Pattern synthesis via convex optimization for microphone arrays

Beampattern synthesis is an important topic in array signal processing. Most of the beampattern synthesis methods are based on quadratic optimization with linear or quadratic constraints. In this paper, we show that a variety of array pattern synthesis problems can be expressed as convex optimization problems, which can be numerically solved with great efficiency by the recently developed interior-point methods (IPM). Various convex functions can be formed as the cost function as well as constraints in optimization. Moreover, we also show that the conventional quadratic optimization problem in beampattern synthesis can be transformed to convex optimization via spatial or temporal sampled approximation. This method can be applied to array design with arbitrary geometry, element directivity and constraints on far and near field patterns over narrow or broad frequency bandwidth, etc.