The design of multi-dimensional acoustic beamformers via window functions

The design of broadband beamformers can be formulated as a semi-infinite programming optimization problem, where the coefficients of the filters are determined such that the actual response of the microphone array is near a given desired response. This problem can be solved by existing optimization solvers after the discretization of the infinite constraints. However, this problem will become large-scale and it's expensive to find the optimal solution, as the discretization points grow and the filter length increases. In this paper, we propose a fast method based on the window functions. First, we formulate a simplified optimization problem to find the limit of the cost function values as the filter length is sufficiently long. Hence, the optimal frequency response vector is obtained and the corresponding filter coefficients can be calculated. Second, we apply the window method to truncate the limiting filter to obtain a finite FIR filters. The performance as well as the computational complexity are investigated to demonstrate the effectiveness and efficiency of the proposed method.