Optimization of array performance subject to multiple constraints on the beam power pattern

Among the performance indices used in the processing of acoustic array signals are those that can be expressed as a ratio of quadratic forms in the beamformer weights. Popular indices in this class include directivity, array gain, efficiency factor, and supergain ratio. This paper presents a technique for optimizing a quadratic-form-ratio performance index over the beamformer weights subject to generally nonzero constraints on the value of the array power pattern in multiple directions. Previous related results include performance optimization with no constraints, with specified pattern nulls, and with constraints on the complex-valued beam pattern. When an array designer is interested just in the beam pattern magnitude, the additional specification of pattern phase required by these techniques generally penalizes the optimization. Only magnitude (power pattern) constraints are imposed in this paper. The optimum weights are shown to satisfy a characteristic equation with quadratic constraints that is related to a more familiar optimization problem for which solution methods exist. Applications include performance maximization with fixed main-lobe beamwidth and/or sidelobe values. Examples are given for a circular array.