l2 and l1 beamformers: recursive implementation and performance analysis

Studies array beamformers as optimal waveform estimators. The authors apply an inverse problem formulation, presenting an integrated design to quadratic (l₂) and least absolute value (l₁ ) beamformers. The general solution of the l₂ beamformers is parameterized by a regularizing parameter that weights the confidence placed by the designer on prior knowledge versus the quality of the measurements. This regularizing parameter is used to establish an equivalence between alternative l₂ beamformers. The authors then develop time-recursive implementations of the l₂ and l₁ beamformers. The performance of these beamformers is studied next. The authors show that 1) in the presence of correlated arrivals, the MMSE beamformer uses constructively the correlation between incoming signals in reconstructing the estimated field, while rejecting the uncorrelated returns, and 2) the l₁ beamformer has the ability to adjust itself to unexpected noise conditions because it is considerably more robust than the l₂ beamformers to unmodeled impulsive noise or to the occurrence of malfunctioning sensors. The analysis is confirmed by simulated studies