  A new approach to solving a class of rankconstrained semi-definite
programming (SDP) problems, which appear in many signal processing applications
such as transmit beamspace design in multiple-input multiple-output (MIMO)
radar, downlink beamforming design in MIMO communications, generalized sidelobe
canceller design, phase retrieval, etc., is presented. The essence of the
approach is the use of underlying algebraic structure enforced in such problems
by other practical constraints such as, for example, null shaping constraint.
According to this approach, instead of relaxing the non-convex rankconstrained
SDP problem to a feasible set of positive semidefinite matrices, we restrict it
to a space of polynomials whose dimension is equal to the desired rank. The
resulting optimization problem is then convex as its solution is required to be
full rank, and can be efficiently and exactly solved. A simple matrix
decomposition is needed to recover the solution of the original problem from
the solution of the restricted one. We show how this approach can be applied to
solving some important signal processing problems that contain null-shaping
constraints. As a byproduct of our study, the conjugacy of beamfoming and
parameter estimation problems leads us to formulation of a new and rigorous
criterion for signal/noise subspace identification. Simulation results are
performed for the problem of rank-constrained beamforming design and show an
exact agreement of the solution with the proposed algebraic structure, as well
as significant performance improvements in terms of sidelobe suppression
compared to the existing methods.
