5.2 Suboptimal filtering methods

It is not always possible to compute the shape of the signal from a source. For instance, there is no computation, numerical or analytical, that reliably gives us the highly relativistic and nonlinear dynamics of gravitational collapse, the supernova that follows it and the emitted gravitational signal. The biggest problem here is the unknown physical state of the pre-supernova star and the complex physics that is involved in the collapse and explosion. Thus, matched filtering cannot be used to detect signals from supernovae.

Even when the waveform is known, the great variety in the shape of the emitted signals might render matched filtering ineffective. In binaries, in which one of the component masses is much smaller than the other, the smaller body will evolve on a highly precessing and in some cases eccentric orbit, due to strong spin-orbit coupling. Moreover, the radiation backreaction effects, which in the case of equal mass binaries are computed in an approximate way by averaging over an orbital time scale, should be computed much more accurately. The resulting motion of the small body in the Kerr spacetime of the larger body is extremely complicated, leading to a waveform that is rather complex and matched filtering would not be a practical approach.

Suboptimal methods can be used in such cases and they have a twofold advantage: they are less sensitive to the shape of the signal and are computationally significantly cheaper than matched filtering. Of course, the price is a loss in the SNR. The best suboptimal methods are sensitive to signal amplitudes a factor of two to three larger than that required by matched filtering and a factor of 10 to 30 in volume.

Most suboptimal techniques are one form of time-frequency transform or the other. They determine the presence or absence of a signal by comparing the power over a small volume in the time-frequency plane in a given segment of data to the average power in the same volume over a large segment of data. The time-frequency transform q(τ, f) of data x(t) using a window w(t) is defined as

∫ ∞ q(τ,f ) = w (t − τ)x (t)e2πiftdt. (90 ) −∞
The window function w(t − τ) is centered at t = τ, and one obtains a time-frequency map by moving the window from one end of a data segment to the other. The window is not unique and the effectiveness of a window depends on the signal one is looking for. Once the time-frequency map is constructed, one can look for excess power (compared to average) in different regions [36Jump To The Next Citation Point], or look for certain patterns.

The method followed depends on the signal one is looking for. For instance, when looking for unknown signals, all that can be done is to look for a departure from averaged behavior in different regions of the map [36]. However, when some knowledge of the spectral and temporal content of the signal is known, it is possible to tune the algorithm to improve efficiency. The wavelet-based waveburst algorithm is one such example [218] that has been applied to search for unstructured bursts in LIGO data [9].

One can employ strategies that improve detection efficiency over a simple search for excess power. For example, chirping signals will leave a characteristic track in the time-frequency plane, with increasing frequency and power as a function of time. Time frequency map of a chirp signal buried in noisy data is shown in Figure 7View Image. An algorithm that optimizes the search for specific shapes in the time-frequency plane is discussed in [189]. These and other methods have been applied to understand how to analyze LISA data [172389].

View Image

Figure 7: Time-frequency maps showing the track left by the inspiral of a small black hole falling into an SMBH as expected in LISA data. The left panel is for a central black hole without spin and the right panel is for a central black hole whose dimensionless spin parameter is q = 0.9.

More recently, there has been a lot of progress in extending burst search algorithms for a network of detectors [114219], as well as exploring new Bayesian-based methods to search for unknown transients [338].

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