The CFAR adaptive subspace detector is a scale-invariant GLRT

The CFAR matched subspace detector (CFAR MSD) is the uniformly-most-powerful-invariant test and the generalized likelihood ratio test (GLRT) for detecting a target signal in noise whose covariance structure is known but whose level is unknown. When the noise covariance matrix is unknown, the CFAR adaptive subspace detector (CFAR ASD) uses instead a sample covariance matrix based on training data. We show that the CFAR ASD is GLRT when the test measurement has a different noise level than the training data. This GLRT differs from the well-known GLRT of Kelly (1986) in two respects: (1) their respective hypothesis testing problems, and (2) their group-transformation invariances. Thus the CFAR ASD is given a formal justification and placed in context with the Kelly GLRT and its variants, which have been called adaptive matched filters (AMFs) in the literature.