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For white noise, where the PSD is constant, we can model that as a simple Gaussian. Is there any way to model the probability density function of other types of noise (pink noise, red noise, blue noise, etc.), and if so, is there any way to derive such a thing?

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    $\begingroup$ first of all PSD and probability density function (p.d.f.) are not the same thing. "Gaussian" has nothing to do with the power spectrum. Gaussian is a property of the probability density function. you can have white noise and uniform p.d.f. or pink noise and a gaussian p.d.f. or some other p.d.f. and some completely different power spectrum. $\endgroup$
    – robert bristow-johnson
    Commented Mar 28, 2015 at 2:00
  • $\begingroup$ Right, I didn't mean to suggest that PSD and PDF are the same thing; poor wording on my part. So white/Gaussian noise can have a non-Gaussian PDF? I think that's where my confusion came from, thanks. $\endgroup$
    – Andrew M
    Commented Mar 28, 2015 at 3:25
  • $\begingroup$ p.d.f. and power spectrum are two different things. power spectrum is directly related to the autocorrelation function, and if you make an assumption of "ergodicity", then the autocorrelation function can be derived from conditional probability functions. conceptually, you can have a random process with any combination of power spectrum and p.d.f. assuming you have a good white and gaussian random number generator, there is an iterative process one can make with filtering (to get the power spectrum you want) and non-linear shaping (to get the p.d.f. you want). $\endgroup$
    – robert bristow-johnson
    Commented Mar 28, 2015 at 4:26
  • $\begingroup$ a regular uniform random number generator (like rand( )) will output a white and uniform p.d.f. random process, not gaussian. to make it approximately gaussian, you can either run the uniform through a properly-defined non-linear curve or, they usually do it by adding 12 or more independent uniform p.d.f. random numbers together. $\endgroup$
    – robert bristow-johnson
    Commented Mar 28, 2015 at 4:29

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This is usually done with filtering of white noise.

  • White noise generator + LP filter give you a red noise generator.
  • White noise generator + HP filter give you a blue noise generator

Design a filter with required frequency response and apply it to a white noise generator.

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  • $\begingroup$ there is more to "red" noise than just low-pass filtered white noise. it's a specific low-pass filter. i dunno how "blue noise" is defined spectrally, but i don't doubt it's some kinda high-pass filtered white. $\endgroup$
    – robert bristow-johnson
    Commented Mar 28, 2015 at 2:02

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