Design of directivity patterns with a unique null of maximum multiplicity

Differential beamforming is one of the most popular beamforming approaches, which has the great potential to form frequency-invariant directivity patterns. In this paper, we study the design of beampatterns with multiple nulls in the same direction, which is clearly different from the design of beampatterns with distinct nulls. Our contributions are as follows. First, we show how to constrain multiple nulls to the same direction and design the desired beampattern with both the traditional and robust approaches. Second, we derive an explicit form of the white noise gain (WNG) of the traditional approach as a function of the frequency, interelement spacing, and null direction, which shows that the cardioid is the optimal beampattern as far as the WNG is concerned. Third, we prove that the WNG improvement of the robust approach rarely depends on the null direction at low frequencies. Finally, considering the fact that the robust differential beamforming approach may produce a frequency-dependent beampattern while improving the WNG, we develop a weighted-norm approach that can make a good compromise between the robustness of differential beamforming with respect to white noise and the frequency-invariant beampattern. The performance of the developed approach is verified by simulations.