On the Asymptotic Performance of Diagonally Loaded Detectors for Large Arrays: To Achieve CFAR and Optimality

This paper addresses two critical limitations in diagonally loaded (DL) adaptive matched filter (AMF) detector: (1) the lack of CFAR property with respect to arbitrary covariance matrices, and (2) the absence of selection criteria for optimal loading factor from the perspective of maximizing the detection probability (Pd). We provide solutions to both challenges through a comprehensive analysis for the asymptotic performance of DL-AMF under large dimensional regime (LDR) where the dimension N and sample size K tend to infinity whereas their ratio N/K converges to a constant c\in(0,1). The analytical results show that any DL detectors constructed by normalizing the random variable |a|2=|sH(R+λIN)-1y0|2 with a deterministic quantity or a random variable that converges almost surely to a deterministic value will exhibit equivalent performance under LDR. Following this idea, we derive two CFAR DL detectors: CFAR DL semi-clairvoyant matched filter (CFAR-DL-SCMF) detector and CFAR DL adaptive matched filter (CFAR-DL-AMF) detector, by normalizing |a|2 with an appropriate deterministic quantity and its consistent estimate, respectively. The theoretical analysis and simulations show that both CFAR-DL-SCMF and CFAR-DL-AMF achieve CFAR with respect to covariance matrix, target steering vector and loading factor. Furthermore, we derive the asymptotically optimal loading factor λ_opt by maximizing the explicit expression of asymptotic Pd. For practical implementation, we provide a consistent estimator for λ_opt under LDR. Based on λ_opt and its consistent estimate, we establish the optimal CFAR-DL-SCMF (opt-CFAR-DL-SCMF) and the optimal CFAR-DL-AMF (opt-CFAR-DL-AMF). Numerical examples demonstrate that the proposed opt-CFAR-DL-SCMF and opt-CFAR-DL-AMF consistently outperform EL-AMF and persymmetric AMF in both full-rank and low-rank clutter plus noise environments.