1. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Experiments on Noise Analysis l Need of noise characterization for  Monitoring the instrument behavior  Provide an estimate of the noise level  Detect deviations from the gaussianity or stationarity l Plan of the seminar: three examples  Classical spectral estimation, based on multi-tapers  Modern spectral estimation, based on AR models  Kharounen-Loeve expansion l All the methods have been tested using LIGO 40m data

2. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Multi-tapering in one slide l Purpose: control both bias and variance of a spectral estimate, over a finite sample. l Perform several spectral estimates with different windows, and average. l Choose the windows so as to be “orthogonal”, at fixed frequency resolution. l Use the so called Discrete Prolate Spheroidal Sequences

3. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré A typical noise spectrum (LIGO 40m) l Wideband l Narrow spectral features  Physical resonances  Harmonics of the line l Need to monitor the spectrum over time.

4. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Comparison of spectral estimates (1) l The Hamming window gives the best resolution. l Lower variance from the multi-taper estimate l Better choice: adapt the coefficients of the different tapers. l Warning: this is actually an harmonic of the 60 Hz line!

5. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Comparison of spectral estimates (2) l The use of several different windows is possible only reducing the frequency resolution: NumTapers <= N  l The f.r. is necessarily more limited, as at the 300 Hz line. l Adapting the tapers helps off-resonance.

6. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Modern spectral estimates l Model the noise as gaussian noise filtered through a linear model. l Estimate the model parameters. l Choose the model order on the basis of the final prediction error.

7. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Low order models l The FPE criterion is not robust enough: it is not sensitive to the narrow spectral features. l The suggested order is definitely too small.

8. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Higher order models l Increasing the order the narrow features are resolved. l Monitoring the values of the coefficients, that is the zeros (and poles, for ARMA models) one can detect non- stationarities. l But: some of the lines are actually discrete components of the spectrum.

9. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Karhounen-Loeve basis l Spectral estimates rely on the statistical independence of the different lines l The KL basis gives statistically independent coefficients also over short samples.

10. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Eigenvalues and RMS noise l The basis elements are the eigenvectors of the correlation matrix R. l The eigenvalues measure how each component contributes to the RMS noise.

11. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré KL as a selector of spectral features l Each KL element actually corresponds to some spectral feature. l They are ordered on the basis of the relative RMS “importance.”

12. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Coefficient monitoring (1) l Pairs of KL basis elements correspond to the same eigenvalue l Coefficients estimated from different samples are uncorrelated and gaussianly distributed. l In other languages they correspond to the Principal Components of the noise spectrum

13. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Coefficient monitoring (2) l Elements of the discrete part of the spectrum (infinitely narrow lines) appear much differently l In a scatter plot they manifest perfect correlation, and their relative fase remains correlated in time. l Without surprise, this component corresponds to an harmonic of the line.

14. Experiments on Noise CharacterizationRoma, March 10,1999Andrea Viceré Coefficient monitoring (3) l Deviations from the predicted model may signal a change of noise level in the specific component. l In the specific example, the damping out of a resonance possibly excited by the lock acquisition.