in the short sample case

O. V. Bolkhovskaya, A. A. Maltsev, L. Lo Presti, F. Sellone

The generalized likelihood ratio (GLR) method uses a
likelihood ratio as a test-statistic, in which all
unknown parameters of a signal and noise are substituted by
them ML-estimates. In general, obtained test-statistic
has a complicate probability distribution. This does not
allow to find analytically the test-statistic threshold
for the given constant level of false alarm
probability . Therefore, various numerical or
asymptotic methods can be adopted. Unfortunately such
methods perform well only when the number of collected
samples is large, hereafter referred as *large sample
case*. In the present work the constant-false-alarm-rate
(CFAR) detection task of multidimensional Gaussian complex
signals with unknown spatial covariance matrix on a
background of additive Gaussian complex noise of a unknown
power is solved for the case where only few samples are
collected, hereafter referred as *small sample case*.
In spite of the fact that the distribution function of the
random variable is not represented in an analytical
form, the exact analytical expressions for statistical
moments of any order for the function were
found. On the next step the approximating series for
probability density function (PDF)of the random variable
was constructed with the help of the beta probability
distribution and orthogonal Jacobi polynomials on the base
of known test-statistic higher orders moments. The
accuracy of the test-statistic cumulative distribution
function (CDF) approximation and the accuracy of threshold
calculations on the given for various
numbers of approximating series terms were checked by
simulation. It is shown, that the employed approximating
series has very high rate of convergence.