Robust Minimum Geometric Power Distortionless Response Beamforming with Sparse Constraint in Heavy-tailed Noise of Unknown Statistics

For impulsive noise environment, the adaptive beamformers based on second-order statistics often degrade severely. In this paper, a robust minimum geometric power distortionless response beamforming with sparse constraint is presented for arbitrary algebraically tailed impulsive signal of unknown statistics. The impulsive signal does not have finite second-order and higher order moments due to its property of heavy tails. Unlike the majority of the existing beamformers based on the second-order statistics, the proposed beamforming method minimizes a constrained mixed geometric power and ℓ
₁
-norm optimization problem. The geometric power effectively characterizes the process strength of the algebraically tailed distribution, while the ℓ
₁
-norm regularized term enforces the sparsity of the beam pattern. Therefore, the proposed algorithm can provide the potential for improved beamforming capability in the presence of impulse signals. Moreover, two adaptive algorithms, namely, the stochastic gradient (LMS)-based and the recursive least-squares (RLS)-based algorithms, are derived to solve the formulated optimization problem. Additionally, computational complexity analysis is conducted and shows that a comparable complexity between the proposed algorithms and the existing tailed impulsive beamformers. Simulation cases without and with mutual coupling are considered to illustrate that the proposed algorithms are more robust than other popular adaptive beamformers for impulsive signals.