Adaptive Matched Direction Detector

We consider the problem of detecting a partially unknown signal, in the presence of unknown noise, using multiple snapshots in the primary data. To account for uncertainties about signal's signature, we assume that the steering vector lies on an unknown line in a known linear subspace. Additionally, we consider a partially homogeneous environment, for which the covariance matrix of the primary and the secondary data have the same structure, but possibly different levels. We study the invariances of the detection problem and derive the maximal invariant. A two-step generalized likelihood ratio test (GLRT) is formulated and compared with a 2-step GLRT which assumes that the steering vector is known