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
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
Multiresolution Analysis LIGO • Optimal time-frequency signal to noise ratio � ∞ 2 | ˜ h ( f ) | 2 h 2 ρ 2 = rss S h ( f ) d f ≃ S h ( f c ) 0 • Only obtained if measurement pixel matches signal • Maximal measurement of burst “energy” � + ∞ | h ( t ) | 2 dt h 2 rss = −∞ • 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
Discrete Dyadic LIGO Wavelet Transform Project x [ n ] onto time-shifted and scaled wavelets. Haar Wavelet s=1 1 s=2 N − 1 � ( n − m ) T � x [ n ] 1 � √ X W [ m, s ] = 2 s ψ 2 s 0 1 2 t n =0 Dyadic Wavelet Decomposition Tree A0 Frequency A1 W1 A2 W2 A3 W3 Time LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 4/23
DWT Example LIGO Discrete Haar wavelet decomposition for simulated burst Haar Wavelet Decomposition Coefficients Approximate Frequency Resolution 8 8192 4096 Amplitude / Sigma 6 2048 1024 4 512 256 2 128 64 0 1 0 −1 0.02 0.04 0.06 0.08 0.1 0.12 time (s) LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 5/23
LIGO Discrete Q Transform Project x [ n ] onto time-shifted windowed sinusoids, whose widths are inversely proportional to their center frequencies. N − 1 � x [ n ] e − i 2 πnk/N w [ m − n, k ] X Q [ m, k ] = n =0 w[m − n, k] Frequency QN/k m n Time LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 6/23
LIGO FAST Q Transform Efficient computation is possible in frequency domain. N − 1 � X [ l + k ] ˜ ˜ W [ l, k ] e − i 2 πml/N X Q [ m, k ] = l =0 • 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
LIGO DQT Energy Conservation The window normalization is chosen to obey a generalized Parseval’s theorem. N − 1 N − 1 N − 1 f s � 2 = 1 | x [ n ] | 2 = σ 2 � � � � � � X Q [ m, k ] x N 2 N m =0 n =0 k =0 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
DQT Example LIGO Simulated sine-gaussian gravitational wave burst Q = 4 spectrogram 3 10 frequency [Hz] 2 10 29.9 29.92 29.94 29.96 29.98 30 30.02 30.04 30.06 30.08 30.1 time [seconds] 0 0.5 1 1.5 2 2.5 3 3.5 signal to noise ratio LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 9/23
SNR Loss due to LIGO 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 0.9 Fraction of SNR 0.8 0.7 50 60 0.6 70 80 90 0.5 1 2 3 4 5 Pixel Q 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
Optimal SNR LIGO Measurement Accuracy Optimal pixel match allows accurate measurement (Sine-Gaussian burst and white Gaussian noise) 2 10 Detected signal to noise ratio 1 10 0 10 0 1 2 10 10 10 Injected 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
LIGO White Noise Statistics DWT: Gaussian DQT: Exponential p(A) p(E) P(A>A 0 ) P(E>E 0 ) A 0 A E 0 E Significance: Significance: � � A � − E � P ( A ) = erfc √ P ( E ) = exp 2 σ A � E � � 1 / 2 RSS = ( E − � E � ) 1 / 2 � E − � E � SNR = � E � LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 12/23
LIGO Event Selection Wavelet Transform Q Transform Frequency Frequency Time Time • threshold on pixel • threshold on pixel significance significance • select vertical “chains” • group overlapping of significant pixels pixels • select most significant pixel in group LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 13/23
Linear Predictor Error LIGO Filter (LPEF) • Linear Prediction: Assume each sample is a linear combination of the previous M samples. M � x [ n ] = ˜ c [ m ] x [ n − m ] m =1 • 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. N e = 1 � σ 2 e [ n ] 2 N n =1 LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 14/23
LPEF Properties LIGO • 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 � f s /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
LPEF Example: Spectra LIGO Uncalibrated amplitude spectra Amplitude spectral density [counts Hz −1/2 ] −2 10 −3 10 −4 10 Raw HPF LPEF 1 2 3 10 10 10 Frequency [Hz] LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 16/23
LPEF Example: LIGO Time Series Uncalibrated time series 1 Signal 0 −1 1 HPF 0 −1 1 LPEF 0 −1 −100 −50 0 50 100 Time [milliseconds] LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 17/23
LPEF Example: Statistics LIGO DWT HPF DQT HPF 1 1 0.8 0.8 0.6 0.6 CDF CDF 0.4 0.4 0.2 0.2 0 0 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Uncalibrated Energy Uncalibrated Energy −7 x 10 DQT LPEF DWT LPEF 1 1 0.8 0.8 0.6 0.6 CDF CDF 0.4 0.4 0.2 0.2 0 0 0 0.002 0.004 0.006 0.008 0.01 0 1 2 3 4 5 Uncalibrated Energy Uncalibrated Energy −9 x 10 LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 18/23
Data Analysis Pipeline LIGO High Pass Filter Linear Predictor Error Filter Discrete Q Transform Discrete Wavelet Transform Thresholding Event Selection Coincidence and Veto LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 19/23
Simulated Gravitational LIGO Wave Data Simulated S2 noise curve for H1 −18 10 • Simulated H1 noise for second LIGO science run −19 Amplitude spectral density [strain Hz −1/2 ] 10 • Shaped gaussian white noise −20 10 • Included major lines • Random injections • Gaussians −21 10 • Sine-gaussians • Caveat: No glitches Actual Simulated −22 10 2 3 10 10 Frequency [Hz] LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 20/23
Preliminary LIGO Detection Efficiencies Wavelet Transform Q Transform 275 Hz Sine−Gaussian with Q of 9 1 1 0.9 0.8 0.8 Detection Efficiency 0.7 efficiency 0.6 0.6 1.5 × 10 −21 Hz −1/2 0.5 0.4 0.4 1.08 Hz false rate 0.3 0.2 0.2 0.1 0 0 -21 -20 1 2 3 4 5 10 10 Signal to noise ratio h [strain/rtHz] rss Sine-Gaussians Gaussians f: 275 Hz Q: 9 σ : 0.5, 1.0, 2.0 ms RSS: 1.5 × 10 − 21 RSS: 3.6, 5.2, 15 × 10 − 21 SNR: 3 SNR: 4.6, 4.3, 4.5 false rate 1.2 Hz false rate: 0.37 Hz LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 21/23
Black Hole Merger Model LIGO • 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. � M � − 1 f ISCO ≃ 2 × 10 3 Hz M ⊙ � M � − 1 f QNM ≃ 10 4 Hz M ⊙ LIGO-G030690-00-Z 8th Gravitational Wave Data Analysis Workshop, December 2003 22/23
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