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Consider a -element antenna array with arbitrary
locations of sensors. The -dimensional input signal
is assumed to be complex random Gaussian vector. We suppose
that samples of the signal
are
statistically independent and identically distributed
(i.i.d.) zero-mean random vectors with spatial covariance
matrix . By reviewing of non-singular case, let us
suppose that the sample size is greater than the number
of the antenna elements . The problem of detection
of some spatially correlated useful signal by the antenna
array is formulated as a classical two-hypothesis
alternative:
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(1) |
where is the identity matrix, and is the
a-priori unknown noise power.
For the problem considered herein, the GLR test-statistic is given by
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(2) |
where
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(3) |
is the likelihood function for complex variables, is the
parameter sub-region corresponding to the null hypothesis in
the parameter space , is the determinant of a
matrix and the superscript represents the transpose
conjugate or Hermitian operator. Note that the
test-statistic obtained accordingly with (2)
can accept values only in interval [0,1].
It can be shown [1] that maximum value of the
GLR (2) denominator is achieved by using
maximum likelihood estimation
of the
covariance matrix :
and this value is equal to
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(4) |
The maximum of the GLR (2) numerator should be
obtained in parameter sub-region () corresponding to the null hypothesis (1).
If hypothesis is true, the likelihood function is given as:
where
is the
likelihood function of the signal from the -th sensor.
From the last expression for the likelihood function the
maximum value of the GLR (2) numerator is
easily obtained
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(5) |
and this maximum is achieved by using maximum likelihood
estimation
of the variance
where
Taking into account the
expressions (4), (5) the GLR
test-statistic (2) for the problem
(1) can be represented as:
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(6) |
The rejection region of the hypothesis is determined by
the inequality , where is the test-statistic
threshold only dependent upon the given false alarm probability level
.
Next: The Exact Analytical Expression
Up: Approximation of distribution function
Previous: Introduction