Spectral Filtering for CW Searches

1. Spectral filtering for CW searches S. D’Antonio*, S. Frasca%&, C. Palomba& * INFN Roma2 % Universita’ di Roma “La Sapienza” & INFN Roma Abstract: In our all-sky search method for periodic sources, after the first incoherent step based on the Hough transform, we have a number of candidates that must be analyzed using a much longer time baseline periodogram. In this step we correct the signal power spread due to the Earth rotation, which becomes relevant for observation times greater than ~1 day, using a bank of matched filters in the frequency domain. The filters depend on three parameters: the polarization angle of the linearly polarized wave component, the percentage of linear polarization, the phase difference between the linearly and circularly polarized wave components. Here we discuss in detail the issue of how to discretize the parameter space in order to minimize the number of filters for a given sensitivity loss. This same step is used also as part of the procedure for targeted searches. GWDAW 13 – San Juan (Portorico) – Jan 19-22, 2009

2. All-sky search scheme On the candidates resulting from the incoherent step of the all-sky search, or when targeting a known source, we apply a coherent procedure (Astone et al. PRD65 022001) Scheme of the hierarchical pipeline • Start from the SFDB • Compute the analytical signal, extracting the frequency band of interest • Correct the phase shift (Doppler, spin-down, timing noise) multiplying by • Compute the power spectrum • Due to the Earth rotation the power of a signal of angular frequency is spread into five bands: • : Earth angular frequency Spectral filtering to correct the spread (Frasca & Palomba CQG 21, S1645, 2004)

3. The spread becomes relevant for observation times and must be corrected to avoid sensitivity loss • For a given detector and source declination, the amount of power in the five bands depends on three parameters: : percentage of linear polarization : polarization angle (of the linearly polarized component) : phase difference between the circular and linear parts

4. We correct for this spread by applying a bank of matched filters in the frequency domain. • A filter is a set of 5 numbers (‘weights’) proportional to the square root of the power content of the five bands (with ) • The filter output would be where • is the square root of the data power spectrum • But, as the Earth rotation frequency is not, in general, a multiple of the frequency bin, it is better to build the filters in the delay domain (with a proper procedure): • where is the Fourier transform of

5. Here is an example of the filter output in the case of signal only (with position and gravitational wave frequency corresponding to the Vela pulsar). The signal frequency and energy are correctly recovered.

6. A key point is the number of filters we have to build. This depends on the sensitivity loss we can allow. • To study this problem it is more convenient to work in the space of the five line amplitudes rather than in the space of physical parameters • We have evaluated the loss factor (Lf) due to the use of a not perfectly matched filter (discretization of the amplitudes space). : signal (i.e. a set of five spectral amplitudes) : template (built using the nearest point on the grid of the amplitudes) • Using n=10 steps for each amplitude, the loss is ~ 2%

7. The choice of n determines the total number of filters we should build to cover the parameter space n=10  Nfilters=1000 • We have found that the number of filters can be drastically reduced exploiting the dependencies among the amplitudes of the spectral lines • By plotting the amplitude of four lines as a function of the (suitably chosen) fifth, we have that very often a simple relation comes out

8. For instance, for a source on the equator the amplitude of sidebands has a perfect linear dependency on the amplitude of the central line. • For a source at the pole, the line at has a linear dependency on the amplitude of the line at (and the others have amplitude zero)  The number of needed filters decreases of a factor of the order of 100.

9. In cases like these, we have a non negligible spread but still smaller than the step width and a polynomial fit for the lines amplitude works very well. • The error introduced using the fit is smaller than (or comparable to) the discretization error.

10. In the regions the situation is worse but a polynomial fit still works reasonably well for most points of the parameter space. • For a small portion of the parameter space the loss factor can be high. • E.g., for we have a loss bigger than 15% in ~7% of the cases

11. In cases like this a parametrization using a combination of two amplitudes works very well.

12. Conclusions • The spectral filtering is an important step for both candidates follow-up in all-sky searches and for targeted searches of continuous gravitational signals. • Here we have shown that the amplitude of the five spectral lines, in which the power of a signal would be spread due to the Earth rotation, are not independent. • This allows to strongly reduce the number of filters needed to do the analysis and, consequently, makes negligible its computationally load.