The directional loudspeaker array has various applications due to its capability to direct sound generation towards the target listener and reduce noise pollution. Differential beamforming has recently been applied to the loudspeaker line array to produce a broadside frequency-invariant radiation pattern. However, the existing methods cannot achieve a compromise between robustness and broadband frequency-invariant beampattern preservation. This paper proposed a robust broadband differential beamforming design to allow the loudspeaker line array to radiate broadside frequency-invariant radiation patterns with robustness. Specifically, we propose a method to determine the ideal broadside differential beampattern by combining multiple criteria, namely null positions, maximizing the directivity factor, and achieving a desired beampattern with equal sidelobes. We derive the above ideal broadside differential beampattern into the target beampattern in the modal domain. We propose a robust modal matching method with Tikhonov regularization to optimize the loudspeaker weights in the modal domain. Simulations and experiments show improved frequency-invariant broadside beamforming over the 250–4k Hz frequency range compared with the existing modal matching and null-constrained methods.

Differential beamforming has attracted much research since it can utilize the array with small aperture size to form frequency-invariant beampatterns and achieve high directional gains. It has recently been applied to the linear loudspeaker array to produce a broadside frequency-invariant radiation pattern. However, designing steerable frequency-invariant beampatterns for the linear loudspeaker array has yet to be explored. This paper proposes a method to design the steerable differential beamformer with the linear loudspeaker array. We first determine the target differential beampatterns according to the desired direction, the main lobe width, and the beampattern order. Then, we transform the target beampattern into the modal domain for representation. The Jacobi-Anger expansion is subsequently used to design the beamformer so that the resulting beam pattern matches the target differential beam pattern. Furthermore, based on the criterion of minimizing the mean square error between the synthesized beam pattern and the ideal one, a multi-constraint optimization problem is formulated to calculate the optimal desired weighting vector. Simulations and experimental results show that the proposed method can achieve steerable frequency-invariant beamforming over 300 Hz – 4 kHz.

Differential beamforming has attracted much research since it can utilize an array with a small aperture size to form frequency-invariant beampatterns and achieve high directional gains. It has recently been applied to the loudspeaker line array to produce a broadside frequency-invariant radiation pattern. However, designing steerable frequency-invariant beampatterns for the loudspeaker line array has yet to be explored. This paper proposes a method to design a steerable differential beamformer with a loudspeaker line array. We first determine the target differential beampatterns according to the desired direction, the main lobe width, and the beampattern order. Then, we transform the target beampattern into the modal domain for representation. The Jacobi-Anger expansion is subsequently used to design the beamformer so that the resulting beampattern matches the target differential beampattern. Furthermore, based on the criterion of minimizing the mean square error between the synthesized beampattern and the ideal one, a multi-constraint optimization problem, which compromises between the robustness and the mean square error, is formulated to calculate the optimal desired weighting vector. Simulations and experimental results show that the proposed method can achieve steerable frequency-invariant beamforming from 300 Hz–4 kHz.

Linear superarrays (LSAs) are linear microphone arrays composed of omnidirectional and first-order directional microphones, and their mixed-sensor manifold supports fully steerable differential beamforming. Building on established null-based target-beampattern theory and WNG-constrained beampattern-error design, this paper develops their joint realization for an LSA. Given a look direction and N prescribed null offsets, the corresponding Nth-order differential reference is obtained through the established null-constrained polynomial relation under the stated nonsingularity condition. The LSA weights are then obtained by minimizing the mean-squared beampattern error (MSBE) over the full array plane, subject to exact distortionless and null constraints and a prescribed white noise gain (WNG) lower bound. For the adopted omnidirectional/first-order-directional LSA model, the angular-integral matrices in the MSBE objective are evaluated analytically, while the final weights are obtained from the resulting convex program. Numerical simulations for the evaluated configurations show lower broadband MSBE than the maximum-WNG NC baseline, stable steering, and more consistent broadband patterns than the tested finite-order Jacobi–Anger designs. Anechoic-chamber measurements with an 11-element prototype further confirm that the main lobe, null structure, and steering behavior are retained in measured free-field responses.