Differential Beamforming With Planar Arrays of Arbitrary-Order Directional Elements

Differential beamforming enables compact arrays to synthesize frequency-invariant, highly directive beampatterns, but existing formulations for arbitrary planar geometries assume omnidirectional or, at most, first-order directional elements. In this letter, we extend differential beamforming theory to planar arrays whose elements exhibit directivities of arbitrary order, modeled through a truncated circular-harmonic expansion. By combining the Fourier-series representation of the directional array response with the Jacobi-Anger expansion, we derive a closed-form expression for the modal coefficients that embed the element directivities into the filter design, leading to a modal-matching solution that synthesizes a prescribed target beampattern and recovers existing zero- and first-order formulations as special cases. Simulations over fixed and randomly generated array geometries show that the proposed design accurately reproduces the target beampattern across frequency, while remaining consistent across different array geometries and element directivities.