Robust minimum variance beamforming with dual response constraints

The minimum variance distortionless response (MVDR) beamformer is a popular method of combining multiple antenna outputs in order to recover a signal of interest (SOI) with known steering vector in the presence of noise and interference. However, in practice the precise value of the SOI steering vector is unknown and only an estimate is used. In hopes of protecting the actual SOI in case of mismatches it has been recently proposed to use a non-attenuation constraint inside a hypersphere centered at the presumed SOI steering vector. In an effort to strike a balance between robustness to steering vector error and interference-plus-noise suppression, we propose in this paper to use two concentric hyperspheres instead of one with different degrees of protection in each. We determine conditions on the user defined parameters to ensure existence of a solution to the resulting constrained optimization problem. The multiply constrained filter solution is demonstrated to be of the diagonally loaded type with adaptive loading factor. We further give necessary conditions for the two-constraint filter to be distinct, in terms of SINR, to the one-constraint case. Numerical simulations show that using two constraints yields improved SINR performance compared to one constraint for small steering vector mismatches or large input SNR.