Random number generator with arbitrary (pink) power spectrum and element-to-element correlations

[DavidSagan]

When using long_term_tracking with ramper elements simulating noise (EG ground motion or Rf noise), the use of the ran() or ran_gauss() function generates noise that has a white spectrum. This is probably not a good approximation for something like ground motion whose spectrum will typically be in the 0 - 10 Hz range.

There are two issues to worry about:

• The noise spectrum at a given lattice element
• The correlation of the noise spectrum between separate lattice elements (useful for, say, ground motion).

In terms of constructing a random number generator with an arbitrary power spectrum see, for example: https://dsp.stackexchange.com/questions/47640/generating-a-timeseries-with-an-arbitrary-power-spectrum The problem with this method is that the number of random values to generate needs to be known beforehand. Another possibility is to use a lowpass LRC filter network.

[DavidSagan]

Thought: Use the algorithm: Timmer, J. and Koenig, M.: On generating power law noise. Astron. Astrophys. 300, 707-710 (1995)

Have number of points Np (default, say, 10000) and frequency rolloff coef beta (default = 1) set by the user (via bmad_com?). Have generator calculate Np points and store for use. After being used, generator rerandomizes and generates a new set of Np points.

[DavidSagan]

And to smooth out the power spectrum. interleave sets of points using triangular weighting.

[DavidSagan]

Some thoughts:

• [ ] Restrict to using 2^N points to take advantage of FFT algorithms (or just tell the user it is best to do this?).
• [ ] The noise should have zero mean. That is: f(w = 0) = 0.
• [ ] The noise should have unit sigma.
• [ ] Have two signal output functions. One for equally spaced time points (uses FFT and is fast) and the other for arbitrary time points which evaluates the inverse Fourier transform (and is slow).