Pattern synthesis for symmetric linear transducer array by bilinear factorization of Hankel matrices

Abstract

We investigate the problem of designing an unequally spaced symmetric linear array with a minimum number of elements. For unequally spaced arrays, simultaneously tuning all the elements' positions and weights leads to a non-convex problem. Inspired by the fact that the matrix pencil method (MPM) can estimate the weights and positions of array elements from a Hankel matrix formed by samples of beampattern, we attack the design problem by finding the Hankel matrix fulfilling specifications with the minimum rank. Fortunately, the low-rank optimization can be addressed by nuclear norm relaxation. In the proposed approach, we minimize the nuclear norm of the Hankel matrix utilizing the low-rank matrix factorization-based augmented Lagrangian alternating direction method. Once the low-rank Hankel matrix is obtained, we finish the array design with the help of MPM. By finding the desired Hankel matrix with a minimum nuclear norm, the proposed approach avoids the NP-hard problem of simultaneously optimizing the weights and positions of elements. Moreover, the proposed method does not need a reference beampattern and makes no assumptions on element positions, making it more flexible in facing real-world problems. We compare the proposed approach with existing methods under different scenarios and prove its effectiveness.