A modified exchange method for the design of near-field broadband beamformer

In this paper, the design of near-field broadband beamformer is considered, where the filter coefficients are designed such that the maximum error between the actual response and the desired response is minimized. This problem is equivalent to a semi-infinite programming problem, where the infinite constraints are defined in a space-frequency domain. This problem is large scale in general, and it is always expensive to solve this problem. We proposed a modified exchange method to solve this problem, where a series of subproblems with finite constraints is constructed and switched. We prove that this algorithm can terminate in a finite number of steps. For each iteration, we show that the problem is equivalent to a linear positive semi-definite programming problem, and then the proposed method can be implemented. Compared with the interior point method, we have illustrated in numerical experiments that the proposed method can be implemented very efficiently.