Comparison of directional and derivative constraints for beamformers subject to multiple linear constraints

Optimum beamformers with multiple linear constraints are usually designed under the assumptions of plane-wave signals and an ideal propagation medium. In acutal operating conditions these ideal assumptions do not hold; for example, signal suppression can arise from beam steering angle errors if the optimum weights are calculated from the signal-plus-noise cross-spectral matrix. To overcome this, it is desirable to broaden the signal acceptance angle (i.e. the width of the main beam) while preseving the beam-former´s ability to reject interference from directions outside this acceptance angle. This can be accomplished by the use of multiple directional constraints or a single directional constraint combined with multiple derivative constraints. The paper compares the performance of these two constraint systems and shows that the signal suppression of a directional constraint system approaches that of a derivative constraint system as the angle between the directional constraints is decreased. The comparison is based upon the analysis of signal suppression effects and the ill conditioning of the equations for the optimum weight vectors. In the examples considered the derivative constraints avoid the ill conditioning problems associated with the use of multiple directional constraints.