Noise and Chaos on the Arduino

Off-topic. I needed to get my head into gear for work-work, and over the weekend I had an odd little idea I wanted to try. So here’s a quick & dirty write-up and video.

After playing with Arduino White Noise the other week, I did a bit of reading up on the Colors of Noise. Particularly interesting is Pink Noise, in which “each octave (halving/doubling in frequency) carries an equal amount of noise energy… This is in contrast with white noise which has equal intensity per frequency interval.”. It occurs a lot in nature, but is not entirely trivial to synthesize either using analog or digital processing. (Here’s a fairly accurate analog pink noise generator circuit).

Mind wandering, this led me onto chaotic signals. These are remarkably easy to slip into in the analog domain, essentially all you need is a non-linear system with feedback (and the right parameters)  – see this old magazine write-up on non-linear circuits. They also easy to generate in the digital. The best known system is probably the Lorenz Attractor,

But there are much simpler discrete systems, notable the Logistic Map. This is just:

x1 = r * x0 * (1-x0)

where r is a constant, x0 is the current value of x, x1 the next value. With values of r between about 3.6 and 4, the thing goes chaotic.

This was pretty easy to plug into the same skeleton code I used for Arduino white noise generation. The result was the same distinctive kind of racket that the analog circuits generate.  To provide a bit of control, I put a pot. on an analog input, scaling the read value between 0-1 and adding it to 3 to provide an interesting value range for r.

But what I wanted to play with wasn’t just this. One way of generating electronic (and mechanic) chaos is to drive an otherwise periodic system with a periodic signal, as in the chaotic double pendulum. But with pink noise on my mind, I was curious to see what would happen if a chaotic system was driven with white noise.

The code, again using the skeleton I already had, was straightforward. I added another pot. to another analog input to determine the level of the noise signal.

My code is a real hacky mess at the moment, mostly due to hopping between integer and float values, and scaling, but the core of it looks like this (effectively inside a loop):

  // Shift register-based random number generator (white noise)
  unsigned lsb = lfsr & 1;   /* Get LSB (i.e., the output bit). */
  lfsr >>= 1;                /* Shift register */
  lfsr ^= (-lsb) & 0xB400u;

  // control values
  noise_level = analogRead(NOISE_LEVEL_PIN); // will be 0 - 1023
  r_value = analogRead(R_VALUE_PIN);

  r = 3 + ((float)r_value) / 1024;

  noise_scale = ((float)noise_level) / 2048;

  x_scale = 1 - noise_scale;

  noise = noise_scale * ((float)lfsr) / 65536;

  x = x_scale * x + noise;

  // logistic map
  x = r * x * (1 - x);


  // the value to output
  temp3 = (uint16_t)(x * 65536); // scale & cast

I’ve no idea where this is going…