A maximal invariant framework for adaptive detection with structured and unstructured covariance matrices

We introduce a framework for exploring array detection problems in a reduced dimensional space by exploiting the theory of invariance in hypothesis testing. This involves calculating a low-dimensional basis set of functions called the maximal invariant, the statistics of which are often tractable to obtain, thereby making analysis feasible and facilitating the search for tests with some optimality property. Using this approach, we obtain a locally most powerful invariant test for the unstructured covariance case and show that all invariant tests can be expressed in terms of the previously published Kelly´s generalized likelihood ratio (GLRT) and Robey´s adaptive matched filter (AMF) test statistics. Applying this framework to structured covariance matrices, corresponding to stochastic interferers in a known subspace, for which the GLRT is unavailable, we obtain the maximal invariant and propose several new invariant detectors that are shown to perform as well or better than existing ad-hoc detectors. These invariant tests are unaffected by most nuisance parameters, hence the variation in the level of performance is sharply reduced. This framework facilitates the search for such tests even when the usual GLRT is unavailable