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 Msof the secondary data is proportional to the covariance matrix Mpof the vector under test (or Ms= gamma/Mp). In this letter, we show that ACE is indeed the GLRT for a broader class of nonhomogeneous environments, more precisely when Msis a random matrix, with inverse complex Wishart prior distribution whose mean only is proportional to Mp. 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 Mp.