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Subject: [music-dsp] high & low pass correlated dither noise question

From: "Alan Wolfe" <alan.wo...@gmail.com>

Date: Thu, June 27, 2019 7:42 am

To: "A discussion list for music-related DSP" <music-dsp@music.columbia.edu>


> I read a pretty cool article the other day:

> https://www.digido.com/ufaqs/dither-noise-probability-density-explained/


> It says that if you have two dice (A and B) that you can roll both dice and

> then...

> 1) Re-roll die A and sum A and B

> 2) Re-roll die B and sum A and B

> 3) Re-roll die A and sum A and B

> 4) repeat to get a low pass filtered triangular noise distribution.


> It says that you can modify it for high pass filtered triangle noise by

> rolling both dice and then...

> 1) Re-roll die A and take A - B

> 2) Re-roll die B and take B - A

> 3) Re-roll die A and take A - B

> 4) repeat to get a high pass filtered triangular noise distribution.


> What i'm wondering is, what is the right thing to do if you want to do this

> with more than 2 dice? (going higher order)


> For low pass filtered noise with 3+ more dice (which would be more gaussian

> distributed than triangle), would you only re-roll one die each time, or

> would you reroll all BUT one die each time.


> I have the same question about the high pass filtered noise with 3+ more

> dice, but in that case I think i know what to do about the subtraction

> order... I think the right thing to do if you have N dice is to sum them

> all up, but after each "roll" you flip the sign of every die.


> What do you guys think?
so, to add to Nigel's comments:
repeatedly rolling a single honest die is like rectangular (or "uniform") 
p.d.f. bandlimited white noise.
adding random variables add their means and adds their variances and makes the 
random variable more gaussian-like..
if you want to stick with gaussian p.d.f. you can filter the noise to your 
heart's content and get the power-spectrum or the auto-correlation you want, 
but the result is still gaussiian.
or, if you stick with independent random variables (which are
bandlimited white), you can take whatever p.d.f. and run that through a 
memortyless non-linear mapping function and get out whatever p.d.f. you're 
desiring. but the result is still bandlimited white.
to get both the p.d.f. and power spectrum desired requires an iterative process 
to design a
filter for gaussian white and passing that through a non-linear curve.
so adding/subtracting 3 dice will get you a piecewise quadratic p.d.f.   
adding/subtracting 4 dice will get you piecewise cubic p.d.f. , 5 dice is 
piecewise quartic.  etc.  these p.d.f. will look more
and more like a gaussian curve.
if you want your dither for dithering quantization, whether is for "mastering" 
(the final quantization to, say, 16 bits) or for internal nodes in an 
algorithm, my understanding is that triangular p.d.f. (what you get from adding 
or subtracting two
dice) is the lowest-power dither noise that you can add that will completely 
decorrelate the first and second moments (the mean and the variance) of the 
quantization error from the audio signal getting quantized.  so i don't think 
you want to go farther than that, which is adding or subtracting
two rectangular random variables.  if you do that subtractive thing (with 
memory), you will have high-pass, triangular p.d.f. dither which might sound 
better than flat dither or low-pass dither.  the net rounding will still be 
white which is added to the dither, however the power spectrum
of that is.

r b-j                         r...@audioimagination.com

"Imagination is more important than knowledge."

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