We first present the false alarm and detection probabilities when the intrinsic parameters of the signal
are known. In this case the
-statistic is a quadratic form of the random variables that are correlations of
the data. As we assume that the noise in the data is Gaussian and the correlations are linear functions of
the data,
is a quadratic form of the Gaussian random variables. Consequently the
-statistic has a
distribution related to the
distribution. One can show (see Section III B in [65
]) that
for the signal given by Eq. (30
),
has a
distribution with
degrees of freedom
when the signal is absent and noncentral
distribution with
degrees of freedom and
non-centrality parameter equal to the square of the signal-to-noise ratio when the signal is
present.
As a result the pdfs and
of the
-statistic, when the intrinsic parameters are known and
when respectively the signal is absent or present in the data, are given by
Next we return to the case in which the intrinsic parameters are not known. Then the statistic
given by Eq. (62
) is a certain multiparameter random process called the random field
(see monographs [5, 6] for a comprehensive discussion of random fields). If the vector
has
one component the random field is simply a random process. For random fields we define the
autocovariance function
just in the same way as we define such a function for a random process:
One can estimate the false alarm probability in the following way [68]. The autocovariance function
tends to zero as the displacement
increases (it is maximal for
). Thus we
can divide the parameter space into elementary cells such that in each cell the autocovariance
function
is appreciably different from zero. The realizations of the random field within a cell
will be correlated (dependent), whereas realizations of the random field within each cell and
outside of the cell are almost uncorrelated (independent). Thus, the number of cells covering the
parameter space gives an estimate of the number of independent realizations of the random
field.
We choose the elementary cell with its origin at the point to be a compact region with boundary
defined by the requirement that the autocovariance
between the origin
and any point
at
the cell’s boundary equals half of its maximum value, i.e.,
. Thus, the elementary cell is defined
by the inequality
To estimate the number of cells we perform the Taylor expansion of the autocovariance function up to the second-order terms:
As If the components of the matrix are constant (i.e., they are independent of the values of the intrinsic
parameters
of the signal), the above equation defines a hyperellipsoid in
-dimensional (
is the
number of the intrinsic parameters) Euclidean space
. The
-dimensional Euclidean volume
of the elementary cell given by Eq. (95
) equals
To estimate the number of cells in the case when the components of the matrix are not constant,
i.e., when they depend on the values of the intrinsic parameters
, one replaces Eq. (97
) by
The concept of number of cells was introduced in [68] and it is a generalization of the idea of an effective number of samples introduced in [46] for the case of a coalescing binary signal.
We approximate the pdf of the -statistic in each cell by the pdf
of the
-statistic when
the parameters are known [it is given by Eq. (84
)]. The values of the
-statistic in each
cell can be considered as independent random variables. The probability that
does not
exceed the threshold
in a given cell is
, where
is given by Eq. (86
).
Consequently the probability that
does not exceed the threshold
in all the
cells is
. Thus, the probability
that
exceeds
in one or more cells is given by
It was shown (see [39]) that for any finite
and
, Eq. (99
) provides an upper bound for the
false alarm probability. Also in [39] a tighter upper bound for the false alarm probability was derived by
modifying a formula obtained by Mohanty [92
]. The formula amounts essentially to introducing a suitable
coefficient multiplying the number
of cells.
When the signal is present in the data a precise calculation of the pdf of the -statistic is very difficult
because the presence of the signal makes the data’s random process non-stationary. As a first approximation
we can estimate the probability of detection of the signal when the intrinsic parameters are unknown by the
probability of detection when these parameters are known [it is given by Eq. (87
)]. This approximation
assumes that when the signal is present the true values of the intrinsic parameters fall within the cell where
the
-statistic has a maximum. This approximation will be the better the higher the signal-to-noise ratio
is.
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Living Rev. Relativity 15, (2012), 4
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