In this paper, colocated MIMO radar waveform design is considered byminimizing the integrated side-lobe level to obtain beam patterns with lowerside-lobe levels than competing methods. First, a quadratic programming problemis formulated to design beam patterns by using the criteria for a minimalintegrated side-lobe level. A theorem is derived that provides a closed-formanalytical optimal solution that appears to be an extension of the Rayleighquotient minimization for a possibly singular matrix in quadratic form. Suchsingularities are shown to occur in the problem of interest, but proofs for theoptimum solution in these singular matrix cases could not be found in theliterature. Next, an additional constraint is added to obtain beam patternswith desired 3 dB beamwidths, resulting in a nonconvex quadraticallyconstrained quadratic program which is NP-hard. A semidefinite program and aGaussian randomized semidefinite relaxation are used to determine feasiblesolutions arbitrarily close to the solution to the original problem.Theoretical and numerical analyses illustrate the impacts of changing thenumber of transmitters and orthogonal waveforms employed in the designs.Numerical comparisons are conducted to evaluate the proposed design approaches.
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