On the Extension of Differential Beamforming Theory to Arbitrary Planar Arrays of First-Order Elements

Small-size acoustic arrays exploit spatial diversity to achieve capabilities beyond those of single-element devices, with applications ranging from teleconferencing to immersive multimedia. A key requirement for broadband array processing is a frequency-invariant spatial response, which ensures consistent directivity across wide bandwidths and prevents spectral coloration. Differential beamforming offers an inherently frequency-invariant solution by leveraging pressure differences between closely spaced elements of small-size arrays. Traditional approaches, however, assume the array elements to be omnidirectional, whereas real transducers exhibit frequency-dependent directivity that can degrade performance if not properly modeled. To address this limitation, we propose a generalized modal matching framework for frequency-invariant differential beamforming, applicable to unconstrained planar arrays of first-order directional elements. By representing the desired beampattern as a truncated circular harmonic expansion and fitting it to the actual element responses, our method accommodates arbitrary planar geometries and element orientations. This approach enables the synthesis of beampatterns of any order and steering direction without imposing rigid layout requirements. Simulations confirm that accounting for sensor directivity at the design stage yields accurate and robust performance across varying frequencies, geometries, and noise conditions.