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Quantization Grid, Split from Topic ID #55966
KMD
post Mar 19 2012, 16:00
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Sampling has nothing to do with it . Sampling is totally benign. The thing under scrutiny is quantisation. details small in duration can also get lost
in an analogue recording if the noise happens to be its mirror image for its duration. Resolution refers to the peak to peak level of siganal that that can be recorded hence resolved
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icstm
post Mar 19 2012, 16:07
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yes, but what makes you say a analogue signal can be resolved to infinite precision with 100% accuracy?
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KMD
post Mar 19 2012, 16:11
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I did not mention accuracy and precision. The subject matter is resolution, the smallest peak to peak signal that can be recorded hence resolved. The resolution in a dithered digital system is infinite because the dither noise takes up the space between quantisation steps and so there is no room for a small peak to peak signal to go undetected.

This post has been edited by KMD: Mar 19 2012, 16:23
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icstm
post Mar 19 2012, 17:14
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resolution is comparible with precision it is the "how many decimal places can I measure this to".
Accuracy is the "and are the significant digits right"
To say somthing has infinite resolution is only useful it that measurement is correct.
In the anologue world this is impacted by the sensitivity of the system. Are you saying you have an infinitely sensitive system?
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2Bdecided
post Mar 19 2012, 19:03
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QUOTE (KMD @ Mar 19 2012, 15:11) *
I did not mention accuracy and precision. The subject matter is resolution, the smallest peak to peak signal that can be recorded hence resolved. The resolution in a dithered digital system is infinite because the dither noise takes up the space between quantisation steps and so there is no room for a small peak to peak signal to go undetected.
Detected by who? Using what?!


btw, useful measures are noise, distortion, and monotonicity.

No one doubts that digital resolution can be infinite, in theory (see above). In practice, the same limits impact both analogue and digital. Pesky electronics. this is The weak point of digital storage. The weak point of analogue storage is elsewhere (the medium, the transduction, etc).

Cheers,
David.
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knutinh
post Mar 19 2012, 23:32
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Being able to recreate infinitely small signals would indicate that there are an infinite number of different, resolvable signals between "1" and "1/inf". I guess that would imply that the system could encode an infinite amount of information on a storage medium of finite capacity?

-k
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2Bdecided
post Mar 20 2012, 10:38
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QUOTE (knutinh @ Mar 19 2012, 22:32) *
Being able to recreate infinitely small signals would indicate that there are an infinite number of different, resolvable signals between "1" and "1/inf". I guess that would imply that the system could encode an infinite amount of information on a storage medium of finite capacity?
...which is tantamount to inventing a perpetual motion machine.

Cheers,
David.
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KMD
post Mar 20 2012, 10:57
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- icstm , you are confusing resolution with signal to noise ratio.
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icstm
post Mar 20 2012, 11:28
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no, I am saying that resolution is only relevant in the context that of SNR.
For an instrument to have high resolution and for that to be of use we are saying it offers more significant digits in its reading than something else. I would suggest that can only be the case if what it is measuring is the real signal.
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KMD
post Mar 20 2012, 11:57
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No, the small variations in the waveform are there in the lower significant bits with noise added
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greynol
post Mar 20 2012, 15:09
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Why do you keep insisting that dither must be present in order for there to be voltages between quantization steps in a reconstructed digital signal?

Tell me KMD, do you have any formal training or is this just from the seat of your pants?

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2Bdecided
post Mar 20 2012, 17:33
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QUOTE (greynol @ Mar 20 2012, 14:09) *
Why do you keep insisting that dither must be present in order for there to be voltages between quantization steps in a reconstructed digital signal?
For a low-ish frequency input signal, ignoring noise etc, most of the instantaneous output voltages will be essentially "on" the quantization steps. You can see this very easily at low amplitudes.

Cheers,
David.
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greynol
post Mar 20 2012, 18:11
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I'm more than a bit skeptical of that.


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pdq
post Mar 20 2012, 18:36
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QUOTE (greynol @ Mar 20 2012, 13:11) *
I'm more than a bit skeptical of that.

I'm surprised that that surprises you. It seems rather obvious to me. Perhaps someone will generate a low-ish frequency sine wave in 8 bits without dither and show us the reconstructed waveform?
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greynol
post Mar 20 2012, 18:46
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Are you trying to tell me inter-sample points don't transition smoothly as if the reconstruction filter doesn't do its job?

And why are we stipulating that the bandwidth be somewhere other than Nyquist?!? It's not as if the OP had also included such a stipulation.

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pdq
post Mar 20 2012, 18:53
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The reconstruction filter only filters out frequencies above fs/2. The steps in the DAC output for a low frequency will contain much lower frequencies than fs/2, which aren't filtered out, unless you have proper dither.

Picture the case of a sine wave that is above a step threshold half the time, and below it the other half. The DAC produces a square wave, which gets low-pass filtered. All of those odd harmonics above fs/2 will be filtered out, leaving a square wave with rounded edges.

Edit: Am I responding to the wrong thing? I though you were skeptical of what David said.


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greynol
post Mar 20 2012, 19:06
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QUOTE (pdq @ Mar 20 2012, 10:53) *
leaving a square wave with rounded edges.

The part I emphasized is at the very crux of my argument, unless you are claiming that dither is required for this rounding to occur(?).

The unnecessary stipulation of lowish frequency only muddies the waters.

This post has been edited by greynol: Mar 20 2012, 19:14


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Arnold B. Kruege...
post Mar 20 2012, 19:23
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QUOTE (KMD @ Mar 19 2012, 09:28) *
Signal to noise ratio, and resolution, are not the same thing.


Not exactly the same, but dynamic range and resolution are related by a simple monotonic algebraic equation according to Shannon (Information Theory).

In most systems dynamic range and SNR differ only due to the presence of nonlinear distortion, which becomes moot if it is small enough. In modern systems they can be very close.

So a first approximation is that in high quality modern systems SNR approximately equals dynamic range which pretty well defines resolution.
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Arnold B. Kruege...
post Mar 20 2012, 19:30
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QUOTE (KMD @ Mar 19 2012, 10:19) *
Hi icstm

An analogue recording system has infinite resolution but not infinite signal to noise ratio

A dithered digital recording system has infinite resolution but not infinite signal to noise ratio


Both statements are false.

No real world system has infinite resolution.

Shannon's information theory provides the relationship between noise+distortion and resolution.

Shannon Information Channel Capacity link
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Arnold B. Kruege...
post Mar 20 2012, 19:33
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QUOTE (KMD @ Mar 20 2012, 05:57) *
- icstm , you are confusing resolution with signal to noise ratio.


They are often close enough. Many of us often interchange the two knowing full well the details about the differences. SNR and Dynamic range of high quality practical digital audio systems are often very similar.
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Arnold B. Kruege...
post Mar 20 2012, 19:36
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QUOTE (pdq @ Mar 20 2012, 13:36) *
QUOTE (greynol @ Mar 20 2012, 13:11) *
I'm more than a bit skeptical of that.

I'm surprised that that surprises you. It seems rather obvious to me. Perhaps someone will generate a low-ish frequency sine wave in 8 bits without dither and show us the reconstructed waveform?


That seems like unecessary work given all the times I've seen this played out with real world equipment.

The output of a DAC contains all of the intermediate voltages even if the signal is undithered. The steps get lost in the reconstruction filter. If the digital signal is undithered there is correlated quantization error but it is smoothed out by the reconstruction filter's low pass effects.
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Arnold B. Kruege...
post Mar 20 2012, 19:41
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QUOTE (greynol @ Mar 20 2012, 13:46) *
Are you trying to tell me inter-sample points don't transition smoothly as if the reconstruction filter doesn't do its job?


Bingo!

The reconstruction filter does do its job and the output voltage changes smoothly.

What puzzles people is what happens when the digital data is not dithered. Yes, the in-band quantization error is there, but it does not take the form of steps. It gets smoothed, too.

I base this on looking at 100s of DACs. I've actually seen old, old audio interfaces that put out stairsteps, reason being they left out the reconstruction filter. Very, very early Soundblasters, for example.
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greynol
post Mar 20 2012, 19:42
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QUOTE (Arnold B. Krueger @ Mar 20 2012, 11:36) *
The output of a DAC contains all of the intermediate voltages even if the signal is undithered.

Thank you!


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2Bdecided
post Mar 20 2012, 19:43
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QUOTE (pdq @ Mar 20 2012, 17:36) *
I'm surprised that that surprises you. It seems rather obvious to me. Perhaps someone will generate a low-ish frequency sine wave in 8 bits without dither and show us the reconstructed waveform?
You can use CoolEdit/Audition/whatever to take 44.1kHz 8-bit to 44.1kHz 16-bit then to 352.8kHz 16-bit (with the best oversampling filter available = simulated near-ideal reconstruction). Those original 8-bit quantisation steps are still easily visible...
Attached Image


You can even see their remnants if the original signal is dithered...
Attached Image



In the first instance, the reconstruction filter is only removing the aliased harmonics of the truncation distortion - turning a square wave into a slightly ringy square wave.

In the second instance, the reconstruction filter is also removing the aliases of the dither, hence the more obvious change in shape.

Cheers,
David.

P.S. EDIT: note amplitude and time scales on those plots. The full vertical scale is about 8 LSBs in 8-bit audio. The frequency of the sweep at that point is about 50Hz. It's still pretty similar at 500Hz though.

This post has been edited by 2Bdecided: Mar 20 2012, 19:46
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2Bdecided
post Mar 20 2012, 19:52
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QUOTE (greynol @ Mar 20 2012, 18:42) *
QUOTE (Arnold B. Krueger @ Mar 20 2012, 11:36) *
The output of a DAC contains all of the intermediate voltages even if the signal is undithered.

Thank you!

You thanked him a bit early.

I'm not sure why anyone thinks a filter at 22kHz is going to change a square-ish low frequency waveform that much. The example I posted was a low amplitude sine wave at 50Hz; at 8-bits it ends up with square-wave-like transitions at ~250Hz due to quantisation. In this example, you can comfortably fit the first 40 harmonics within the transition band. That's more than you need to make a square wave look something like a square wave.

Cheers,
David.

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