The Adaptive Coherence Estimator is the Generalized Likelihood Ratio Test for a Class of Heterogeneous Environments

The adaptive coherence estimator (ACE) is known to be the generalized likelihood ratio test (GLRT) in partially homogeneous environments, i.e., when the covariance matrix M_s of the secondary data is proportional to the covariance matrix M_p of the vector under test (or M_s = gamma/M_p). In this letter, we show that ACE is indeed the GLRT for a broader class of nonhomogeneous environments, more precisely when M_s is a random matrix, with inverse complex Wishart prior distribution whose mean only is proportional to M_p. Furthermore, we prove that, for this class of heterogeneous environments, the ACE detector satisfies the constant false alarm rate (CFAR) property with respect to gamma and M_p.