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So I'm trying to get some sample time-series data from pink noise (PSD: $S(f) = 1/f$). However, when I try to evaluate the Inverse Fourier Transform integral

$\int \frac{1}{\omega} e^{j \omega t} dt$

from negative infinity to positive infinity, I get a non-converging integral. I tried a Taylor Expansion, but I'm running into the same issue. Is there some other way around this?

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Even if you had succeeded in deriving the inverse Fourier transform of $1/|f|$ (note that you need the absolute value) then you still wouldn't have a time-series with a $1/f$ behavior. What you would get is the auto-correlation sequence.

A good way to generate pink ($1/f$) noise is to generate (approximately) white noise with a random number generator, and filter that noise with a filter whose frequency response approximates $1/\sqrt{f}$ over a large frequency range. Of course the frequency response needs to have a finite value at $f=0$, but it is possible to design a filter approximating $1/\sqrt{f}$ for $f>f_{min}$ with a relatively small value for $f_{min}$. If you have a question as to how to design such a filter you should probably formulate this as a new question. You can also have a look at this site.

And by the way, the inverse Fourier transform of $1/|f|$ does exist in terms of distributions: check this.

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