Introduction

The generalized likelihood ratio (GLR) test is widely used for a solution of a detection task of signals with unknown statistical characteristics in case of noise present with unknown parameters too. The GLR- method uses a likelihood ratio as a test-statistic, in which all unknown parameters of a signal and noise are substituted by them maximum-likelihood (ML) estimates obtained from samples [1]. For example, in the case of signal propagation in random inhomogeneous channel there is a problem of a detection of a useful signal with an unknown wavefront and/or incomplete spatial coherence on the background of an additive noise with unknown intensity [2,3]. However, in general (for any sample size) the GLR test-statistic has complicate probability density function (PDF) and cumulative distribution function (CDF). The last fact does not allow to analytically find the test-statistic threshold values for a given constant level of a false alarm probability $P_{FA}$ (in accordance with Neyman-Pearson criterion). Therefore, for threshold calculation various numerical methods or asymptotic analytical methods good for large-sample tests only are used. In the present work a new method for the analytical calculation of the threshold for a given constant false alarm probability level in the case of small-sample test-statistic is described. Accuracy of the threshold evaluation by this method is investigated on the example of detection of complex multi-dimensional Gaussian signals with an a-priori unknown spatial covariance matrix on the background of complex additive spatially white noise with unknown power.