Noise and the Detection of Gravitational Waves
Professor Daniel Bessis
Research Assistant Professor Luca Perotti
We simulated a search for Gravitational Waves Bursts: the aim was to detect a weak short signal embedded in a single very long sequence of highly correlated noise and to determine the time at which it happened. R.Grosso has provided us with simulated data where a ``burst" is injected on a background ``noise" accurately reproducing the correlations found in the data from gravitational waves interferometric detectors; the ``burst" peak amplitude is approximately equal to the background amplitude once the lowest frequencies, known to contain no signal, have been filtered out  .The key points of the procedure we have devised are:1) To determine the time of the event, we use a variant of the time-frequency methods (see e.g. refs. [2,3]) which frequency analize small subsequent time intervals. The basic difference is that the Z-transform is used instead of the Fourier transform in the frequency analysis.Given a time-series a1, a2, a3,.....an,....... its Z-transform isdefined as:
In the presence of noise, the Z-transform of a finite number of damped oscillations is, an analytic function outside the unit disk, plus a rational function has a finite number of poles strictly inside of the unit disk, each of them representing one damped oscillation, plus couples of zeros and poles (Froissart doublets) concentrated around the unit disk and approaching the roots of unity, representing the noise. Signal and noise are thus clearly separated (see also our mathematical physics and computational physics research pages). 2) Because in this case the ``noise" also includes strong fake signals, it is highly correlated; this results in a non uniform distribution of the poles and zeros of the Padé approximant on the unit circle. Signals resulting in poles of the Z-transform Padé approximant too close to the unit circle, or appearing more than once are therefore considered due to the ``noise" and neglected. 3) We undersample each time sequence using what we call ``interlaced sampling" to obtain multiple time sequences from a single one and thus improve detection through coincidence. Figure 1 shows the noisy time sequence in the time interval where the signal has been detected, the poles of the Z-transform Padé approximant corresponding to the signal and a comparison of the original signal to the reconstructed one.
Supported by sub-award CREST to the center for Gravitational Waves, Texas University at Brownsville, Texas USA. For more details, follow the link to our arXiv preprint: http://arxiv.org/abs/0905.2000v1
Figure 1: a) The time sequence for block 76 of the simulated ``burst" data produced by R. Grosso (signal plus noise). b) Poles (black crosses) and zeros (red circles) remaining after cleaning the most likely Froissart doublets; to guide the eye, the unit circle is drawn in black. The highlighted square is magnified in the inset and shows the two poles in the upper complex planes individuated as corresponding to the signal (the complex conjugate poles in the lower complex plane can be seen in the full picture). c) the reconstructed signal (red dash) compared to the original one injected into the noise (black full line). The reconstructed signal is obtained using ρ= 0.97331 and φ= 0.65711 given by the average of the positions of the two poles shown in Figure 1 above. The vertical scale is arbitrary.