[jcx]

 The name Nyquist–Shannon sampling theorem honors Harry Nyquist and Claude Shannon. The theorem was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others. It is thus also known by the names Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, and cardinal theorem of interpolation.

isn't the only useful result, "Shannon-Hartley Channel Capacity Theorem" gives a relation including bit depth and Signal to Noise
 
https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem
 
since all physical systems are both Bandwidth and Noise limited we use both the Sampling and Channel Capacity Theorems for Digital Audio, add Analog filters to ADC and DAC signal processing to make the real better fit the Math
 
all of physical acoustics, microphone, electronic, recording and listening room noise need to be considered, as well as human hearing thresholds - all by frequency too
 
 
quantization is mathematically approximated as added noise - for some purposes - for human hearing however the correlation of quantization error to the signal is easily heard - which leads to discussions of masking and the unobvious different (and psychoacousticaly better) use of dither(s)

 

[RRod]

  quantization is mathematically approximated as added noise - for some purposes - for human hearing however the correlation of quantization error to the signal is easily heard - which leads to discussions of masking and the unobvious different (and psychoacousticaly better) use of dither(s)

 
It's a rare track of mine where truncating without dither to 16 bits produces easily-audible correlated errors. Many times the examples I see are for low bit depths, where in all fairness even dithering can be easily audible depending on the particulars.

 

[jcx]

there are situations where the recording noise floor may well mask quantization errors, yet Bob Katz devotes a chapter in his 3rd ed Mastering Audio book, blog entries to dither
 
many (most?) pros do use dither today when creating 16 bit releases, DAW software, their manuals, workshops all incorporate, document, teach the use of dither in Digital Audio Mastering
 
understanding all the pieces of noise, quantization, bit depth and perceptual limits could let you drop dithering in some recording/editing/mastering situations
 
but it really seems a minority position today with TPDF flat/white dither inside DAW when any manipulation is taking place and results only saved to 32 or 24 bits and when the perceptual "cost" of advanced psychoaucousticaly shaped final dither to 16 bits is so low

 

[RRod]

  there are situations where the recording noise floor may well mask quantization errors, yet Bob Katz devotes a chapter in his 3rd ed Mastering Audio book, blog entries to dither
 
many (most?) pros do use dither today when creating 16 bit releases, DAW software, their manuals, workshops all incorporate, document, teach the use of dither in Digital Audio Mastering
 
understanding all the pieces of noise, quantization, bit depth and perceptual limits could let you drop dithering in some recording/editing/mastering situations
 
but it really seems a minority position today flat with dither inside DAW when any manipulation is taking place and results only saved to 32 or 24 bits and when the perceptual "cost" of advanced psychoaucousticaly shaped final dither to 16 bits is so low

 
Well there's obviously no reason NOT to do it; I'm just speaking to audibility of the last dither+shaping to 16 bits. Pros are in quieter environments than I (or most people) and are probably just a tad more nit-picky and thus do things like amp-up reverb tails beyond normal listening levels. That doesn't change the fact that for little people like me, truncation to 15 or even 14 bits is inaudible for the vast majority of my material in my normal listening environments at my optimal listening level. That would probably hold for a goodly number of albums honed by Mr. Katz himself. Heck I can get some stuff down to 8 bits (yay loudness wars). But for my more dynamic classical tracks, with minimal RMS around -65dBFS, 14-bits without dither typically holds up just fine in my non-treated listening room. 

 

[gregorio]

Originally Posted by SoundAndMotion /img/forum/go_quote.gif
 
If quantisation error (noise) damages the information, you don't have 100% anymore.

 
Quantisation error (noise) does not damage the information, 100% of the information is still there. This is a basic tenet of the proof developed by Shannon! It's also why SACD works with only one bit, even though the quantisation error (noise) is relatively massive.
 

This requires infinite time for complete accuracy when you look at the differential equations but obviously approximations are acceptable once they remain below audibility.

 
Although this statement is unusually (for you) correct, it is unfortunately irrelevant. I was addressing Biggerhead's assertion that infinite bit depth was required for complete accuracy, not infinite time!
 
G

 

[mmerrill99]

If quantisation error (noise) damages the information, you don't have 100% anymore.

Quantisation error (noise) does not damage the information, 100% of the information is still there. This is a basic tenet of the proof developed by Shannon! It's also why SACD works with only one bit, even though the quantisation error (noise) is relatively massive.

If you want to just deal with mathematics rather than the physical realisation of the maths then this paper from Stanley P. Lip****z and John Vanderkooy proves mathematically that 1-bit sigma-delta converters cannot be dithered 100% correctly because if done so they are in constant overload. "Single-stage, 1-bit sigma-delta converters are in principle imperfectible. We prove this fact. The reason, simply stated, is that, when properly dithered, they are in constant overload. Prevention of overload allows only partial dithering to be performed. The consequence is that distortion, limit cycles, instability, and noise modulation can never be totally avoided."

This requires infinite time for complete accuracy when you look at the differential equations but obviously approximations are acceptable once they remain below audibility.

Although this statement is unusually (for you) correct, it is unfortunately irrelevant. I was addressing Biggerhead's assertion that infinite bit depth was required for complete accuracy, not infinite time!

G

Well, mathematically you are wrong with regard to 1-bit depth as the above paper shows & fundamentally you are incorrect that we can realise a 100% accurate recreation of the analogue waveform. In reality, we are dealing with an approximation of the waveform.

 

[coli]

1-bit sigma-delta converters sucks, lost almost $1000 on one. And yeah, the noise level was ridiculous on it.
 
Multi-bit is the future (and the past), now I realize why the industry moved in the direction it did. Most DAC today are hybrid multi-bit R2R/delta sigma implementation, best of both worlds, all while maintaining accuracy and low noise. Another words, cheap DAC chips are better than the supposedly audiophile ones... Now they just need to offer the option to turn off digital filter/upsampling, and it would be perfect. And a lot of new ones by the major manufacturers are starting to offer that!

 

[sonitus mirus]

  1-bit sigma-delta converters sucks, lost almost $1000 on one. And yeah, the noise level was ridiculous on it.
 
Multi-bit is the future (and the past), now I realize why the industry moved in the direction it did. Most DAC today are hybrid multi-bit R2R/delta sigma implementation, best of both worlds, all while maintaining accuracy and low noise. Another words, cheap DAC chips are better than the supposedly audiophile ones... Now they just need to offer the option to turn off digital filter/upsampling, and it would be perfect. And a lot of new ones by the major manufacturers are starting to offer that!

 
To me, muli-bit is similar to vinyl with regards to any future.  It will be a niche product sold to a relatively few for a heck of a lot.  It seems too expensive to implement properly with no real benefits to my ears, only my eyes.

 

[ralphp@optonline]

  1-bit sigma-delta converters sucks, lost almost $1000 on one. And yeah, the noise level was ridiculous on it.
 
Multi-bit is the future (and the past), now I realize why the industry moved in the direction it did. Most DAC today are hybrid multi-bit R2R/delta sigma implementation, best of both worlds, all while maintaining accuracy and low noise. Another words, cheap DAC chips are better than the supposedly audiophile ones... Now they just need to offer the option to turn off digital filter/upsampling, and it would be perfect. And a lot of new ones by the major manufacturers are starting to offer that!

 
 

   
To me, muli-bit is similar to vinyl with regards to any future.  It will be a niche product sold to a relatively few for a heck of a lot.  It seems too expensive to implement properly with no real benefits to my ears, only my eyes.

Now these two quotes seem to contradict each other. So which one is correct?
 
Let's start with the easy one: are multi-bit converters inexpensive or expensive? And are multi-bit converters the past or the future?
 
I don't know the answers myself but seeing these two conflicting posts, I just had to ask. Any help would be appreciated.

 

[sonitus mirus]

   
 
Now these two quotes seem to contradict each other. So which one is correct?
 
Let's start with the easy one: are multi-bit converters inexpensive or expensive? And are multi-bit converters the past or the future?
 
I don't know the answers myself but seeing these two conflicting posts, I just had to ask. Any help would be appreciated.

 
Schiit sells versions of their DACs that have a more expensive multi-bit option, so I assume that multi-bit is more expensive, at least from a development perspective.  Multi-bit was generally superseded by sigma-delta, so it definitely had a past.  The audiophile world has brought back multi-bit to the limelight as a new, superior product, but nobody has shown that they make any appreciable difference that can be heard.  Even Schiit won't say which sounds better, and they claim it is a policy, but it speaks volumes.  If it was obvious, why not state it, especially if you are going to charge significantly more for multi-bit?
 
There was a giant discussion about it not too long ago.  The thread probably got locked for the usual reasons from the usual suspects.

 

[ralphp@optonline]

   
Schiit sells versions of their DACs that have a more expensive multi-bit option, so I assume that multi-bit is more expensive, at least from a development perspective.  Multi-bit was generally superseded by sigma-delta, so it definitely had a past.  The audiophile world has brought back multi-bit to the limelight as a new, superior product, but nobody has shown that they make any appreciable difference that can be heard.  Even Schiit won't say which sounds better, and they claim it is a policy, but it speaks volumes.  If it was obvious, why not state it, especially if you are going to charge significantly more for multi-bit?
 
There was a giant discussion about it not too long ago.  The thread probably got locked for the usual reasons from the usual suspects.

Thanks. Now let's hope this doesn't lead to another flame war - which is why I posted (hopefully) simple questions with (hopefully) simple answers.

 

[sonitus mirus]

 
Thanks. Now let's hope this doesn't lead to another flame war - which is why I posted (hopefully) simple questions with (hopefully) simple answers.

 
This was the thread I was referencing:
 
http://www.head-fi.org/t/779572/r2r-multibit-vs-delta-sigma-is-there-a-measurable-scientific-difference-thats-audible
 
There is a lot of good information, especially in the beginning.  Even developers joined in and provided some measurements and commentary.