In order to detect signals we search for threshold crossings of the -statistic over the intrinsic parameter space. Once we have a threshold crossing we need to find the precise location of the maximum of in order to estimate accurately the parameters of the signal. A satisfactory procedure is the two-step procedure. The first step is a coarse search where we evaluate on a coarse grid in parameter space and locate threshold crossings. The second step, called a fine search, is a refinement around the region of parameter space where the maximum identified by the coarse search is located.
There are two methods to perform the fine search. One is to refine the grid around the threshold crossing found by the coarse search [94, 92, 134, 127], and the other is to use an optimization routine to find the maximum of [65, 78]. As initial values to the optimization routine we input the values of the parameters found by the coarse search. There are many maximization algorithms available. One useful method is the Nelder–Mead algorithm , which does not require computation of the derivatives of the function being maximized.
Usually the grid in parameter space is very large and it is important to calculate the optimum statistic as efficiently as possible. In special cases the -statistic given by Eq. (62) can be further simplified. For example, in the case of coalescing binaries can be expressed in terms of convolutions that depend on the difference between the time-of-arrival (TOA) of the signal and the TOA parameter of the filter. Such convolutions can be efficiently computed using FFTs. For continuous sources, like gravitational waves from rotating neutron stars observed by ground-based detectors  or gravitational waves form stellar mass binaries observed by space-borne detectors , the detection statistic involves integrals of the general form,  for details).
Living Rev. Relativity 15, (2012), 4
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