# Windowing and Leakage in the Cross-Correlation Search for Periodic Gravitational Waves

Document #:
LIGO-T1200431-v1
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T - Technical notes
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Abstract:
We consider the impact of spectral leakage and windowing on the sensitivity of the cross-correlation search for periodic gravitational waves. We consider the modification to the expected signal-to-noise ratio ($$\propto h_0^2$$, so perhaps better thought of as SNR-squared) in the detection statistic relative to the {\naive} formula which assumes rectangular windows and signal frequency always in the center of an SFT bin. On average we expect the SNR associated with a search of rectangular-windowed data to be $$77.4\%$$ of the {\naive} value. This is still better than the average expected from Hann-windowed data ($$60.1\%$$) and data processed with a half-Hann/half-rectangular Tukey window ($$69.9\%$$). Even though the Hann and Tukey windows leak a smaller fraction of their best-case SNR out of the best bin, the best-case scenarios are not as good--$$66.7\%$$ and $$81.8\%$$, respectively, of the {\naive} SNR is obtained even if the Doppler-shifted frequency always falls in the center of a bin.

The sensitivity of the search can be improved by including contributions from multiple SFT bins. In general this requires accounting for correlations between bins, but for rectangularly-windowed data those correlations vanish and the combination is simpler, and results in an improvement of SNR from $$77.4\%$$ to $$90.3\%$$ of the {\naive} value when the two closest bins from each SFT are included in the search, and to $$93.1\%$$ with the three closest bins. These values all come from an assumption that the sum over SFT pairs effects an average over the fractional offset of the signal frequency from the SFT bin center, an assumption which we investigate for several choices of search parameters.

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