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

Summary of optimal signal filtering 341where õ(ω) and ˜q(ω) are the Fourier transforms of o(t) and q(t). The expectationvalue (or ensemble average) of this ratio defines the filtered signal-to-noise ratioρ[q](t) = σ [q](t) =∫ +∞−∞ 2π dω ˜h(ω) ˜q ∗ (ω)e iωt( ∫ ) +∞1/2. (17.6)−∞ 2π dω S n(ω)|˜q(ω)| 2The optimal filter (or Wiener filter) which maximizes the signal-to-noise (17.6) ata particular instant t = 0 (say), is given by the matched filtering theorem as˜q(ω) = γ ˜h(ω)S n (ω) , (17.7)where γ is an arbitrary real constant. The optimal filter (17.7) is matched on theexpected signal ˜h(ω) itself, and weighted by the inverse of the power spectraldensity of the noise. The maximum signal to noise, corresponding to the optimalfilter (17.7), is given by( ∫ +∞ρ =−∞dω2π| ˜h(ω)| 2 ) 1/2=〈h, h〉 1/2 . (17.8)S n (ω)This is the best achievable signal-to-noise ratio with a linear filter. In (17.8), wehave used, for any two real functions f (t) and g(t), the notation〈 f, g〉 =∫ +∞−∞dω2π˜ f (ω) ˜g ∗ (ω)S n (ω)(17.9)for an inner scalar product satisfying 〈 f, g〉 =〈f, g〉 ∗ =〈g, f 〉.In practice, the signal h(t) or ˜h(ω) is of known form (given, for instance,by (17.27)–(17.32) later) but depends on an unknown set of parameters whichdescribe the source of radiation, and are to be measured. The experimenters musttherefore use a whole family of filters analogous to (17.7) but in which the signalis parametrized by a whole family of ‘test’ parameters which are a priori differentfrom the actual source parameters. Thus, one will have to define and use a latticeof filters in the parameter space. The set of parameters maximizing the signalto noise (17.6) is equal, by the matched filtering theorem, to the set of sourceparameters. However, in a single detector, the experimenters maximize the ratio(17.5) rather than the signal to noise (17.6), and therefore make errors on thedetermination of the parameters, depending on a particular realization of noise inthe detector. If the signal-to-noise ratio is high enough, the measured values of theparameters are Gaussian distributed around the source parameters, with variancesand correlation coefficients given by the covariance matrix, the computation ofwhich we now recall. Since the optimal filter (17.7) is defined up to an arbitrarymultiplicative constant, it is convenient to treat separately a constant amplitudeparameter in front of the signal (involving, in general, the distance of the source).We shall thus write the signal in the form˜h(ω; A,λ a ) = A ˜k(ω; λ a ), (17.10)