, BigShot is right. Shannon-Nyquist deals with sample rate and bandwidth, not bit depth. (@bigshot-I never claimed Nyquist deals with bit depth). You have mistaken the complete lack of bit depth in Shannon-Nyquist to mean bit depth is irrelevant. That is not what it means. Sampling theory deals with periodically taking samples... but garbage-in, garbage-out... the quality (resolution) of the sample determines the quality of the reconstruction. Perfect reconstruction requires perfect samples. I think BigShot may want to add: "But good enough is good enough". Perhaps a dictionary is needed to agree on the meaning of "perfect"! ...?
2a. When talking about the math and theory of Shannon-Nyquist, we are talking analytic, not noisy world. IF the original signal is analog (which I didn't say), then talking about analog electronics can be "straight wire with gain". I was actually thinking of a double- or extended-precision float signal. Convert it to 16-bit integer (scaling properly, of course), and then convert back to the same floats, dither, filter, scale as needed, and the signal will be different. The difference is 5 or 6 orders of magnitude down, but different. If you do the same with 24-bit, the difference will more than 7 orders of magnitude down.
But you say any bit depth is perfect... 1 or 2 bit LPCM will really suck...

Need a break, will answer the rest of your post later.

 

[gregorio]

I understand how some find weird that @gregorio would insist on some perfect aspects of digital audio so often while it's obvious that he himself recognizes the limits of a practical application as he just did. but here is why I believe he does that and is right to do so (maybe I'm wrong, he'll tell me^_^) ...

You're not wrong at all. In fact you've done a very good job of explaining why "I do that". I would like to expand it with a couple of points, which admittedly might add confusion rather than reduce it:

1. Shannon stated: "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." (emphasis is mine) - We have to remember that this statement isn't just an assumption, opinion, idea or suggestion, Shannon proved it mathematically. Furthermore, there's been no hint in the 70 years since he published his proof that it's in anyway incorrect, in fact quite the opposite, his proof forms the basis for Information Theory and therefore of all digital technology. Hence why Shannon is often described as "the father of the digital age". In short, Shannon's statement is logically incontestable, effectively it's an unquestionable absolute truth/fact. It's vital to fully appreciate this, Shannon's statement does not require any additional conditions in order to be true, it is true, period!! For example, note that it does not mention bit depth, that's because it does not need to, Shannon's statement is correct as it stands, period (IE. Regardless of bit depth). In effect, all "bit depth" does is define the amount of noise which accompanies our "complete determination" but the "complete determination" is always there and to suggest otherwise effectively means disproving what Shannon has already proven.

2. In addition to the previous point, another reason for "visualising" the issue as: "A signal that is perfectly captured (completely determined) plus some noise", is that this "noise" is a unique case. This noise is generated by/the result of a mathematical process, so within that process this noise can be manipulated independently from the signal, it can be "Shaped". All other noise cannot be "shaped", it's already part of the signal (for example, noise floor of the recording venue or thermal noise from the mic's and other analogue equipment in the recording chain) and cannot be manipulated independently.

[1] Man you just love saying "No", "Not correct". [1a] Maybe I just got the timelines a little wrong??
[2] Well, the information I posted, that you claim is "not correct", is being retransmitted all over the web, on my Facebook pages and blogs.
[2a] And guess what Gregorio: I'm getting people to change both how they buy music and what they listen to it on!
[3] They, like I, are tired of having our chains jerked.

1. That is NOT true! I do not love saying "no" or "not correct" and you obviously don't like being told you are incorrect/wrong, so why don't you do us BOTH a favour and not post incorrect information/assertions in the first place??
1a. No, it wasn't just the timeline, it was your whole "guess". Except in a tiny number of special cases, production could not "remaining at the same specs as Redbook, as it did through most of the eighties" because production has never been "at the same specs as redbook", not during the '80's or any other time. Production was analogue and when it switched to digital it was at higher than 16bit.

2. Good choice, Facebook probably is the best place for your information, due to it's reputation for disseminating "Fake News".
2a. Great! BTW, when you've got say about twenty million people "to change" be sure to let the major record labels know!

3. I don't like having my chains jerked either but for me, "fake news" IS having my chains jerked. I realise though that many people seem to prefer "fake news" to the actual facts, presumably because it doesn't take much effort or intellectual ability. So I hope "they, like you" are very happy together.

G

 

[SoundAndMotion]

Aha, no response to my post... understood. Being embarrassed is not conducive to answering. No point for me to continue on that post...

In short, Shannon's statement is logically incontestable, effectively it's an unquestionable absolute truth/fact. It's vital to fully appreciate this, Shannon's statement does not require any additional conditions in order to be true, it is true, period!! For example, note that it does not mention bit depth, that's because it does not need to, Shannon's statement is correct as it stands, period (IE. Regardless of bit depth). In effect, all "bit depth" does is define the amount of noise which accompanies our "complete determination" but the "complete determination" is always there and to suggest otherwise effectively means disproving what Shannon has already proven.

Not sure, to whom that's directed. I'm not contesting Shannon; I'm contesting your understanding of several issues related to digital audio, including Shannon (my underline in your text). "Complete determination" plus a bunch of noise destroys the "complete determination", so it's wrong. It is not perfect.
Should we make it interesting?
-I will make a signal with a bandwidth less than 20kHz and duration of 1 second.
-You tell me exactly how you want me to prepare it, consistent with a simple test of your contention that bit depth doesn't alter the ability for "complete determination". Anything you want, ending with me sending you a file that is encoded at 4-bits and sampled at 44.1kHz. I will document each step with an intermediate, modified data file.
-You recreate my signal, showing a "complete determination" of my signal. If some of the important features of my signal are buried in noise missing, that is not a success.
-We post the results here.
-If you do succeed... what should we do? Have a bet? You succeed, I send you money; you don't succeed, you send me money? Or we send to a charity of the winner's choice? Or? How much? 10, 100, 1000? Euros, dollars, pounds?

I'm confident. Are you?

Edit: strikethrough buried in noise; replaced with "missing"

 

[71 dB]

2. No, BigShot is right. Shannon-Nyquist deals with sample rate and bandwidth, not bit depth. (@bigshot-I never claimed Nyquist deals with bit depth). You have mistaken the complete lack of bit depth in Shannon-Nyquist to mean bit depth is irrelevant. That is not what it means. Sampling theory deals with periodically taking samples... but garbage-in, garbage-out... the quality (resolution) of the sample determines the quality of the reconstruction. Perfect reconstruction requires perfect samples. I think BigShot may want to add: "But good enough is good enough". Perhaps a dictionary is needed to agree on the meaning of "perfect"! ...?.

Output = original signal + noise. How do you define signal and noise? Original signal is what you put in (input signal) and noise is everything else. So, by definition all inaccuracies of the output IS noise while the signal part is perfect. That's why bit depth doesn't make the signal any less perfect. Lower bit depth just increases inaccuracies => increases noise.

Original signal is bandlimited => Shannon-Nyquist applies to it
Noise is bandlimited => Shannon-Nyquist applies to it
Output signal = original signal + noise is bamdlimited => Shannon-Nyquist applies to it

Output signal is such a bandlimited signal that is able to be fitted into the bit depth while being as close to the original signal as possible. So in the quantization process noise was added to the input signal so that the sum becomes output signal and can be stored to the bit depth available. The signal is perfect bandlimited itself all the time. Only noise is added to it. When the bit depth is large enough, this noise is below the threshold of hearing and only signal is heard, and that signal is perfect (in theory, in practice ADCs and DACs are not perfect and can introduce audible errors, especially during the early years of digital audio).

 

[RRod]

Doesn't seem to me that greg is trying to say that Q(x) = Q(x + n), if Q is a quantifier and n is noise we add intentionally. I read this as "x + n still has x in it, and we can choose n so that we can hear x audibly perfect after quantization."

*edit: @71 dB was on it

 

[71 dB]

-I will make a signal with a bandwidth less than 20kHz and duration of 1 second.
-You tell me exactly how you want me to prepare it, consistent with a simple test of your contention that bit depth doesn't alter the ability for "complete determination". Anything you want, ending with me sending you a file that is encoded at 4-bits and sampled at 44.1kHz. I will document each step with an intermediate, modified data file.
-You recreate my signal, showing a "complete determination" of my signal. If some of the important features of my signal are buried in noise missing, that is not a success.
-We post the results here.
-If you do succeed... what should we do? Have a bet? You succeed, I send you money; you don't succeed, you send me money? Or we send to a charity of the winner's choice? Or? How much? 10, 100, 1000? Euros, dollars, pounds?

I'm confident. Are you?

At 4-bit there is about 24 dB of dynamic range, so of course all the stuff below that is likely to be lost in the noise. So, what are these important features? However, it's perfect signal + really loud noise.

 

[gregorio]

[1] But your loophole does not only use the "perceptual" (noise-shaped dither) part of the definition, your argument includes the idea that RBCDs use "lossy compression".
[2] Cool, that's a neat trick. [2a] Indeed, I didn't read that part. [2b] If you put that in your sig, you never have to worry about being wrong again.

1. My argument includes the idea of selectively removing data, namely the last 8 LSBs, not specifically that RBCDs use "lossy compression". With a noise-shaped RBCD the data we're removing only represents unwanted noise, with lossy compression the data being removed doesn't only represent unwanted noise, it represents actual musical signals (which are masked according to a perceptual model). And BTW, that was just one loophole off the top of my head, I can think of others!

2. Yes, it is but I can't take any credit for it. Creating an over-simplified "visualisation" (which isn't precisely accurate) to help someone understand some basic concept has been around for millennia, probably since humans first developed complex language skills.
2a. Or apparently several other parts/posts, where I made it abundantly clear I was talking in very simplified terms. My initial response (here) to this topic of the thread spelt it out in some detail.
2b. That is an idea but I haven't done it because I'm not always posting "over-simplified visualisations". Instead, I do it the other way around, I try to make it clear when I am posting in over-simplified terms. However, some level of simplification is usually necessary, because this is a forum of mostly laypeople and if I used accurate pro audio engineering terminology (as I would with other professional engineers) some/many would have trouble understanding/following.

1a. I don't even get how you can make this mistake.
1. Yay! I'm so glad you have this brand new perspective that corrects so very many of your previous posts, e.g.
among so many dozens more over the years. We don't measure/convert frequency. We measure/convert amplitude as a function of the independent variable, time. ...
[2] As for helping "him understand some of the fundamental basics", how can you explain the need for anti-aliasing filters, the idea of frequency folding, the need for anti-imaging (reconstruction) filters, even really fundamental stuff like bandwidth and frequency response without the frequency domain?

1a. What mistake, quoting you? "The analog signal from a microphone, the signal on an analog interconnect, the output of a DAC or the signal on the speaker wire from an analog amplifier are all voltage amplitude vs. time (time domain). Frequency information is not available unless one transforms the signal, using a spectrum analyzer for analog data or a Fourier transform for digital data." - Are you now saying it's not "all voltage amplitude vs time"?
1. So you're saying that "amplitude vs time" does not contain frequency information? Presumably you're not saying that, otherwise the second part of your quote is incorrect. Sound waves can be defined by "Frequency + Amplitude" or "Amplitude vs Time" (and separated into the Frequency and Time domains as audio engineering text books do) both are interchangeable and valid, Frequency, by definition, contains time and "Amplitude vs Time" contains frequency. Looking at it one way can help with some explanations and looking at it the other way can help with other explanations. In other words, I do not "have this brand new perspective", I have various different perspectives of the same thing and none of them are new to me!

2. Again, that's the point, I was not trying to explain bandwidth, frequency response, anti alias/imaging, decimation or any other type of filter, just amplitude/bits.

Consistently in this line of "discussion" you are failing to account for the context of my responses or even that they are in fact responses!

2. No, BigShot is right. Shannon-Nyquist deals with sample rate and bandwidth, not bit depth.
2a. I was actually thinking of a double- or extended-precision float signal. Convert it to 16-bit integer (scaling properly, of course), and then convert back to the same floats, dither, filter, scale as needed, and the signal will be different.
[3] But you say any bit depth is perfect... 1 or 2 bit LPCM will really suck...

2. I believe I dealt with this in my additional point #1 in my previous message (response to castleofargh).
2a. Of course it will be different, as castleofargh explained, if you're going to remove a bunch of bits, the information in those bits are gone for good. Converting 16bit fixed back to 32 or 64bit float isn't ever going to give us that exact same data back. We can, with noise-shaped dither, expose more of the "completely determined" signal (beneath the noise floor imposed by 16bit with TDPF dither) so that in practical application the 16bit conversion sounds exactly the same as the 32/64 bit float original but the data will not be exactly the same (because we're adding noise-shaped dither).

3. It IS effectively perfect but with just one or two bits it would "really suck" because you'd barely be able to hear any of that perfect signal buried in the huge amount of quantisation error/noise. Of course, if you applied noise-shaped dither during the quantisation process then more of our perfect ("completely determined") signal would be exposed, as the quantisation noise is redistributed away from our range of hearing. This is the reason why SACD does not "really suck"!

G