(ed.). Gravitational waves (IOP, 2001)(422s).

96 Resonant detectors for gravitational waves and their bandwidthwhere S(ω) and N(ω) as already specified are, respectively, the Fourier transformof the signal and the power spectrum of the noise at the end of the electronic chainwhere the measurement x(t) is taken.Let us apply the above result to the case of measurements x(t) done at theend of a chain of two filters with transfer functions W a (representing the bar) andW e (representing the electronics).Let S uu be the white spectrum of the Brownian noise entering the bar andS ee the white spectrum of the electronics noise. The total noise power spectrumisN(ω) = S uu |W a | 2 |W e | 2 + S ee |W e | 2 . (8.23)The Fourier transform of the signal isS(ω) = S g (ω)W a W e . (8.24)where S g is the Fourier transform of the GW signal at the bar entrance.The optimum filter will have, applying equation (8.21), the transfer functionW(ω) = S∗ g e− jωt 0S uu1W a W e11 + Ɣ|W a | 2 (8.25)whereƔ = S ee. (8.26)S uuUsing equations (8.23) and (8.24), from equation (8.22) we obtainSNR = 12π S uu∫ ∞−∞|S g (ω)| 2 dω1 + Ɣ|W a | 2 . (8.27)We now apply the above result to a delta GW with Fourier transform S gindependent of ω. The remaining integral of equation (8.27) can be easily solvedif we make use of a lock-in device which translates the frequency, bringing theresonant frequency ω 0 to zero. Then the total noise becomes [7]with the bar transfer function given byN(ω) ⇒ N(ω − ω 0 ) + N(ω + ω 0 ) (8.28)W a = β 1β 1 + iω . (8.29)We now estimate the signal and noise just after the first transfer function, beforethe electronic wide-band noise. For the signal due to a delta excitation we haveV (t) = 12π∫ ∞−∞β 1β 1 + iω S g dω = S g β 1 e −β 1t = V s e −β 1t(8.30)