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
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
[58, 59, 78, 76], and the other is to use an optimization routine to find the
maximum of
[40
, 50
]
. As initial value 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
[51]
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 Equation (33
) 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 Fast Fourier Transforms (FFTs). For
continuous sources, like gravitational waves from rotating
neutron stars observed by ground-based detectors
[40
]
or gravitational waves form stellar mass binaries observed by
space-borne detectors
[50
], the detection statistic
involves integrals of the general form
In order to test the performance of the
maximization method of the
statistic it is useful to perform Monte Carlo simulations of the
parameter estimation and compare the variances of the estimators
with the variances calculated from the Fisher matrix. Such
simulations were performed for various gravitational-wave
signals
[45, 16, 40
]
. In these simulations we observe that above a certain
signal-to-noise ratio, that we call the threshold signal-to-noise
ratio, the results of the Monte Carlo simulations agree very well
with the calculations of the rms errors from the inverse of the
Fisher matrix. However, below the threshold signal-to-noise ratio
they differ by a large factor. This threshold effect is
well-known in signal processing
[82]
. There exist more refined theoretical bounds on the rms errors
that explain this effect, and they were also studied in the
context of the gravitational-wave signal from a coalescing
binary
[63
]
. Here we present a simple model that explains the deviations
from the covariance matrix and reproduces well the results of the
Monte Carlo simulations. The model makes use of the concept of
the elementary cell of the parameter space that we introduced in
Section
4.5.2
. The calculation given below is a generalization of the
calculation of the rms error for the case of a monochromatic
signal given by Rife and Boorstyn
[69]
.
When the values of parameters of the template
that correspond to the maximum of the functional
fall within the cell in the parameter space where the signal is
present, the rms error is satisfactorily approximated by the
inverse of the Fisher matrix. However, sometimes as a result of
noise the global maximum is in the cell where there is no signal.
We then say that an
outlier
has occurred. In the simplest case we can assume that the
probability density of the values of the outliers is uniform over
the search interval of a parameter, and then the rms error is
given by
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