Multiresolution time-frequency searches for gravitational wave - - PowerPoint PPT Presentation

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Multiresolution time-frequency searches for gravitational wave - - PowerPoint PPT Presentation

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LIGO Multiresolution time-frequency searches for gravitational wave bursts Shourov K. Chatterji Lindy Blackburn Gregory J. Martin Erik Katsavounidis shourov@ligo.mit.edu Massachusetts Institute of Technology LIGO-G030690-00-Z 8th

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SLIDE 1

LIGO

Multiresolution time-frequency searches for gravitational wave bursts

Shourov K. Chatterji Lindy Blackburn Gregory J. Martin Erik Katsavounidis

shourov@ligo.mit.edu

Massachusetts Institute of Technology

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 1/23

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SLIDE 2

LIGO

Presentation Outline

  • Multiresolution analysis
  • The discrete wavelet transform
  • The discrete Q transform
  • Gaussian white noise statistics
  • Selection of events
  • Linear predictor error filters
  • Analysis Pipeline
  • Simulated gravitational wave data
  • Burst detection efficiencies
  • Black hole mergers

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 2/23

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SLIDE 3

LIGO

Multiresolution Analysis

  • Optimal time-frequency signal to noise ratio

ρ2 = ∞ 2|˜ h(f)|2 Sh(f) d f ≃ h2

rss

Sh(fc)

  • Only obtained if measurement pixel matches signal
  • Maximal measurement of burst “energy”

h2

rss =

+∞

−∞

|h(t)|2 dt

  • Minimize background energy
  • Bursts are signals with Q 10
  • Tile the time frequency plane to maximize the

detectability of bursts with a particular Q

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 3/23

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SLIDE 4

LIGO

Discrete Dyadic Wavelet Transform

Project x[n] onto time-shifted and scaled wavelets.

XW[m, s] =

N−1

  • n=0

x[n] 1 √ 2sψ (n − m)T 2s

  • 1

2 1 s=2 s=1 Haar Wavelet t

Time Frequency

W3 A3 A2 A1 A0 W1 W2 Dyadic Wavelet Decomposition Tree

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 4/23

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SLIDE 5

LIGO

DWT Example

Discrete Haar wavelet decomposition for simulated burst

Haar Wavelet Decomposition Coefficients 8192 4096 2048 1024 512 256 128 64 0.02 0.04 0.06 0.08 0.1 0.12 −1 1 time (s) Amplitude / Sigma 2 4 6 8

Approximate Frequency Resolution

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 5/23

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SLIDE 6

LIGO

Discrete Q Transform

Project x[n] onto time-shifted windowed sinusoids, whose widths are inversely proportional to their center frequencies.

XQ[m, k] =

N−1

  • n=0

x[n]e−i2πnk/Nw[m − n, k]

Time Frequency w[m − n, k] QN/k m n

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 6/23

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SLIDE 7

LIGO

FAST Q Transform

Efficient computation is possible in frequency domain.

XQ[m, k] =

N−1

  • l=0

˜ X[l + k] ˜ W[l, k]e−i2πml/N

  • One time FFT of signal: ˜

X[l]

  • Frequency domain window: ˜

W[l]

  • Inverse FFT for each frequency bin
  • Only for frequency bins of interest
  • Only for samples in proximity of window
  • Length determines overlap in time

Fast dyadic wavelet transform is also possible.

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 7/23

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SLIDE 8

LIGO DQT Energy Conservation The window normalization is chosen to obey a generalized Parseval’s theorem.

fs N2

N−1

  • m=0

N−1

  • k=0
  • XQ[m, k]
  • 2 = 1

N

N−1

  • n=0

|x[n]|2 = σ2

x

The square root of the reported pixel energy yields the sum of the background noise amplitude spectral density and the signal root sum square in units of Hz−1/2.

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 8/23

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SLIDE 9

LIGO

DQT Example

Simulated sine-gaussian gravitational wave burst

Q = 4 spectrogram 29.9 29.92 29.94 29.96 29.98 30 30.02 30.04 30.06 30.08 30.1 time [seconds] 10

2

10

3

frequency [Hz] signal to noise ratio 0.5 1 1.5 2 2.5 3 3.5

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 9/23

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SLIDE 10

LIGO

SNR Loss due to Pixel Mistmatch

Mismatch between a signal and the nearest time frequency pixel will result in a loss in measured signal to noise ratio. (Sine-Gaussian burst and white Gaussian noise)

1 2 3 4 5 0.5 0.6 0.7 0.8 0.9 1 Pixel Q Fraction of SNR 50 60 70 80 90

This is similar to the problem of selecting discrete template banks in a matched filtering analysis.

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 10/23

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SLIDE 11

LIGO

Optimal SNR Measurement Accuracy

Optimal pixel match allows accurate measurement (Sine-Gaussian burst and white Gaussian noise)

10 10

1

10

2

10 10

1

10

2

Injected signal to noise ratio Detected signal to noise ratio

Error due to statistical fluctuation in background noise and error in mean background energy measurement.

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 11/23

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SLIDE 12

LIGO

White Noise Statistics

DWT: Gaussian

A0 p(A) A P(A>A

0)

Significance:

P(A) = erfc

  • A

√ 2 σA

  • SNR =

E − E E 1/2

DQT: Exponential

E0 p(E) E P(E>E

0)

Significance:

P(E) = exp

  • − E

E

  • RSS = (E − E)1/2

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 12/23

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SLIDE 13

LIGO

Event Selection

Wavelet Transform

Time Frequency

  • threshold on pixel

significance

  • select vertical “chains”
  • f significant pixels

Q Transform

Frequency Time

  • threshold on pixel

significance

  • group overlapping

pixels

  • select most significant

pixel in group

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 13/23

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SLIDE 14

LIGO

Linear Predictor Error Filter (LPEF)

  • Linear Prediction: Assume each sample is a linear

combination of the previous M samples.

˜ x[n] =

M

  • m=1

c[m]x[n − m]

  • Prediction Error: We are interested in the unpredictable

signal content.

e[n] = x[n] − ˜ x[n]

  • Training: Choose c[m] to minimize the mean squared

prediction error.

σ2

e = 1

N

N

  • n=1

e[n]2

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 14/23

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SLIDE 15

LIGO

LPEF Properties

  • Linear least squares optimal filter problem
  • Robust efficient algorithms exist to train and apply
  • Levinson-Durbin recursion
  • Produces minimum phase FIR filter
  • Frequency domain autocorrelation and filtering
  • Zero-phase implementation exists
  • Filter order, M, can compensate for features

∆f fs/M

  • Training time, T can learn about features

∆f 1/T

  • Performance depends upon detector stationarity

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 15/23

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SLIDE 16

LIGO

LPEF Example: Spectra

10

1

10

2

10

3

10

−4

10

−3

10

−2

Frequency [Hz] Amplitude spectral density [counts Hz−1/2] Uncalibrated amplitude spectra Raw HPF LPEF

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 16/23

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SLIDE 17

LIGO

LPEF Example: Time Series

−1 1 Signal Uncalibrated time series −1 1 HPF −100 −50 50 100 −1 1 Time [milliseconds] LPEF

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 17/23

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SLIDE 18

LIGO

LPEF Example: Statistics

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1 Uncalibrated Energy CDF DWT HPF 1 2 3 4 5 x 10

−7

0.2 0.4 0.6 0.8 1 Uncalibrated Energy CDF DQT HPF 0.002 0.004 0.006 0.008 0.01 0.2 0.4 0.6 0.8 1 Uncalibrated Energy CDF DWT LPEF 1 2 3 4 5 x 10

−9

0.2 0.4 0.6 0.8 1 Uncalibrated Energy CDF DQT LPEF

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 18/23

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SLIDE 19

LIGO

Data Analysis Pipeline

Discrete Q Transform Linear Predictor Error Filter High Pass Filter Discrete Wavelet Transform Thresholding Event Selection Coincidence and Veto

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 19/23

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SLIDE 20

LIGO

Simulated Gravitational Wave Data

  • Simulated H1 noise

for second LIGO science run

  • Shaped gaussian

white noise

  • Included major lines
  • Random injections
  • Gaussians
  • Sine-gaussians
  • Caveat: No glitches

10

2

10

3

10

−22

10

−21

10

−20

10

−19

10

−18

Frequency [Hz] Amplitude spectral density [strain Hz−1/2] Simulated S2 noise curve for H1 Actual Simulated

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 20/23

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SLIDE 21

LIGO

Preliminary Detection Efficiencies

Wavelet Transform

[strain/rtHz]

rss

h

  • 21

10

  • 20

10 efficiency 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Gaussians

σ: 0.5, 1.0, 2.0 ms

RSS: 3.6, 5.2, 15 ×10−21 SNR: 4.6, 4.3, 4.5 false rate: 0.37 Hz Q Transform

1 2 3 4 5 0.2 0.4 0.6 0.8 1 Detection Efficiency Signal to noise ratio 275 Hz Sine−Gaussian with Q of 9 1.5 × 10−21 Hz−1/2 1.08 Hz false rate

Sine-Gaussians f: 275 Hz Q: 9 RSS: 1.5 ×10−21 SNR: 3 false rate 1.2 Hz

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 21/23

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SLIDE 22

LIGO

Black Hole Merger Model

  • Equal mass black holes with no spin
  • Optimally oriented with isotropic emission
  • Fraction of rest mass energy emitted, ǫ = 0.01
  • Detectable amplitude signal to noise ratio, ρ = 5
  • Dimensionless Kerr spin parameter, a = 0.9
  • Energy distributed uniformly in frequency between the

ISCO and QNM frequencies.

fISCO ≃ 2 × 103 M M⊙ −1

Hz

fQNM ≃ 104 M M⊙ −1

Hz

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 22/23

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SLIDE 23

LIGO

Black Hole Mergers Predicted Range

Predicted from published detector noise spectra for second LIGO science run and simple merger model.

10 10

1

10

2

10

3

10

−2

10

−1

10 10

1

10

2

10

3

Detectable range for second LIGO science run Individual black hole mass [Msolar] Detectable range [Mpc] LIGO I Andromeda Virgo L1 H1 H2

LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 23/23