Sparse microphone array design for frequency-invariant beamforming via group [formula omitted]-norm optimization

Abstract

Frequency-invariant (FI) beamforming aims at recovering signals without any distortion and maintaining spatial selectivity over the entire bandwidth. However, most existing FI beamforming (FIB)methods consider the weighted ℓ1-norm or modified ℓ1-norm of the filters as the objective function for sparse FIB design, which cannot assure the optimal sparse solution. To deal with this drawback, an optimization problem for the FIB design of sparse microphone array in terms of ℓp-norm (0<p<1) minimization is formulated, where distortion-less response, mainlobe and sidelobe constraints are considered. Due to the existence of the ℓp-norm objective function, the resultant problem is nonconvex, and therefore, an alternating direction method of multipliers (ADMM) algorithm is devised. Specifically, the corresponding problem is decomposed into multi-block subproblems to determine all unknown variables separately. Then, the most dominant sensor positions for each frequency are determined by using the principal component analysis (PCA) and K-mean clustering algorithm. Numerical examples show that the proposed design achieves good directivity factor, frequency invariance and better sparsity.