Creating Bluish Noise With Minimal Compute

Often random numbers are referred to as "noise", and sometimes we attach colours to refer to the different characteristics of that noise. The colours reference power characteristics at at different frequencies, but those power characteristics are the ultimately the result of how the random values are distributed.

Blue noise is a type of noise with a power density that increases with frequency, giving it a high-pitched, hissy sound. Visually, it manifests as a pattern with minimal low-frequency components and a relatively even distribution of points, and is frequently used for dithering. Blue noise is often used for its aesthetically pleasing randomness, since numbers don't clump together as they would with truly random (white noise) sources.

Saarna's Bluing Algorithm

Blue noise is typically created by generating random values, and massaging them in various ways to not be as clumpy. This isn't a great approach for older platforms that don't have a lot of compute horsepower or ram.

An alternative approach, is to blue the noise by keeping track of recent values with an exponentially decaying running average, and ensuring that new random numbers aren't near the decaying average...

A=GetRandom() ; any masking or range-reduction needs to be done here, or else
B=GetRandom() ; it will reintroduce clumpiness into the value distribution
if Absolute(B-C) > Absolute(A-C) then A=B
C=(C+A)/2 ; exponentially decaying running average
; "A" is holding your next bluish noise value

The algorithm creates the blue noise characteristic +3 dB per octave power increase, and it works equally well with LFSR noise sources.

The algorithm could be modified to produce bluer noise by adding more random values into the comparison. The algorithm as presented produces noise that is strikingly blue, as can be seen in the following Spectral Analysis section, so additional random number steps are probably not worth the additional effort for most applications.

Spectral Analysis

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  Spectral Analysis of a 16-bit LFSR after application of the bluing algorithm.

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  Spectral Analysis of a 16-bit LFSR without application of the bluing algorithm.

Authorship

The novel algorithm described here was created by Mike Saarna. If you're going to cite, please refer to it as "Saarna's Bluing Algorithm".