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I am writing an app to work with synthetic time series data from a physics experiment. In our experiments we always have $1/f$ noise in our time series, but I haven't been able to find code/packages to generate $1/f$ noise in synthetic time series data. Can anyone tell me how to do this? I have a feeling it should be pretty simple and that I'm missing something obvious, but I don't know what it is. Any advice?

Also, I looked through the list of questions and don't see anything too on point apart from this: making pink noise (1/f) using list of frequencies

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this is an old resource, but good. in fact, i have a very old contribution (another 3 decade old design) that i will repeat here at SE:

A method that Orfanidis (Introduction to Signal Processing mentions came from an old comp.dsp post of mine. (here's a pdf, check Problem B.9) It's just a simple "pinking" filter to be applied to white noise. since the rolloff is -3 dB/octave, -6 dB/octave (1st order pole) is too steep and 0 dB/octave is too shallow.

An equiripple approximation to the ideal pinking filter can be realized by alternating real poles with real zeros. A simple 3rd order solution that i obtained is:

$$ H(z) = \frac{(z-q_1)(z-q_2)(z-q_3)}{(z-p_1)(z-p_2)(z-p_3)} $$

where

$p_1$ = 0.99572754

$p_2$ = 0.94790649

$p_3$ = 0.53567505

$q_1$ = 0.98443604

$q_2$ = 0.83392334

$q_3$ = 0.07568359

The response follows the ideal -3 dB/octave curve to within $\pm$0.3 dB error over a 10 octave range from 0.0009$\times \pi$ to 0.9$\times \pi$. ("Nyquist" = $\pi$.)

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  • $\begingroup$ Do you know this filter? Its poles and zeros are very close to yours. Not sure how J.O. Smith came up with that filter. How did you optimize the pole and zero locations? $\endgroup$ – Matt L. Apr 9 '15 at 9:02
  • $\begingroup$ R B-J's filter worked great for me in a quick and dirty implementation: gain = 1.0/23.1188232529392; out[t] = in[t]*gain + in[t-1]*(-1.89404297*gain) + in[t-2]*(0.9585641562817477*gain) + in[t-3]*(-0.0621320035261672*gain) - out[t-1]*-2.47930908 - out[t-2]*1.985012853639686 - out[t-3]*-0.505600430025288; $\endgroup$ – Olli Niemitalo Apr 9 '15 at 15:09
  • $\begingroup$ @MattL., it's possible that JOS got it from me. or independently. i have never "collaborated" with JOS, but i have corresponded with him dozens of times. usually he credits accurately, but i just don't know where he got this. if it's third order and good for 10 octaves (20 to 20kHz), then there's about only one set of real and alternating poles and zeros that does it. when i first did this (in the 80's) i took the -3 dB straight slope and prewarped it (regarding bilinear transform). then i fit asymptotes to it with either -6 dB slope or 0 dB slope to get the analog corner frequencies. $\endgroup$ – robert bristow-johnson Apr 9 '15 at 15:52
  • $\begingroup$ one thing that Sophicles points out is that, without edge effects, the frequencies of the alternating poles and zeros are equally spaced in log-frequency. so that might be a good first guess. and then just manually tweek some of the side corner frequencies. if i were to do it over again, i would probably do it with 5 poles and 4 zeros. $\endgroup$ – robert bristow-johnson Apr 9 '15 at 15:58
  • $\begingroup$ @robertbristow-johnson: OK, thanks. Do you have a reference for the poles and zeros being equally spaced in log-frequency? $\endgroup$ – Matt L. Apr 9 '15 at 16:45

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