GW SEMINAR: FINAL EXAM: FINDING A SIGNAL IN NOISE
GABRIELA GONZÁLEZ
We will use the waveform that was “blindly” injected in the LIGO data, as described in:
http://www.ligo.org/science/GW100916/index.php
You can find more details about this injection in Search for Gravitational Waves from
Low Mass Compact Binary Coalescence in LIGO’s Sixth Science Run and Virgo’s Science
Runs 2 and 3, Phys. Rev D85 (2012) 082002 and arXiv:1111.7314 and in Parameter
estimation for compact binary coalescence signals with the first generation gravitationalwave detector network, arXiv:1304.1775, and in the recent story in Science 15 March 2013:
Vol. 339 no. 6125 p. 1260.
Waveform
The injected waveform corresponded to a binary system with m1 = 24.81Ms and m2 =
1.71Ms , where Ms is the mass of the Sun.
(1) Using formulas 3.190 in Section 3.5 in Creighton-Wiseman textbook, calculate the
time series before coalescence of the ”Newtonian” waveform h(t) for a system optimally
aligned at 1Mpc away from Earth, from the time it has a frequency of 40 Hz (see formulas
3.178). Compare the waveform with the actual waveform injected in H1 in the website
below:
http://www.phys.lsu.edu/faculty/gonzalez/Teaching/GWseminar/FinalExam/H1-timeTemplateStrain.dat
I include in Fig.1 the plot you should see (but you should produce your own plot).
Notice the large differences in amplitude, and the differences in the envelope - comment on
possible reasons for those differences.
(2) Calculate numerically your Newtonian waveform’s Fourier transform1. Compare the
absolute value of the Fourier transforms with the analytical expression in Problem 3.7 in
the textbook, and with the Fourier transform h(f ) of the injected waveform given in
http://www.phys.lsu.edu/faculty/gonzalez/Teaching/GWseminar/FinalExam/H1-freqTemplateStrain.dat
Comment on the differences you see (amplitude at low and high frequencies, sharp
corners, ripples,). I include in Fig. 2 the plot of the absolute values of the waveforms you
should see.
1Hints: choose a number of points that is a power of 2 dropping some points at the beginning; normalize
appropriately, for example in Matlab x(f ) = f f t(x)dt, with dt the sampling time for points in the vector
x.
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GABRIELA GONZÁLEZ
Figure 1. Waveforms of gravitational waves, using a ”newtonian chirp”
formula (top), and the waveform injected in the H1 detector in 2010.
Data Fourier transform and noise PSD
For the next exercise, download the first 8 seconds of data s(t) for the H1 detector
including the time of the injection from
http://www.ligo.org/science/GW100916/H-strain hp30-968654552-10.txt
Also, generate 8 seconds of simulated strain random data sr (t) with zero mean and mean
square 10−21 .
Fig. 3 shows the injected waveform strain time series compared with the actual LIGO
H1 data and the simulated data.
(3) Calculate the Fourier transform s(f ) of the H1 data (using 8 seconds) using a Hann
window, and a noise power spectral density Sn (f ) also using a Hann window. Do the same
for the simulated random data (the spectral density of white noise should be approximately
GW SEMINAR: FINAL EXAM: FINDING A SIGNAL IN NOISE
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Figure 2. Fourier transforms of gravitational waveforms.
R
constant). The PSD integral Sn (f )df should be approximately the same as the mean
square of the (windowed) data. You may want to resample some of these quantities to
have them all correspond to the same frequency vector.
p
Figure 4 shows the amplitude spectral density Sn (f ) of
√ the windowed real data, as
well as the amplitude of the data Fourier transform |s(f )| f (with similar units to the
ASD), and the waveform Fourier transform scaled in the same way.
Large Signal to noise ratio time series
(4) If you use the injected waveform Fourier transform h(f ), you should expect a peak in
SNR at the beginning of the template (see Fig1). To shift the peak to the largest amplitude
point, use the template multiplied by e2iπf ∆t , where ∆t is the length of the waveform time
series used.
Using the shifted waveform Fourier transform h(f ) and the noise PSD Sn (f ) to calculate
a template normalization:
Z ∞
|h(f )|2
df
σ2 = 4
Sn (f )
0
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GABRIELA GONZÁLEZ
Figure 3. Injected waveform time series compared with actual H1 data and
simulated data. Notice the different vertical scales, and that the injected
waveform in the top panel is multiplied by 10,000.
Compare the result with what you would get in white noise with rms 10−21 .
(5) Calculate the signal to noise ratio time series ρ = |z(t)|/σ, where
Z ∞
s(f )h∗ (f ) i2πf t
z(t) = 4
e
df
Sn (f )
0
Fig 5. shows the SNR time series I obtained with the procedure above. Find the time
of the maximum SNR (within a fraction of a second) - how does that compare with the
GPS time quoted in the website below? What do you think the envelope of the time series
is due to? For comparison, I show in Fig 6 the SNR time series using the same procedure
on random white data with 10−21 rms (without windowing). Do you think the differences
in SNR peak and time series are consistent with expectations?
http://www.ligo.org/science/GW100916/index.php
GW SEMINAR: FINAL EXAM: FINDING A SIGNAL IN NOISE
Figure 4. H1 amplitude spectral density, and Fourier transform of H1 data
and injected waveform. Notice that the ASD of white noise with 10−21 rms
strain sampled at 1 kHz is constant, about 3 × 10−21 .
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GABRIELA GONZÁLEZ
Figure 5. SNR time series for the injected waveform in H1 (windowed) data.
GW SEMINAR: FINAL EXAM: FINDING A SIGNAL IN NOISE
Figure 6. SNT time series of the waveform injected in H1 in random data
with rms 10−21 .
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