"In fact, people have actually made spatial pattern generators that allow you to input the frequency profile that you want, and get the corresponding point pattern out. It’s really quite neat, and I highly recommend reading this paper so you can see some other possible noise parameters, like anisotropy."

I had to hunt this 'custom colour' noise paper out since that link was dead. It is quite neat. Here it is on ACM in case anyone else is interested:

Point Sampling with General Noise Spectrum (2012) https://dl.acm.org/doi/10.1145/2185520.2185572