  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 by decomposing the presumed covariance of the desired
signal into a product between a matrix and its Hermitian, and putting an error
term into the matrix and its Hermitian. In the literature, the norm of the
matrix errors often is the Frobenius norm in the maximization problem. Herein,
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
first derived. Then, the worst-case SINR maximization problem is reformulated
into the maximization of the difference between an $l_2$-norm function and a
$l_q$-norm function, subject to a convex quadratic constraint. It is shown that
for any $q$ in the set of rational numbers greater than or equal to one, the
maximization problem can be approximated by a sequence of second-order cone
programming (SOCP) problems, with the ascent optimal values. The resultant
beamvector for some $q$ in the set, corresponding to the maximal actual array
output SINR, is treated as the best candidate such that the RAB design is
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 subject to
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, in terms of the actual array output
SINR and the CPU-time for the sequential SOCP approximation algorithm.

Signal Processing