A synthetic noise generator M. Hueller LTPDA meeting, AEI Hannover 27/04/2007

2 Purpose Simulate noise data with given continuous spectrum Choose between  input the model parameters (developing and modeling)  fit experimental data Use as a tool for system identification: data simulation

3 Input parameters: available features LP filters HP filters f -2 noise, by a LP filter with roll-off at very low frequency f -1 noise, by a cascade of LP filters with very low roll-off frequencies (not yet implemented) Mechanical resonances Mechanical forcing lines (not yet implemented)

4 The approach (1) x(t) is the output of a filter, with transfer function H(  ), with a white noise  (t) at input, with PSD=S 0 Assuming that the transfer function H(  ) has the form then the process x(t) can be seen as the process x(t) is equivalent to N p correlated processes

5 The approach (2) Once defined A powerful recursive formula One can calculate cross correlation of the innovation processes And for the starting values

6 Matlab implementation (1) Vector of starting values, with the given statistics Propagate through time evolution, adding contributions from innovation processes Innovations are evaluated starting from N p uncorrelated random variables, transformed according to: Eventually, add up the contribution from all correlated processes:

7 Matlab implementation (2) The base changing matrix A kj contains the eigenvectors of the cross-correlation matrix (diagonalization) Additionally, a phase factor must be applied to each eigenvector, to allow the sum of all the Np contribution to be real ↓ Force the first element of each eigenvector to be real Call from the command line (or other routines): [t_res2,x_res2] = syntetic_noise(1e6,10,'lp',1,'res',[1e-2 0.5],[ ],'notalk',‘nopl');

8 Numeric approach: some (precision?) problem associated with the calculation of the eigenvalues, impacting on the eigenvectors, being investigated Imaginary part of the output process x(t) is not zero This disagreement is associated with resonances (complex values in the cross correlation matrix) Disagreement increases with the number of resonances Compared with Mathematica evaluation, “zero” is bigger by a factor ~10 6 Workaround using Symbolic Math Toolbox? Coding not finished yet High-precision calculation in Mathematica passing the eigenvectors matrix to Matlab routine? This is also being considered

9 Some results LP, 1 Hz

10 Some results mHz, Q= points, evaluated in 60s

11 Some results mHz, Q= points, evaluated in 60s Normalization problem, under investigation !

12 Some results mHz, Q= points, evaluated in 60s

13 Some results mHz and 0.5 Hz, Q= 10 3 and points, evaluated in 63s Normalization problem, under investigation !

14 What comes next: Get the fitting features to work Pick the best solution for numerical precision Include into the AO architecture Use it as the tools for system identification …