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
.