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On the resonant detectors sensitivity to continuous signals
For a continuous (periodic or quasi-periodic) signal of amplitude $\bar{h}$ at the frequency $\bar \nu$ the squared modulus of the Fourier Transform provides ${\bar{h}^2}$ with a noise contribution of $2~S_h(\bar{\nu})~\delta\nu$, being $S_h(\bar{\nu})$ the two-sided noise power spectrum of the detector (measured in 1/Hz). Thus the SNR for periodic signals is:

\begin{displaymath}
SNR={{\overline{h}^2 t_{obs}} \over{2 S_h(\overline{\nu})}}
\end{displaymath}

The Eq. holds if the instantaneous frequency of the continuous signal at the detector is known. The analysis procedure in this case is ``coherent'', since the phase information contained in the data is used and the sensitivity (in amplitude) increases with the square-root of the time. For periodic waves the sensitivity of a bar detector at its resonances is given by:

\begin{displaymath}\overline{h}=2.04 \cdot 10^{-25} {\sqrt{ {T\over{0.05~K}}
{...
...{10^7\over{Q}} {900~Hz\over{\nu_0}} {1~day\over{t_{obs}}}
}}
\end{displaymath}

$T$ is the bar temperature; $M$ its mass; $Q$ the merit factor; $\nu_0$ the resonance frequency of the mode; $t_{obs}$ the observation time. After one year of effective observation, the minimum detectable $\overline{h}$ (amplitude detectable with SNR=1), using the nominal parameters of the Explorer detector ($T=2$ $K$, $M=2300$ $kg$, $Q=10^6$), is

\begin{displaymath}
\overline{h}=2 \cdot 10^{-25}
\end{displaymath}

For the NAUTILUS detectors, cooled at $T=0.1~K$, the expected sensitivity is: $\overline{h}\simeq 1.5 \cdot 10^{-26}$ at the resonances. In some cases it may be impossible, for various reasons, to perform a single Fourier Transform over all the data. This means that the observation time has to be divided in $M$ sub-periods, such that the spectral resolution of the spectra becomes $\delta \nu ^{'}=M / t_{obs}$ and the corresponding $SNR$ is $M$ times smaller than that given by Eq. 1. The $M$ spectra can be combined together, for example, by incoherent summation, that is by averaging the square modulus, or using pattern-tracking procedures (variuos papers have been done on this problem ) In this case the final spectral resolution is again $\delta\nu ^{'}$ but there is still some gain as the averaging reduces the variance of the noise in each bin. We obtain:

\begin{displaymath}
{SNR}^{\prime}={{\overline{h}^2 t_{obs}} \over{2 S_h(\overline{\nu}) \sqrt{M}}}
\end{displaymath}

We have developed a procedure for the search of signals from periodic sources in the data of gravitational wave detectors. We have analyzed data of the resonant detector Explorer, searching for sources located in the Galactic Center (GC). No signals with amplitude greater than $\overline{h}=
2.9~10^{-24}$, in the range 921.32-921.38 Hz, were observed using data collected over a time period of 95.7 days, for a source located at $\alpha=17.70 \pm 0.01$ hours and $\delta=-29.00 \pm 0.05$ degrees. The procedure we have used can be extended for any assumed position in the sky and for a more general all-sky search, with the proper frequency correction to account for the spin-down and Doppler effects. The results are being printed on PRD (2001).


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Pia Astone
2001-08-22