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 [79], 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 [65
] or gravitational waves form stellar mass
binaries observed by space-borne detectors [78
], the detection statistic
involves integrals of the general
form
http://www.livingreviews.org/lrr-2012-4 |
Living Rev. Relativity 15, (2012), 4
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