  In this paper, colocated MIMO radar waveform design is considered by
minimizing the integrated side-lobe level to obtain beam patterns with lower
side-lobe levels than competing methods. First, a quadratic programming problem
is formulated to design beam patterns by using the criteria for a minimal
integrated side-lobe level. A theorem is derived that provides a closed-form
analytical optimal solution that appears to be an extension of the Rayleigh
quotient minimization for a possibly singular matrix in quadratic form. Such
singularities are shown to occur in the problem of interest, but proofs for the
optimum solution in these singular matrix cases could not be found in the
literature. Next, an additional constraint is added to obtain beam patterns
with desired 3 dB beamwidths, resulting in a nonconvex quadratically
constrained quadratic program which is NP-hard. A semidefinite program and a
Gaussian randomized semidefinite relaxation are used to determine feasible
solutions arbitrarily close to the solution to the original problem.
Theoretical and numerical analyses illustrate the impacts of changing the
number of transmitters and orthogonal waveforms employed in the designs.
Numerical comparisons are conducted to evaluate the proposed design approaches.
