A CFAR Adaptive Subspace Detector for First-Order or Second-Order Gaussian Signals Based on a Single Observation

In this paper, we consider the problem of detecting a signal in Gaussian noise with unknown covariance matrix, in partially homogeneous environments where the test and training data samples share the same noise covariance matrix up to an unknown scaling factor. One solution to this problem is the adaptive subspace detector (ASD) with a single or multiple observations. However, the probabilities of false alarm and detection of this ASD have not been obtained yet. In this paper, these expressions are derived on the basis of a single observation, which are confirmed with Monte Carlo simulations. It is shown that the ASD has a constant false alarm rate property with respect to both the shared noise covariance matrix structure and the independent scaling of the noise in the test data. In addition, we prove that for the First-Order model where the signal of interest is assumed to be a deterministic but unknown vector, the ASD derived with the generalized likelihood ratio test is consistent with that derived with an ad hoc two-step design procedure.