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 as
From the expression (19) 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 [28] . This property is independent of the probability distribution of the noise.
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