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

8.2 Sensitivity for various GW signalsSensitivity for various GW signals 95Let us consider a signal s(t) in the presence of noise n(t) [8]. The availableinformation is the sumx(t) = s(t) + n(t) (8.15)where x(t) is the measurement at the output of the low-noise amplifier and n(t) isa random process with known properties. Let us start by applying to x(t) a linearfilter which must be such to maximize the SNR at a given time t 0 (we emphasizethe fact that we search the signal at a given time t 0 ).Indicating the impulse response of the filter as w(t) (to be determined) andwith y s (t) = s(t) ∗ w(t) and y n (t) = n(t) ∗ w(t), respectively, the convolutionsof the signal and of the noise, we haveSNR = |y s(t 0 )| 2E[|y n (t 0 )| 2 ] . (8.16)The expectation of the noise after the filter, indicating with N(ω) the powerspectrum of the noise n(t), isE[|y n (t 0 )| 2 ] = 1 ∫ ∞N(ω)|W(ω)| 2 dω (8.17)2π −∞where W(ω) is the Fourier transform of the unknown w(t).At t = t 0 the output due to s(t) with Fourier transform S(ω) is given byy s (t 0 ) = 1 ∫ ∞S(ω)W(ω)e jωt 0dω. (8.18)2π −∞We now apply [8] the Schwartz’ inequality to the integral (8.18) and usingthe identityS(ω)W(ω) =S(ω) √ N(ω)W(ω) √ N(ω) (8.19)we obtainSNR ≤ 1 ∫ ∞|S(ω)| 2d f. (8.20)2π −∞ N(ω)It can be shown [8] that the equals sign holds if and only ifW(ω) = constant S(ω)∗N(ω) e− jωt 0. (8.21)Applying this optimum filter to the data we obtain the maximum SNRSNR = 1 ∫ ∞|S(ω)| 2dω (8.22)2π −∞ N(ω)