Enhanced robust adaptive beamforming designs for general-rank signal model via an induced norm of matrix errors

The robust adaptive beamforming (RAB) problem for general-rank signal model with an uncertainty set defined through a matrix induced norm is considered. The worst-case signal-to-interference-plus-noise ratio (SINR) maximization RAB problem is formulated. First, the closed-form optimal value for a minimization problem of the least-squares residual over the matrix errors with an induced l_p,q-norm constraint is derived. Then, the maximization problem is reformulated into the maximization of the difference between an l₂-norm function and an l_q-norm function, with a convex quadratic constraint. It is shown that for any q≥1 in the set of rational numbers, the maximization problem can be approximated by a sequence of second-order cone programming problems, with the ascent optimal values. The resultant beamvector for some q in the set with the maximal actual array output SINR, is treated as the candidate making the RAB design improved the most. In addition, a generalized RAB problem of maximizing the difference between an l_p-norm function and an l_q-norm function with the convex quadratic constraint is studied, and the actual array output SINR is further enhanced by properly selecting p and q. Simulation examples are presented to demonstrate the improved performance of the robust beamformers for certain matrix induced l_p,q-norms.