Asymptotically Optimal CFAR Detectors

This paper investigates the asymptotic optimality of the Constant False Alarm Rate (CFAR) tests obtained using the Minimal Invariant Group (MIG) reduction. We show that the CFAR tests obtained after MIG reduction using the Wald test is a Separating Function Estimation Test (SFET) and that the Generalized Likelihood Ratio Test (GLRT) and the Rao test are asymptotically SFET using Maximum Likelihood Estimation (MLE) under some mild conditions. Thus, they are asymptotically optimal. In order to find an improved test and motivated by the invariance property of the MLE of induced maximal invariant, we maximize the asymptotic Probability of Detection of the SFET using the MLE after reduction. We propose a systematic method allowing to derive the asymptotically optimal Separating Function (AOSF). This AOSF is obtained as the Euclidean distance of the transformed parameters under two hypotheses such that the gradient of the transformed parameters is the Cholesky decomposition of the Fisher Information Matrix (FIM), i.e., the FIM is transformed into an identity matrix. Interestingly, the AOSF Estimation Test (AOSFET) using MLE simplifies to the Wald-CFAR wherever the FIM does not depend on the unknown parameters. The simulation results show that the proposed AOSFET usually outperforms the GLRT, Wald test and Rao test.