Adaptive Detection in Elliptically Distributed Noise and Under-Sampled Scenario

The problem of adaptive detection of a signal of interest embedded in elliptically distributed noise with unknown scatter matrix R is addressed, in the specific case where the number of training samples T is less than the dimension M of the observations. In this under-sampled scenario, whenever R is treated as an arbitrary positive definite Hermitian matrix, one cannot resort directly to the generalized likelihood ratio test (GLRT) since the maximum likelihood estimate (MLE) of R is not well-defined, the likelihood function being unbounded. Indeed, inference of R can only be made in the subspace spanned by the observations. In this letter, we present a modification of the GLRT which takes into account the specific features of under-sampled scenarios. We come up with a test statistic that, surprisingly enough, coincides with a subspace detector of Scharf and Friedlander: the detector proceeds in the subspace orthogonal to the training samples and then compares the energy along the signal of interest to the total energy. Moreover, this detector does not depend on the density generator of the noise elliptical distribution. Numerical simulations illustrate the performance of the test and compare it with schemes based on regularized estimates of R.