[sdiy] A new shade of pink (noise)

[Richie Burnett]

Hi Tom,

I got to the bottom of this eventually after we were discussing this offline 
a few years back.  What he does is he runs the "top octave" noise generator 
at the full sample rate, then runs the next highest octave noise generator 
at half of the full sample rate, but linearly interpolates this.  This 
really just means inserting one new sample between each pair of samples, and 
this is done by increasing the bit-depth.  Essentially all he does is uses 
linear-interpolation to interpolate all of the successively lower speed 
noise generators up to the full output sample rate, and then mix them 
together.  This is like a Zero-Order-Hold followed by an integrator for each 
successive noise source.  So the next octave down runs at a quarter of the 
sample rate, and 3 new levels are inserted between each pair of samples to 
bring the output up to the full output sample rate.  So every noise source 
is effectively upsampled to the final output rate using linear 
interpolation.

The result of this is that each of the many noise sources running at the 
different sample rates produces an output that ramps linearly between two 
levels (with the amount of "in between" levels being determine by the sample 
rate of this noise source relative to the final output sample rate.)

When all of the different noise sources are mixed together to get the final 
output (pink noise) it gives less ripple in the power spectrum than using a 
set of Zero-Order-Hold noise sources because the sinc-like spectral envelope 
of each noise source rolls off faster with linear interpolation than with 
ZOH.  (You get an additional 6dB/oct rolloff due to the conceptual 
integration.  Or you can think of it that the discontinuities are now in the 
gradient dy/dt rather than in the value y.)

I remember he says something in his paper about the very top part of the 
spectral profile not having quite the right slope though, and he proposed 
correcting this with some sort of FIR filter?  When I implemented it, I just 
ran the whole thing at a faster sample rate so that the part of the power 
spectrum that wasn't quite "pink" was moved above the audio band where human 
ears wouldn't notice.  But for test/measurement usage it might be worth 
correcting the response with the FIR.

I have to say that I didn't code this up on a PIC or dsPIC or anything Tom. 
Just wrote some equations in Goldwave expression evaluator to generate the 
relevant noise sources, mixed them together, and then looked at the power 
spectrum using MATLAB.  The performance is really good, but you have to run 
the power spectrum evaluation on several minutes of noise in order to really 
get a picture for how well the power spectrum converges to 1/f.  I think I 
was generating 10 minute long pink noise recordings with 96kHz sample rate 
in order to assess the residual ripple in the power spectrum!

-Richie,



-----Original Message----- 
From: Tom Wiltshire
Sent: Tuesday, November 24, 2020 12:57 PM
To: Benjamin Tremblay via Synth-diy
Subject: [sdiy] A new shade of pink (noise)

There’s an interesting paper by Stefan Stenzel on Github, describing a new 
digital pink noise generation algorithm:


https://github.com/Stenzel/newshadeofpink/blob/master/newshadeofpink.pdf


Has anyone seen this? I’ve read it, but there’s one thing I don’t 
understand. In the paper, he talks about taking multiple 1-bit noise sources 
(as you would in the Voss0-McCartney algorithm) but instead of using a “zero 
order hold” (e.g. “stretching” each sample) to decrease the sample rate, he 
uses linear interpolation.
This is the bit I don’t get - how do you linearly interpolate a 1-bit 
signal? There’s nothing in between!
He mentions at one point that the digital signal is to be interpreted as -1 
or +1, which would mean that there is a 0 between the two values, but I 
still don’t understand how that makes sense when it’s a digital signal and 
not a bit of signal processing maths formula.

Any clarifications appreciated. I’d like to understand this method better, 
but the paper is very brief, assumes quite a lot of background I don’t have, 
and doesn’t provide any worked examples for illustration.

Many thanks,
Tom


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