Constraints for maximally flat optimum broadband antenna arrays

Frequency response shaping for the direct form pre-steered broadband (PB) antenna array processor is often achieved by imposing look direction constraints on the weights of the processor. This results in a linearly constrained optimization problem. To ensure a maximally flat spatial response of a specified order in the look direction of the PB processor, additional constraints known as derivative constraints can be further imposed on the weights. In general, derivative constraints corresponding to necessary and sufficient (NS) conditions for a maximally flat spatial power response can result in a quadratic equality constrained optimization problem. In this paper, we transform the quadratic NS derivative constraints to parameterized linear forms. These parameterized linear forms allow the global optimum of the quadratic equality constrained optimization problem to be obtained easily. They also provide a general framework for deriving new sets of derivative constraints which correspond only to sufficient conditions for a maximally flat spatial power response. These sufficient derivative constraints are useful for real-time processing because of their reduced computational requirements and because they can deliver performance comparable to the NS derivative constraints.