Why We Dither

[panatrope]

Originally Posted by oky****

hi,

thanks for posting information panatrope. however, the above paragraph is kind of unintelligible to me, just because i can't figure out the sentence structure, where the parentheses or quotation marks go, and so forth.

when you get a minute, could you clarify it so i can better understand what you are trying to say. i'm interested, but i don't know if i am reading it correctly.

thanks.



right.

I've edited the original post, to try and improve the clarity. Let me know if I haven't succeeded ...

[okydoky]

Originally Posted by panatrope

No, you can't 'average' sampling rate - that's like averaging amplitude readings. The whole issue with sampling is that you sample at a point (the mathematical concept of a location whose area is infinitesimally small), and that point must be accurately located both in amplitude AND time. Any deviation from that ideal is a distortion of the representation of that waveform.

hi,

i totally understand what you are saying, and i am not really disagreeing, but i know i read something a while back about using an "averaging scheme" for sample points. seems like it was from some reputable source [whatever that means ].

but its not worth talking about unless i find the information that i am talking about. if i find it, i will post it at some point.

[edit]
actually, now i think i remember. the averaging thing is the basis of the ability to reconstruct musical waveforms. nyquist / shannon actually only pertains strictly to periodic signals, as does the pertinent fourier stuff, apparently. music is generally comprised of non-periodic signals. i will look for more info on the issue, or perhaps one of the other posters has it handy.
[edit]


right.

[David Rick]

Originally Posted by oky****

what i am trying to get at is the following question:

"if one were able to magically remove only the quantization error signal from the output of an undithered sampler [leaving everything else untouched], would one be left with

(a.) a perfect linear representation of the amplitude of the input signal, or,

(b.) a representation equaling the amplitude of the input signal minus the amount / amplitude of the quantization error."

If you are willing to posit an ideal sampler with infinite resolution, then the answer is (a). But its output could not be limited to a finite number of digits, or there would still be quantization error. "Ideal samplers" exist as abstract objects in Linear Systems Theory, but they don't exist in the real world. Also, the output of such a sampler includes spectral replicates, so it can be called a "perfect representation" only if there is no aliasing.

Answer (b) makes no algebraic sense.

i am trying to use a logical thought process to establish the following fact:

even if the quantization error gets "handled" by the dither [or even if it were somehow possible to actually "surgically" remove the quantization error signal], the remaining signal would still not be a perfect linear copy of the input, because the quantizer's bit depth prevents such a signal from actually being represented.

You're asking that we remove all the spectral energy (below Nyquist) that isn't part of the original signal. OK, then all you have left is the original signal (plus spectral replicates), and therefore it can't be quantized anymore. Those stairsteps and the spectral distortion spurs (or noise) are not two distinct things; they are different mathematical representations of the same thing.

David L. Rick

[David Rick]

Originally Posted by oky****

actually, now i think i remember. the averaging thing is the basis of the ability to reconstruct musical waveforms. nyquist / shannon actually only pertains strictly to periodic signals, as does the pertinent fourier stuff, apparently. music is generally comprised of non-periodic signals. i will look for more info on the issue, or perhaps one of the other posters has it handy.

Please erase all these ideas from your head. Every single one of them is wrong! Whoever taught you these things doesn't know enough of the relevant math to be expounding on the subject to begin with.

Fourier analysis works just fine with non-periodic signals. But one needs to switch to the Fourier Transform instead of the Fourier Series. The frequency-domain representation is then continuous instead of having discrete spectral lines, but it still exists. Now it is true that any proper musical piece will be strictly limited in time, and therefore can't be strictly limited in bandwidth. But it can be practically limited in bandwidth because the frequency-domain representation of a finite-length time window falls off at 6 dB per octave. Eventually it falls below the noise floor and you can stop integrating. (Unless, of course, you think the noise floor is part of the music, in which case I have some John Cage albums to sell you.)

Even if you believe Nyquist, Shannon, Kotelnikov, and Whittaker all deserve to rot in hell, that needn't keep you from digitizing audio. There exist various "generalized" sampling theories that don't require the band-limiting assumptions digital audio has grown up with. We could build perfectly servicable A/D/A chains without brickwall filters that would still allow proper reconstruction of audio. The trouble is, such systems would be incompatible with the existing digital audio infrastructure, and would have to exist in their own audio ghettos, connected to the rest of the world by analog "bridges" -- pretty much like those Korg DSD recorders.

David L. Rick

[MichaelPatrick]

Originally Posted by David Rick

(Unless, of course, you think the noise floor is part of the music, in which case I have some John Cage albums to sell you.)

At first I thought I couldn't keep up with you guys. Heck, I can follow this part just fine !

[okydoky]

Originally Posted by panatrope

I've edited the original post, to try and improve the clarity. Let me know if I haven't succeeded ...

hi,

yes, thanks panatope, i think i understand the paragraph now, so hopefully i will be able to understand it. why can't the anti-aliasing filter remove the distortion products?

i'm kind of slammed right now, so i haven't had time to grok everything that's been posted lately. i may have some questions yet.


right.

[d_fu]

Originally Posted by oky****

why can't the anti-aliasing filter remove the distortion products?

Because anti-aliasing has to do with the sample rate, not the resolution/bit depth/word length.

[okydoky]

Originally Posted by d_fu

Because anti-aliasing has to do with the sample rate, not the resolution/bit depth/word length.

hi,

no, come on now. that's irrelevant. a filter is a filter, and it can filter frequencies generated by any process.

some other explanation. the distortion products must be above or below the filter's cut-off or something. ?


right.

[okydoky]

Originally Posted by oky****

hi,

here's another question [basic] for mr. rick:

is it contended [theoretically] that, when quantizing a signal during sampling, the quantizer's output is:

1.) a perfect linear representation of the input signal's amplitude, plus a signal representing the quantization error, or,

2.) a signal that is defective to the extent of the amount of the quantization error plus a signal representing the quantization error?


thanks.



right.

Originally Posted by oky****

what i am trying to get at is the following question:

"if one were able to magically remove only the quantization error signal from the output of an undithered sampler [leaving everything else untouched], would one be left with

(a.) a perfect linear representation of the amplitude of the input signal, or,

(b.) a representation equaling the amplitude of the input signal minus the amount / amplitude of the quantization error."


do you [or anyone] know the answer to that question? the answer would be either (a.) or (b.).



right.

Originally Posted by David Rick

If you are willing to posit an ideal sampler with infinite resolution, then the answer is (a). But its output could not be limited to a finite number of digits, or there would still be quantization error. "Ideal samplers" exist as abstract objects in Linear Systems Theory, but they don't exist in the real world. Also, the output of such a sampler includes spectral replicates, so it can be called a "perfect representation" only if there is no aliasing.

Answer (b) makes no algebraic sense.



You're asking that we remove all the spectral energy (below Nyquist) that isn't part of the original signal. OK, then all you have left is the original signal (plus spectral replicates), and therefore it can't be quantized anymore. Those stairsteps and the spectral distortion spurs (or noise) are not two distinct things; they are different mathematical representations of the same thing.

David L. Rick

hi,

david, i do not understand why you say (b.) makes no "algebraic sense".

i am asking if the undithered output of a sampled signal would be equal to the input signal minus the quantization error, if the signal representing that quantization error had somehow been removed from the output [forget about sampling error or anything of that nature].

at the risk of posing a question with an obvious answer, i am just trying to establish whether or not there may be any significant third factor that may somehow be operative. i believe the answer to my question would be yes.

also, exactly what do you mean by "spectral energy", and "spectral replicates"?

also, what dBfs was the 3.6kHz signal recorded at in your graph example?

thanks.



right.

[Audiop]

Originally Posted by oky****

i am asking if the undithered output of a sampler signal is equal to the input signal minus the quantization error,

Or plus the error.


/Peter

[David Rick]

Originally Posted by oky****

david, i do not understand why you say (b.) makes no "algebraic sense".

i am asking if the undithered output of a sampled signal would be equal to the input signal minus the quantization error, if the signal representing that quantization error had somehow been removed from the output [forget about sampling error or anything of that nature].

at the risk of posing a question with an obvious answer, i am just trying to establish whether or not there may be any significant third factor that may somehow be operative. i believe the answer to my question would be yes.

also, exactly what do you mean by "spectral energy", and "spectral replicates"?

also, what dBfs was the 3.6kHz signal recorded at in your graph example?

The output of a quantizer is always equal to its input plus or minus the magnitude of the quantization error. (This is true on a sample-by-sample basis.) So if you could magically remove the quantization error, you'd have the original signal again. It would be exact. Too bad we can't do magic.

When I speak of spectral replicates, I mean the copies of the spectrum at multiples of the sample rate. Real-world systems generally do sampling and quantization at the same time, so I was assuming sampling in my answer. But one could at least imagine an ideal quantizer that worked in continuous time. In fact, a single-bit one is trivial to build with a comparator chip. In that case there would be no spectral copies, but there would still be extra stuff in the spectrum due to the quantization.

The 3.6 kHz signal was simulated at 0 dBFS.

David L. Rick
Seventh String Recording

[okydoky]

Originally Posted by David Rick

Please erase all these ideas from your head. Every single one of them is wrong! Whoever taught you these things doesn't know enough of the relevant math to be expounding on the subject to begin with.

hi,

not to worry. it is unecessary for me to erase the ideas from my head. that's because most of these ideas are not even written to, or stored in, my head, to begin with.

i have a clone [mr. kerry oky****] that i have been using to deal with all of these potentially confusing ideas and explanations. i have access to the ideas and explanations at all times, but i do not have to be responsible for their upkeep [or accuracy]. kareoky**** has been doing a great job, though.

Originally Posted by david rick

Fourier analysis works just fine with non-periodic signals. But one needs to switch to the Fourier Transform instead of the Fourier Series. The frequency-domain representation is then continuous instead of having discrete spectral lines, but it still exists. Now it is true that any proper musical piece will be strictly limited in time, and therefore can't be strictly limited in bandwidth. But it can be practically limited in bandwidth because the frequency-domain representation of a finite-length time window falls off at 6 dB per octave. Eventually it falls below the noise floor and you can stop integrating. (Unless, of course, you think the noise floor is part of the music, in which case I have some John Cage albums to sell you.)

well, i can confirm that i did not really understand most of that, and neither did kareoky****, but i will probably think about it and figure it out.

i am talking about the fact that most people seem to think that every single complex musical sound can be easily reduced to a bunch of component sine waves, which is apparently not the case. i think dan lavry is one of the people who has frequently mentioned this. something about just considering the preceding samples back to infinity as zero.

and whatever you guys say, mehhh, i do know that i saw something about being able to reconstruct signals from samples taken at irregular intervals by averaging the samples. i cannot say more about it because i have not taken the time to locate the information i am "citing", and i don't want to talk about stuff when i am more clueless than typically.

Originally Posted by david rick

Even if you believe Nyquist, Shannon, Kotelnikov, and Whittaker all deserve to rot in hell, that needn't keep you from digitizing audio.

i do not think they should rot in hell, and that seems like an extreme punishment for merely ruining all music and the recording industry generally.

Originally Posted by david rick

There exist various "generalized" sampling theories that don't require the band-limiting assumptions digital audio has grown up with. We could build perfectly servicable A/D/A chains without brickwall filters that would still allow proper reconstruction of audio. The trouble is, such systems would be incompatible with the existing digital audio infrastructure, and would have to exist in their own audio ghettos, connected to the rest of the world by analog "bridges" -- pretty much like those Korg DSD recorders.

David L. Rick

hmmm. i would like to know more about all that.



right.

[okydoky]

Originally Posted by panatrope

Also bear in mind that distortion generated by truncating at a certain bit number, while being signal-related, is going to generate distortion components above Fs/2 (which cannot be subsequently removed by any anti-aliasing filter).

hi,

i don't understand why a filter cannot remove distortion components above Fs2.

i am assuming that the truncation and generation of the distortion products occurs before the filter.

i seems to me that the aliases would be what the filter would be unable to remove.

what's up with that? maybe i am misinterpreting what you are saying.



right.

[Audiop]

Originally Posted by oky****

and whatever you guys say, mehhh, i do know that i saw something about being able to reconstruct signals from samples taken at irregular intervals by averaging the samples. i cannot say more about it because i have not taken the time to locate the information i am "citing", and i don't want to talk about stuff when i am more clueless than typically.

I have a USB PC oscilloscope that has such a feature. By ofsetting the time of some samples slightly the result is higher effective bandwith.

Don't know how the samples are moved in time, pattern/randomness and I have not really used the function.


/Peter

[okydoky]

hi,

ta da! i found the reference i was talking about.

wikipedia: nyquist-shannon sampling theorum

"Nonuniform sampling

The sampling theory of Shannon can be generalized for the case of nonuniform samples, that is, samples not taken equally spaced in time. Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition[4]. Therefore, although uniformly spaced samples may result in easier reconstruction algorithms, it is not a necessary condition for perfect reconstruction." [Nonuniform Sampling, Theory and Practice (ed. F. Marvasti), Kluwer Academic/Plenum Publishers, New York, 2000]


ipso fatso, oky**** is not crazy. mehh.



right.

[okydoky]

Originally Posted by oky****

hi,

ta da! i found the reference i was talking about.

wikipedia: nyquist-shannon sampling theorum

"Nonuniform sampling

The sampling theory of Shannon can be generalized for the case of nonuniform samples, that is, samples not taken equally spaced in time. Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition[4]. Therefore, although uniformly spaced samples may result in easier reconstruction algorithms, it is not a necessary condition for perfect reconstruction." [Nonuniform Sampling, Theory and Practice (ed. F. Marvasti), Kluwer Academic/Plenum Publishers, New York, 2000]


ipso fatso, oky**** is not crazy. mehh.



right.

and here's an except from an interview with paul frindle in "audio design lline" magazine about cumulative dither noise that you may find interesting or helpful.


"Another similar problem can exist when you dither signals, because of relative correlation. When you dither a number of channels, the dither noise of any one channel should be unrelated and uncorrelated to any other, or you risk the relative correlation of the dither noise starting to become evident. You can end up with statistically partially 'mono' dither-noise, which can close in the stereo effect, especially on fade-outs.

Again, a single channel works fine, a stereo channel apparently sounds fine whilst the music is playing, but as channels build up with the same dither, the stereo begins to close in. We spent a fortune on the R3 finding 256 independent noise sources, because it was a strange problem to solve. It is hard to measure, and the sound problems are something which you often need to be working on day after day before they become clearly apparent.

You feel a sort of unease, that something is wrong, but you can't quite explain what it is. At one time, before we found an easier solution, 30% of the processor of the R3 was dealing with this problem. [And the R3 had about 3000 times the processing power of a current (2005) Pro Tools system. P.N.]"






right.

[David Rick]

Originally Posted by oky****

hi,
"Nonuniform sampling
Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition[4]. Therefore, although uniformly spaced samples may result in easier reconstruction algorithms, it is not a necessary condition for perfect reconstruction." [Nonuniform Sampling, Theory and Practice (ed. F. Marvasti), Kluwer Academic/Plenum Publishers, New York, 2000]

Yes, this is true, but it states a constraint on the average sample rate, which isn't the same thing as saying that one can just average the samples, which one cannot. I think the applicable reconstruction algorithms are probably related to Lagrange interpolators.

The required reconstruction algorithms are so onerous that systems architects use non-uniform sampling only in very special circumstances. (There are a few advanced oscilloscopes that can do a version of this.) If you must digitize a signal that you know will cause severe aliasing, I think you can use non-uniform sampling to spread the alias energy around so it looks more like noise. But I haven't ever studied this topic, so I can say for certain. If you happen to know a lot about the spectrum of the signal you're digitizing, you can often pick a particular uniform sample rate so that the aliases land someplace you aren't worried about. This is the basis of so-called sub-sampling radio architectures.

Sometimes researchers get stuck analyzing non-uniformly-sampled data because that's the only data available, but they don't generally have to process it in real time.

David L. Rick
Seventh String Recording

[okydoky]

Originally Posted by David Rick

Yes, this is true, but it states a constraint on the average sample rate, which isn't the same thing as saying that one can just average the samples, which one cannot. I think the applicable reconstruction algorithms are probably related to Lagrange interpolators.

The required reconstruction algorithms are so onerous that systems architects use non-uniform sampling only in very special circumstances. (There are a few advanced oscilloscopes that can do a version of this.) If you must digitize a signal that you know will cause severe aliasing, I think you can use non-uniform sampling to spread the alias energy around so it looks more like noise. But I haven't ever studied this topic, so I can say for certain. If you happen to know a lot about the spectrum of the signal you're digitizing, you can often pick a particular uniform sample rate so that the aliases land someplace you aren't worried about. This is the basis of so-called sub-sampling radio architectures.

Sometimes researchers get stuck analyzing non-uniformly-sampled data because that's the only data available, but they don't generally have to process it in real time.

David L. Rick
Seventh String Recording

hi,

so you admit that the average sample rate could be calculated on a sample by sample basis, and that i do not have to wear a uniform.


right.

[David Rick]

Originally Posted by oky****

and here's an except from an interview with paul frindle in "audio design lline" magazine about cumulative dither noise that you may find interesting or helpful.

"Another similar problem can exist when you dither signals, because of relative correlation. When you dither a number of channels, the dither noise of any one channel should be unrelated and uncorrelated to any other, or you risk the relative correlation of the dither noise starting to become evident. You can end up with statistically partially 'mono' dither-noise, which can close in the stereo effect, especially on fade-outs.

There exists some literature on how to reduce the processing load when making several uncorrelated dithers. The late Michael Gerzon worked on this problem. If I remember correctly, he assumed that only first-moment decorrelation was needed between channels, so you could reuse a particular RPDF dither stream to make TPDF dither for two different channels. With four independent RPDF random sequences, you could make 4! / (4-2)!2! = 6 TPDF dithers. No doubt there are some more advanced tricks that I don't know.

David L. Rick
Seventh String Recording

[okydoky]

Originally Posted by David Rick

There exists some literature on how to reduce the processing load when making several uncorrelated dithers.

David L. Rick
Seventh String Recording

hi,

seems like this issue may possibly have some relevance with respect to the pro tools dithered mixer thing. do you use pro tools at all?


right.

[David Rick]

Originally Posted by oky****

so you admit that the average sample rate could be calculated on a sample by sample basis, and that i do not have to wear a uniform.

LOL

[David Rick]

Originally Posted by oky****

seems like this issue may possibly have some relevance with respect to the pro tools dithered mixer thing. do you use pro tools at all?

No.

Bob Olhsson knows a lot about such issues, having used PT (very carefully!) for mastering before it really deserved such use. I'm certain he could shed a lot of light on the issue.

By the time Digi finally cleaned up their audio act, I was firmly entrenched in the Samplitude/Sequoia camp. (Bob has now switched, as has Bob Katz.)

Please, let's not turn this thread into an argument about the merits of specific DAW's! Each has strengths and weaknesses. A skilled and informed operator can overcome the weaknesses of most any modern DAW. The main barrier is transparency -- not audio transparency, but algorithmic transparency. The programmers and manual writters don't give us enough information about what they are actually doing "under the hood". If they did, an engineer with a sufficiently good technical background could stay out of trouble. But instead, most software companies just say "trust us".

David L. Rick
Seventh String Recording

[okydoky]

Originally Posted by David Rick

No.

Bob Olhsson knows a lot about such issues, having used PT (very carefully!) for mastering before it really deserved such use. I'm certain he could shed a lot of light on the issue.

By the time Digi finally cleaned up their audio act, I was firmly entrenched in the Samplitude/Sequoia camp. (Bob has now switched, as has Bob Katz.)

Please, let's not turn this thread into an argument about the merits of specific DAW's! Each has strengths and weaknesses. A skilled and informed operator can overcome the weaknesses of most any modern DAW. The main barrier is transparency -- not audio transparency, but algorithmic transparency. The programmers and manual writters don't give us enough information about what they are actually doing "under the hood". If they did, an engineer with a sufficiently good technical background could stay out of trouble. But instead, most software companies just say "trust us".

David L. Rick
Seventh String Recording

hi,

ha, ha. you definitely don't have to worry about me starting one of those daw fights.

yeah, they all tend to obscure the details of the implementation, and its a drag for people who could use the knowledge to make better recordings.

i use pro tools [and other stuff], and digi has two different mixers now. the manual and other documentation is ambiguous as to whether or not the "undithered mixer" does actually add dither at the outputs, even if not at other places in the channels. it says one thing one place and another thing in another. lovely.

you don't happen to know about that, do you? i hear sequoia is good.

do you think previous dither is destroyed by redithering? i agree with dithering whenever bit depth is reduced, but when it is going back and forth between 24 to 48 to 24 to 48.....to 24 i wonder if it is not better to just add it at the last reduction. ? wouldn't that still 86 the quantization distortion?

but than again, if the intermediate reductions were done without dither, would the file be accumulating low level inaccuracy on the way to the final dithered reduction, thus yielding a slightly more inaccurate file, but with no actual quantization distortion?


right.

[Bob Olhsson]

Truncation distortion accumulates 6 dB./process. Dither accumulates 3 dB./process.

It's your choice!...

[okydoky]

Originally Posted by Bob Olhsson

Truncation distortion accumulates 6 dB./process. Dither accumulates 3 dB./process.

It's your choice!...

hi bob,

thanks.

are you using pro tools for anything now? i'm trying to sort out a couple of things, and it seems to me that you probably have a handle on this stuff from the standpoint of the whole process mixing through mastering.

i'm looking at the documentation from digidesign which appears to say about 3 different things in as many places, so .............. ??????

how do you feel about not using the dithered mixer and just dithering once during mastering?

also, do you think the 48 channel regular mixer dithers at the outputs anyway? it seems to say that in couple of places, but then again in others it says something like "when the dithered mixer is used, dither is added at the outputs when the signal is placed back on the tdm bus" [something like that].

also, it seems to me that any truncation going on when the audio goes back and forth from the 24 bit bus to the 48 mixer would mostly be just a bunch of zeros getting chopped off, and i wonder if that would even create any distortion artifacts anyway.

i think paul frindle said something somewhere about redithering not causing problems because the previous dither gets destroyed. not really sure what he meant. ???

jay has some info about temporal resolution actually being improved by dither also, and i've been trying to figure that out.

i get some of this from you guys telepathically, i believe.

anyhow, i have no problem with the idea of dither, i just don't see any reason to glom any more of it on that is actually useful, and it would be great to know exactly what the daw is doing.

if you have time, maybe we could discuss some of this at your convenience. thanks again.


right.