We first present the false alarm and
detection pdfs 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
-statistic has a distribution related to the
distribution. One can show (see Section III B in
[40
]) that for the signal given by Equation (12
),
has a
distribution with 4 degrees of freedom when the signal is absent
and noncentral
distribution with 4 degrees of freedom and non-centrality
parameter equal to signal-to-noise ratio
when the signal is present.
As a result the pdfs
and
of
when the intrinsic parameters are known and when respectively
the signal is absent and present are given by
Next we return to the case when the intrinsic
parameters
are not known. Then the statistic
given by Equation (33
) is a certain generalized multiparameter random process called
the
random field
(see Adler’s monograph
[4]
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 can 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
[41]
. 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 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. The
correlation hypersurface is a closed surface defined by the
requirement that at the boundary of the hypersurface the
correlation
equals half of its maximum value. The elementary cell is defined
by the equation
Let
be the number of the intrinsic parameters. If the components of
the matrix
are constant (independent of the values of the parameters of the
signal) the above equation is an equation for a hyperellipse. The
-dimensional Euclidean volume
of the elementary cell defined by Equation (55
) 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 parameters, we write Equation (57
) as
The concept of number of cells was introduced in [41] and it is a generalization of the idea of an effective number of samples introduced in [30] for the case of a coalescing binary signal.
We approximate the probability distribution of
in each cell by the probability
when the parameters are known [in our case by probability given
by Equation (44
)]. 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 Equation (46
). Consequently the probability that
does not exceed the threshold
in
all
the
cells is
. The probability
that
exceeds
in
one or more
cell is thus given by
It was shown (see
[25]) that for any finite
and
, Equation (59
) provides an upper bound for the false alarm probability. Also
in
[25]
a tighter upper bound for the false alarm probability was
derived by modifying a formula obtained by Mohanty
[59
]
. The formula amounts essentially to introducing a suitable
coefficient multiplying the number of cells
.
When the signal is present a precise
calculation of the pdf of
is very difficult because the presence of the signal makes the
data random process
non-stationary. As a first approximation we can estimate the
probability of detection of the signal when the parameters are
unknown by the probability of detection when the parameters of
the signal are known [given by Equation (47
)]. This approximation assumes that when the signal is present
the true values of the phase parameters fall within the cell
where
has a maximum. This approximation will be the better the higher
the signal-to-noise ratio
is.
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