Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Experiments on Noise Analysis Need of noise characterization for

Monitoring the instrument behaviorProvide an estimate of the noise levelDetect deviations from the gaussianity or stationarity

Plan of the seminar: three examplesClassical spectral estimation, based on multi-tapersModern spectral estimation, based on AR modelsKharounen-Loeve expansion

All the methods have been tested using LIGO 40m data

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Multi-tapering in one slide Purpose: control both

bias and variance of a spectral estimate, over a finite sample.

Perform several spectral estimates with different windows, and average.

Choose the windows so as to be “orthogonal”, at fixed frequency resolution.

Use the so called Discrete Prolate Spheroidal Sequences

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

A typical noise spectrum (LIGO 40m) Wideband

Narrow spectral features Physical resonances Harmonics of the line

Need to monitor the spectrum over time.

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Comparison of spectral estimates (1) The Hamming

window gives the best resolution.

Lower variance from the multi-taper estimate

Better choice: adapt the coefficients of the different tapers.

Warning: this is actually an harmonic of the 60 Hz line!

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Comparison of spectral estimates (2) The use of several

different windows is possible only reducing the frequency resolution: NumTapers <= N

The f.r. is necessarily more limited, as at the 300 Hz line.

Adapting the tapers helps off-resonance.

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Modern spectral estimates Model the noise as

gaussian noise filtered through a linear model.

Estimate the model parameters.

Choose the model order on the basis of the final prediction error.

2

1

11

P

tP

P

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Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Low order models The FPE criterion is not

robust enough: it is not sensitive to the narrow spectral features.

The suggested order is definitely too small.

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Higher order models Increasing the order the

narrow features are resolved.

Monitoring the values of the coefficients, that is the zeros (and poles, for ARMA models) one can detect non-stationarities.

But: some of the lines are actually discrete components of the spectrum.

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Karhounen-Loeve basis Spectral estimates rely on

the statistical independence of the different lines

The KL basis gives statistically independent coefficients also over short samples.

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N

n

ntnt

ccE

cx

1

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Eigenvalues and RMS noise The basis elements are

the eigenvectors of the correlation matrix R.

The eigenvalues measure how each component contributes to the RMS noise.

N

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nt

NM

nnt

ntn

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cx

R

1

2

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~

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Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

KL as a selector of spectral features Each KL element

actually corresponds to some spectral feature.

They are ordered on the basis of the relative RMS “importance.”

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Coefficient monitoring (1) Pairs of KL basis

elements correspond to the same eigenvalue

Coefficients estimated from different samples are uncorrelated and gaussianly distributed.

In other languages they correspond to the Principal Components of the noise spectrum

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Coefficient monitoring (2) Elements of the discrete

part of the spectrum (infinitely narrow lines) appear much differently

In a scatter plot they manifest perfect correlation, and their relative fase remains correlated in time.

Without surprise, this component corresponds to an harmonic of the line.

Experiments on Noise Characterization Roma, March 10,1999Andrea Viceré

Coefficient monitoring (3) Deviations from

the predicted model may signal a change of noise level in the specific component.

In the specific example, the damping out of a resonance possibly excited by the lock acquisition.