I'm trying to figure out a way to construct (numerically) time series of random variables given some spectral density $S_{xx}(f)$. I'm interested in various types of spectral densities (flat, lorentzian, 1/f) so if there is a general method that starts from an expression of $S_{xx}(f)$ and generates a suitable time series that would be ideal of course, but I am not sure if there is.
Honestly I am not so sure where to start. My guess is that you compose your random signal as a sum of different components with amplitudes related to your spectral density at a certain frequency. I understand that if you want your noise to have a complete frequency range you need infinitely many terms, so I suppose you cut off your noise at some frequency $f_{max}$, giving you some maximum amount of terms $K$. You'll also need some step size $\Delta f$ for the different frequency components, but after that I get kind of stuck. How do you incorporate your spectral density into this?
I understand that the question is perhaps a bit too general, so I'm also very much interested in advice on some sources on the subject. The ones I have been able to find are very technical, while for now I'm simply trying to create some time series in Mathematica given some spectral density as input, after which I want to see if I can reconstruct the spectral density from the output. That would get me started quite well, and I can think about what features are essential from there on.