A synthetic noise generator

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A synthetic noise generator

• M. Hueller
• LTPDA meeting, AEI Hannover 27/04/2007

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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

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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)

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The approach (1)

• x(t) is the output of a filter, with transfer
  function H(w), with a white noise e(t) at input,
  with PSDS0

• Assuming that the transfer function H(w) has the
  form

• then the process x(t) can be seen as

• the process x(t) is equivalent to Np correlated
  processes

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The approach (2)

• A powerful recursive formula

• Once defined

• One can calculate cross correlation of the
  innovation processes

• And for the starting values

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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 Np
  uncorrelated random variables, transformed
  according to
• Eventually, add up the contribution from all
  correlated processes

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Matlab implementation (2)

• The base changing matrix Akj 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 0.5,1000 10000,'notalk',nopl')
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• 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 106
• 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

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Some results

• LP, roll-off _at_ 1 Hz

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Some results

• Resonance _at_10 mHz, Q 103
• 106 points, evaluated in 60s

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Some results
Normalization problem, under investigation!

• Resonance _at_10 mHz, Q 103
• 106 points, evaluated in 60s

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Some results

• Resonance _at_10 mHz, Q 103
• 106 points, evaluated in 60s

13
Some results
Normalization problem, under investigation!

• Resonances _at_10 mHz and 0.5 Hz, Q 103 and 104
• 106 points, evaluated in 63s

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• 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