Determination of threshold values of the generalized likelihood ratio in the problem of detecting partially coherent spatial signals in the case of short samples

Abstract

In this paper, we solve the problem of detecting multidimensional Gaussian complex signals with an a priori unknown spatial covariance matrix against the background of spatially nonuniform Gaussian noise with unknown power in the case of a fixed false-alarm probability. For an arbitrary sample size, exact analytical expressions are obtained for the moments of a decision statistic represented in the form of a generalized likelihood ratio raised to power reciprocal of a positive integer. The series expansion of the probability density function of the decision statistic in terms of orthogonal Jacobi polynomials is obtained by the method of moments. We use numerical simulation to demonstrate the high accuracy of approximating the probability density and finding the threshold value of the decision statistic. The obtained results hold for the case of short samples whose sizes are comparable with the number of elements of a receiving antenna.