Sparse Design of Polynomial Beamformers by Jointly Sparsifying Sensor Locations and Farrow Structures

Polynomial beamformers for microphone arrays, which employ the well-known Farrow structure in digital filters, have drawn interest in recent years due to their capability of dynamic beam steering via simple online parameter tuning. Nevertheless, the computational complexity of the polynomial beamformers is higher than that of the nonsteerable counterpart. Moreover, the computational burden will become more demanding when used with the conventional uniform-spaced arrays for high-quality sound signal acquisition, because a large number of sensors are required due to the limitation imposed by the spatial Nyquist criterion. To address the problem, in this article, we propose to design sparse polynomial beamformers by jointly sparsifying sensor locations and Farrow structures. However, the joint sparse design problem is rather challenging due to the complex structure of the polynomial beamformers. We propose an efficient algorithm to solve the high-dimensional joint sparse design problem using the alternating direction method of multipliers (ADMM). Under the ADMM framework, we first reduce the original high-dimensional optimization problem into a set of subproblems much easier to solve. Then, we theoretically derive the analytical solutions to the subproblems, which are the key to the proposed ADMM algorithm. It is shown that the proposed design algorithm has a much lower computational complexity than the convex programming-based optimization approach widely employed in sparse array (SA) design. The effectiveness of the proposed joint sparse design is evaluated by the design examples as well as through its application in speech enhancement.