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What happens if you take a white noise generator, run it through a one-pole LPF set to a low cutoff frequency, and then mix the two together? Do you get pink noise?

Say you have this code:

float noise = noiseGenerator();
float pinkNoise = pinkingFilter(noise); //3dB/oct filter
float brownNoise = onePoleLPF(noise); //6dB/oct filter

Would then pinkNoise roughly equal a mix of noise and brownNoise?

Or going further, can you get intermediate colors by just mixing the three noise signals in varying amounts? Like could a mix between pinkNoise and brownNoise create a 4.5dB/oct slope noise?

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  • $\begingroup$ "mixing" is an ambiguous term! Do you mean mixing in the multiplicative sense (as in e.g. a frequency mixer in a radio device) or in the additive sense (as in audio mixer)? $\endgroup$ – Marcus Müller Jan 4 '20 at 12:04
  • $\begingroup$ and: it really makes a difference here whether you use the same noise for both your pinkingFilter and your onePoleLPF, so are these actually the same noise sequences, or are they different sequences, but with the same stochastic properties? $\endgroup$ – Marcus Müller Jan 4 '20 at 12:06
  • $\begingroup$ Marcus I meant if you mix them like (0.5*brownNoise)+(0.5*pinkNoise). I am curious. What would happen if you use the exact same noise source to be filtered as shown in the code of my question vs. if you used separate white noise signals for each? Thanks. $\endgroup$ – mike Jan 5 '20 at 4:56
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    $\begingroup$ You can always simply add up the powers of uncorrelated signals, i.e. if the noises are actually independent, you get simply the sum PSD. When they are identical, it simply looks as if you're filtering the input with a filter that has the sum of the individual filter impulse responses as impulse response. Let's make a model: when you have one "filter" that does nothing but delay the signal by $T$, and another that delays by $2T$, both would have perfectly flat passband, right? Now, add up the results of both when you feed in the same thing: you suddenly built a length-2 moving average; an LPF. $\endgroup$ – Marcus Müller Jan 5 '20 at 11:45
  • $\begingroup$ So, what exactly happens depends on the filters you've designed. What we can say for certain, though, is that you could have combined the two filters into one if both are of the same type. $\endgroup$ – Marcus Müller Jan 5 '20 at 11:48
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Does mixing brown noise and white noise create pink noise?

No.

Pink noise has a spectrum of that falls with 3dB/octave (or 10dB/decade). The spectrum of the sum of white and brown noise will be "brown" at low frequencies and "white" at high frequencies. The spectrum will have two slopes: below the transition frequency it will be -6dB/octave and above it, it will just be flat.

Somewhere in the transition region will be a frequency range that could be considered approximately "pink" but it's not really a constant slope.

That transition frequency is simply a function of the mix ratio (and the initial energies) of the two noise signals.

If you define "mixing" as a multiplication and not a summation, the result will just be white.

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