### 3.2 The matched filter in Gaussian noise

Let be the gravitational-wave signal we are looking for and let be the detector’s noise. For convenience we assume that the signal is a continuous function of time and that the noise 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 can be written as
The autocorrelation function of the noise is defined as
where E denotes the expectation value.

Let us further assume that the detector’s noise 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

where is the time interval over which the data was collected and the function is the solution of the integral equation

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

Spectral properties of stationary noise are described by its one-sided spectral density, defined for non-negative frequencies through relation
For negative frequencies , by definition, . The spectral density can also be determined by correlations between the Fourier components of the detector’s noise

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

where 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 (40) can be written as
From the expression (46) we see immediately that the likelihood ratio test consists of correlating the data with the signal 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 defined by

By means of Eq. (45) it can be written as
We see in the following that 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.