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Topic: Getting familiar with dither. (Read 7124 times) previous topic - next topic
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Getting familiar with dither.

Hi there,

Now that I am reading and understanding the basic principles of dither, I come across something I do not understand. Summarizing, proper dither is used to make the first two moments of the stochastic error signal zero and q^2/4 respectively. The first moment is the mean and the second moment represents the variance. Both are statistically independent of the input signal. I can follow the proof that leads to this.

However, what I don't understand is the statement that a zero mean imply that there is no distortion present in the output signal. What do they mean with "distortion"? To clarify my question, suppose that the quanitizer produces a second order distortion product when the input signal is a sine wave (since the quantizer is a non-linear element, this is not wrong to assume). If I take the mean (expectation{}) of this signal, I will observe that this is zero, but there is distortion. Where do I go wrong?

Regards,
Jacco
Logical reasoning brings you from a to b, imagination brings you everywhere.

Getting familiar with dither.

Reply #1 –
I'm not really sure that it's true to state that the distortion is zero - it's just decorrelated, so that the error function of the the quantized signal vs the input signal is a random variable. Where did you see it described as "zero distortion"?

Getting familiar with dither.

Reply #2 –
Quote
To clarify my question, suppose that the quanitizer produces a second order distortion product when the input signal is a sine wave


Quantizer output is not directly related to the input signal. There's usually a sigma-delta modulator built around it. The so-called distortion a quantizer produces are all forms of limit-cycles (birdies) and other predictable output patterns. By adding dither, birdies are prevented and the supersonic power spectrum is spread more evenly.

Getting familiar with dither.

Reply #3 –
I can quote from: A Theory of Nonsubtractive Dither, Robert A. Wannamaker, Stanley P. Lipshitz, Member, IEEE, John Vanderkooy, and
J. Nelson Wright, Senior Member, IEEE, IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 2, FEBRUARY 2000

Quote
For audio signal processing purposes, there seems to be little
point in rendering any error moments other than the first and
second independent of the input. Variations in higher moments
are believed to be inaudible, and this has been corroborated by a
large number of psycho-acoustic tests conducted by the authors
and others [14], [22]. These tests involved listening to a large
variety of signals (sinusoids, sinusoidal chirps, slow ramps, various
periodically switched inputs, piano and orchestral music,
etc.), which had been very coarsely requantized (from 16 to 8
bits) in order to render the requantization error essentially independent
of low-level nonlinearities in the digital-to-analog
conversion system used for listening purposes. In addition, the
corresponding error signals (output minus input) were used in
listening tests in order to check for any vestiges of audible dependence
on the input. Using undithered quantizers resulted in
clearly audible distortion and noise modulation in the output and
error signals. Rectangular-pdf dither of one LSB peak-to-peak
amplitude eliminated all distortion, but the residual noise level
was found to vary audibly in an input-dependent fashion. When
triangular-pdf dither of two LSB peak-to-peak amplitude (either
white or highpass) was employed, no instance was found
in which the error was audibly distinguishable from a steady
random noise entirely unrelated to the input. Admittedly, these
tests were informal, and there remains a need for formal psychoacoustic
tests of this sort involving many participants under controlled
conditions.


But also from a more recent one: Dithered Noise Shapers and recursive Digital Filters, Stanley P. Lipshitz, Robert A. Wannamaker and John Vanderkooy, AES November 2004
(Section 2.1)
Quote
Note that Eq. (21) implies that there is no distortion of the input signal because the mean error is zero,...


Regards,
Jacco
Logical reasoning brings you from a to b, imagination brings you everywhere.

Getting familiar with dither.

Reply #4 –
Your point?

Signal modulated limit-cycles are artefacts of the input signal. But that is not the quantizer's fault alone, it's predictable behavior of the entire loop when not decorrelated by dither.

I once urged you to read P. Naus's bitstream signal processing course, section 3.3.2 introduces stability and noise modulation.

Getting familiar with dither.

Reply #5 –
Quote
Your point?
How can it be that zero mean means no distortion (mathematical prove please), or in other words what is the definition of distortion in this sense. As can be read in my original question by the way.

I will get the book you recommend. As far as I can see it now, there are different ways of defining "distortion" as such.

Regards,
Jacco
Logical reasoning brings you from a to b, imagination brings you everywhere.

Getting familiar with dither.

Reply #6 –
The output of an SD modulator is

Y(z) = X(z) + {1 – H(z)} E(z).

if E(z) is uncorrelated to X(z), i.e. by dither, and H(z) is sufficient in magnitude (which every decent modulator designer fullfills in the passband), there will be no distortion. If E(z) is not due to dither, it bears a signal dependency and thus there will be distortion.

Getting familiar with dither.

Reply #7 –
Hi,

Quote
if E(z) is uncorrelated to X(z)
I can understand that part. Suppose a = function(z)*X(z) then there will be no distortion if function(z) is independant of X(z). In reality this is not completely true: only in the two first moments. And what I don't understand is the statement that only the mean is sufficient to jurge there is no distortion. I can understand that one has to take a mean since we are dealing with noisy signals, but an added second order component will then also be zero.

Another argument can be (if I am thinking right now), suppose that the output of a quantizer is y = a + b*x + c*x^2 + d*x^3 etc, or that it can be approximated by this. Having an E(z) which is independant of X(z) doesn't imply that the coefficients a to d are zero, it implies that they are non-dependant of the input signal and can be a constant and thus can have distortion.

Regards,
Jacco
Logical reasoning brings you from a to b, imagination brings you everywhere.

Getting familiar with dither.

Reply #8 –
Quote
suppose that the output of a quantizer is y = a + b*x + c*x^2 + d*x^3


NO. It just isn't. There's a time-discrete feedback loop in a noise shaper. I cannot help you with this as good as, say, Norsworthy's book. Nor will I try.

Proper dither controls the output errors statistical properties and hence output error spectrum. Zero mean prevents DC modulation (signal independent).

Getting familiar with dither.

Reply #9 –
Quote
NO. It just isn't.
Ok, let us forget this since it is beyond the scope of my question anyway.

Quote
Proper dither controls the output errors statistical properties and hence output error spectrum.
Right, that is also my idea about applying dither. But my question has to do with the statement that if that error signal has zero mean, the distortion is absent. I still don't see why exactly.

Regards,
Jacco
Logical reasoning brings you from a to b, imagination brings you everywhere.

Getting familiar with dither.

Reply #10 –
Non-zero mean implies the modulator has to overcome an injected (average) offset, risking falling into another limit cycle.

Getting familiar with dither.

Reply #11 –
You've hit on a basic problem, I think, in that "distortion" is not a well-defined word. SNR is, Error is, Mean Squared Error is, but "distortion" is not.

Many people use "distortion" to refer to signal-correlated components alone, i.e harmonic, IM, and the like kinds of distortion.

An undithered (i.e. wrong) quantizer introduces distortions that are signal correlated. A first-order dithered quantizer introduces uncorrelated noise, but the power in the noise is level-correlated. The second-order dither raises the noise level, but decorrelates both the noise and the envelope of the noise from the original signal.

If we chose to include all signal-correlated or signal-related components as "distortion" then proper dithering means there are no such components.

The problem, though, is that "distortion" is not a well-defined word.
-----
J. D. (jj) Johnston

Getting familiar with dither.

Reply #12 –
Quote
Many people use "distortion" to refer to signal-correlated components alone, i.e harmonic, IM, and the like kinds of distortion.
Yes, I would like to use this kind of definition also.

By the way, I decided to raise the exact same question to one of the editors of the article, Stanley P. Lipshitz. It took me some time to find his email address. Surprisingly I got an answer and I don't know if I have permission to quote him. However, the highlights are:

Quote
I agree that, if you just lift the "Note" in the middle of the first
column on p. 1129 of our paper out of context, it is misleading, but IN
CONTEXT, it is correct.
And this is referring to the same matter as described above:
Quote
But also from a more recent one: Dithered Noise Shapers and recursive Digital Filters, Stanley P. Lipshitz, Robert A. Wannamaker and John Vanderkooy, AES November 2004
(Section 2.1)
QUOTE
Note that Eq. (21) implies that there is no distortion of the input signal because the mean error is zero,...


The conclusion can be drawn that indeed the statement as such is incorrect. The other way around is true: suppose that the outcome of the dithered quantizer is uncorrelated with the input signal, there is no distortion. That is, it has become a random variable (noise) with zero mean.

Another highlight from the answer of Lipshitz:
Quote
The power spectrum of epsilon is noise-like, with no line spectral
components; that is, there is "no distortion" (correlated error), either
harmonic, intermodulation, or of any other form.


Personally, the subject is closed since my question has been answered. I hope that more people will benefit from this answer from S.P. Lipshitz.

Regards,
Jacco
Logical reasoning brings you from a to b, imagination brings you everywhere.