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Abstract:
When the raw output of a gravitational wave detector is correlated with the matched filter of a coalescing binary wave form the filtered output shows a periodic behavior—it rings at a certain frequency. This phenomenon could be worrisome since the signal peak in the filtered output might be reduced if it falls in the ``trough'' of the sinusoid. In this paper we address this question and present a detailed examination of the ringing which is caused by the effective narrow banding by the matched filter of detector noise. We first solve the problem for an idealized ``box'' filter and show that the ringing frequency is roughly the central frequency of the box if the box is not too wide. For an idealized coalescing binary filter we show that the expected value of this frequency is 1.27 fs where fs is the seismic noise cutoff of the detector. The ringing implies that there is some redundancy in the filtered ouput. Also the autocorrelation function of the filtered output resembles the sinc function, and hence adjacent sample points are correlated, i.e., the filtered output is colored. These two phenomena are related and have a bearing on the setting up of thresholds and also suggest that we resample the filtered output at a coarser rate. We investigate the problem of thresholds when the filtered output is colored and obtain relations between the false alarm probabilities and threshold levels. Finally we suggest optimal sampling rates so that the resampled filtered output is uncorrelated.
