Beamforming: Directivity and steerability

Beamforming is a technique which uses an array consisting of multiple elements to enhance the transmission or reception of signals.  A beamformer can transmit / receive signals corresponding to particular desired directions, and attenuate signals corresponding to other directions.  This is done by creating a beampattern incorporating constructive and destructive interference at the relevant directions.  Applications include radar, sonar, wireless communications, biomedical imaging, and speech processing.  Many factors influence the beampattern; these include the location of the elements, their individual directivity responses, and their complex-valued weights.

The directivity factor (DF) measures the spatial selectivity of the beampattern with larger DFs generally corresponding to higher spatial resolution.  It is often desirable to achieve high DFs while maintaining the capability of steering the beampattern to any desired direction.  We present a generalized theorem showing that when the optimal DF is averaged over all directions, the result equals the number of array elements minus the number of null constraints.  This result holds for arbitrary array constellations and propagation regimes.  Judicious design can improve the average DF over a certain region of interest and establish robustness to errors.