3.2 The matched filter in Gaussian noise

Let h be the gravitational-wave signal we are looking for and let n be the detector’s noise. For convenience we assume that the signal h is a continuous function of time t and that the noise n is a continuous random process. Results for the discrete-in-time data that we have in practice can then be obtained by a suitable sampling of the continuous-in-time expressions. Assuming that the noise is additive the data x can be written as
x(t) = n(t) + h(t). (38 )
The autocorrelation function of the noise n is defined as
Kn (t,t′) := E [n(t)n(t′)], (39 )
where E denotes the expectation value.

Let us further assume that the detector’s noise n is a zero-mean and Gaussian random process. It can then be shown that the logarithm of the likelihood function is given by the following Cameron–Martin formula

∫ To ∫ To logΛ [x] = q(t)x(t)dt − 1- q(t)h (t)dt, (40 ) 0 2 0
where ⟨0;T ⟩ o is the time interval over which the data was collected and the function q is the solution of the integral equation
∫ T o ′ ′ h(t) = 0 Kn (t,t)q(t )dt. (41 )

For stationary noise, its autocorrelation function (39View Equation) depends on times t and t′ only through the difference t − t′. It implies that there exists then an even function κn of one variable such that

′ ′ E [n (t)n (t )] = κn (t − t). (42 )
Spectral properties of stationary noise are described by its one-sided spectral density, defined for non-negative frequencies f ≥ 0 through relation
∫ ∞ Sn (f) = 2 κn(t)e2πiftdt. (43 ) −∞
For negative frequencies f < 0, by definition, Sn(f) = 0. The spectral density Sn can also be determined by correlations between the Fourier components of the detector’s noise
1 E [&tidle;n(f)&tidle;n∗(f′)] = -Sn(|f|)δ(f − f′), − ∞ < f,f ′ < ∞. (44 ) 2

For the case of stationary noise with one-sided spectral density Sn, it is convenient to define the scalar product (x|y ) between any two waveforms x and y,

∫ ∞ &tidle;x(f)&tidle;y∗(f) (x|y) := 4ℜ ---------df, (45 ) 0 Sn(f)
where ℜ denotes the real part of a complex expression, the tilde denotes the Fourier transform, and the asterisk is complex conjugation. Making use of this scalar product, the log likelihood function (40View Equation) can be written as
1- log Λ[x] = (x|h) − 2(h|h). (46 )
From the expression (46View Equation) we see immediately that the likelihood ratio test consists of correlating the data x with the signal h that is present in the noise and comparing the correlation to a threshold. Such a correlation is called the matched filter. The matched filter is a linear operation on the data.

An important quantity is the optimal signal-to-noise ratio ρ defined by

----- ρ := ∘ (h|h). (47 )
By means of Eq. (45View Equation) it can be written as
∫ ∞ 2 2 |&tidle;h-(f-)|- ρ = 4 0 Sn (f) df. (48 )
We see in the following that ρ determines the probability of detection of the signal. The higher the signal-to-noise ratio the higher the probability of detection.

An interesting property of the matched filter is that it maximizes the signal-to-noise ratio over all linear filters [44]. This property is independent of the probability distribution of the noise.

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