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=S0

2

, 0x xS H S

1

1

1

z

p

p

N

Nkk lN

l ll

l

i sA

Hi s

i s

Assuming that the transfer

function H() has the form

then the process x(t) can be seen as

1

pN

ll

x t y t

'

0' d 'ls t

l ly t A e t t t

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

5

The approach (2)

Once defined

A powerful recursive formula

ls Tl l ly t T y t e t T

'

0' d 'l

T s tl lt A e t t t

One can calculate cross correlation of the innovation processes

*

*

*0

* 1j k

jk j k

s s Tj kmn

j k

C mT nT

S A Ae

s s

*

0

*0

j k

j ky y

j k

S A AR

s s

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 Np uncorrelated random variables, transformed according to:

Eventually, add up the contribution from all correlated processes:

1

2

0

00

...

0pN

y

yy

y

1

2

0 ... 0

0 ... 01 1

... ... ... ...

0 0 ... Np

s T

s T

s T

e

ey n y n n

e

1

pN

ll

x n y n

1

pN

k kj jj

A

7

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 @ 1 Hz

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Some results

Resonance @10 mHz, Q= 103

106 points, evaluated in 60s

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Some results

Resonance @10 mHz, Q= 103

106 points, evaluated in 60s

Normalization problem, under investigation!

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Some results

Resonance @10 mHz, Q= 103

106 points, evaluated in 60s

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Some results

Resonances @10 mHz and 0.5 Hz, Q= 103 and 104

106 points, evaluated in 63s

Normalization problem, under investigation!

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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 …