Let be the gravitational-wave signal and let be the detector 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 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 aszero-mean, stationary, and Gaussian random process, the log likelihood function is given by one-sided spectral density of the noise in the detector, which is defined through equation
From the expression (21) 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
An interesting property of the matched filter is that it maximizes the signal-to-noise ratio over all linear filters . This property is independent of the probability distribution of the noise.
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