The performance of the GLRT for the spatial signals detection with a small number of observations

In this letter, the generalized likelihood ratio test (GLRT) is used for the detection of multidimensional Gaussian complex signals with unknown spatial covariance matrix on a background of additive Gaussian complex noise of a unknown power is solved. The exact analytical expressions for test-statistical moments were found for any numbers of observations. The expansion in a series of probability density functions of a resolving statistic for orthogonal Jacobi polynomials were obtained by method of moments. The accuracy of the GLRT-statistic cumulative distribution function approximation and the accuracy of threshold calculations for the given false alarm probability for various numbers of approximating series terms were checked by simulation. The results are correct for any number of samples.