Gravitational waves, data analysis
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Inspiral events: short durations, all frequencies
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Make short duration filters, correlate with the data
Continuous events: long duration, small frequency band
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Make filter in narrow band, apply to all frequency bands in the data.
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Exercise: find an “inspiral” event in 256 seconds of mock data.
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Signal shape: Chirp. Frequency rises during inspiral.
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Red: signal shape for 2 10-solar mass Bhs. Blue: 2 1.4 solar mass NS.
PSD : in noise.h
/home/henkjan/TOPICAL/include/noise.h
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PSD noise is given, to be used in optimal filter
The signal has much less power than the noise. For the optimal filter, divide
the signal strength by the amplitude of the noise for that frequency bin.
Fourier transforms: from numerical recipes (on vectors with a length of a
power of 2)
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four1(VecComplex,1) transforms from time->frequency
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four1(VecComplex,-1) transforms from frequency->time
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VecComplex can be used as a std::vector of type complex
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e.g. real(vec[i]) is the real part, abs(vec[i]) the amplitude, arg(vec[i]) the phase etc.
To make one: VecComplex xxx(N) with N the amount of entries.
The transforms should be renormalized with a square root of the number of bins.
Exercise Fourier transforms
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Data file, 256 seconds, 4096 samples per second
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Sqrt(PSD) : maximally 10-14 The PSD (power spectral density) is the power in the noise, it is the
fourier-transform of the data squared.
double noise_psd(double freq) { // returns the norm of the expected amplitude, the square root of the PSD
if (freq>2048) freq=4096-freq; // Nyquist frequency at 2048 Hz
if (freq<1) return 1e-14;
double psd= (1e-25*freq+1e-9/pow(freq,8)+5e-20/freq);
if (psd>1e-14) return 1e-14;
return psd
};
PSD(f)=10−25+185/
if(PSD)−>1014=
Signal shape: in signal.h
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/home/henkjan/WINDOWS/CompMeth/CompMeth2017/EXERCISES/ex6/signal.h
inspiral of equal-mass compact binaries, the result will make a black hole
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The signal can be calculated back from the coalescence time. The time you give
in signal.h is the time before coalescence.
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The frequency increases until coalescence. A heavier object merges at lower
frequency.
strategy
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Data contains a signal, with a total power many orders of magnitude smaller than the
data.
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Collapse time is somewhere between 50 and 256 seconds
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Can be found with optimal filter
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Make filters that collapse at a certain time in seconds, e.g. at t=256 (end of filter)
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Fourier transform them
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Normalize power to 1
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Divide bins in Fourier spectrum by power spectral density
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Multiply the complex conjugate of the filter with the data
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Transform back to time domain to get array of lags
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Determine maximum - give lag and mass used in filter
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Determine the uncertainty on those parameters: find the mass range that give correlations 1
standard deviation lower
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Test your procedure by making e.g. a filter for a mass 10 solar masses and coalescence time of 70
seconds and see if you find it back with the right normalization!
Exercise 6
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Due Monday March 13.
256 seconds of data sampled at 4096 Hz is provided. The data
contains an equal-mass inspiral event. Find the mass and the
coalescence time by optimal filtering; try to get the result to within 1
standard deviation.
The coalescence time is between 50 and 256 seconds. The masses
of the inspiral event are between 1.4 solar masses (neutron stars)
and 35 solar masses (heavy stellar black holes).