A two-stage method for the design of near-field broadband beamformer

Abstract

In this paper, the design of near-field broadband beamformer is considered. The idea is to design the beamformer filter coefficients such that the error between the actual response and the desired response is minimized. This problem can be formulated as a minimax optimization problem and can be transformed into a linear semi-definite programming problem. Interior point algorithm can then be applied after discretization. However, if the dimension of the problem is high, the number of discrete points of the specified region and consequently the dimension of the resultant constraint matrix is high. This high dimension provides a difficulty when directly employing interior point algorithm. Since it results in a large size problem, this problem is considered in this paper. To reduce the computational complexity and memory usage, a two-stage method has been proposed. In the first stage, by using an infinite length filter formulation, an infinite length limit for the original problem is obtained. Then, based on that limit, a reduced problem with a finite filter length is found. The computational complexity of this method is derived. It provides a significant reduction compared to a direct solution of the original problem. It is shown that as the filter length increases, the approximative problem comes close to the optimal infinite solution. Furthermore, we demonstrate with examples the close correspondence between proposed design and optimal design. Also the superiority of the suggested method is illustrated in a three-dimensional beamformer design, where the direct method can only design broadband beamformer of very short filter lengths.