Analysis of Optimal Diagonal Loading for MPDR-Based Spatial Power Estimators in the Snapshot Deficient Regime

The minimum power distortionless response (MPDR) beamformer minimizes the output power while passing the look direction signal with unity gain. To alleviate the performance degradation caused by estimating the spatial correlation matrix with a relatively small number of snapshots of the received signal compared to the number of sensors, a regularization implemented via diagonal loading of the estimated correlation matrix is used. This paper presents a study for the optimal diagonal loading that minimizes the estimation mean square error (MSE) of two diagonally loaded MPDR beamformer-based spatial power estimators in the snapshot deficient regime. First, the asymptotic behavior of the power estimators for fixed diagonal loading is analyzed and the approximate characterization of their expectations is derived. Second, it is conjectured that because of the snapshot deficient sample support, the squared bias is the factor that primarily controls the optimal diagonal loading. Finally, the respective performances of the two power estimators are compared using MSE as the metric and it is shown that one outperforms the other. The analytical results are validated using simulation data.