Deriving Time-Series Data from PSD when the Inverse Fourier Transform doesn't converge
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 $f&gt;f_{min}$ 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.