On the Average Directivity Factor Attainable with a Beamformer Incorporating Null Constraints

The directivity factor (DF) of a beamformer describes its spatial selectivity and ability to suppress diffuse noise which arrives from all directions. For a given array constellation, it is possible to select beamforming weights which maximize the DF for a particular look-direction, while enforcing nulls for a set of undesired directions. In general, the resulting DF is dependent upon the specific look- and null directions. Using the same array, one may apply a different set of weights designed for any other feasible set of look- and null directions. In this contribution, we show that when the optimal DF is averaged over all look directions, the result equals the number of sensors minus the number of null constraints. This result holds regardless of the positions and spatial responses of the individual sensors and regardless of the null directions. The result generalizes to more complex wave-propagation domains (e.g., reverberation).