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

Official MATLAB implementation of the article

F. Miotello, D. Albertini and A. Bernardini, "On the Extension of Differential Beamforming Theory to Arbitrary Planar Arrays of First-Order Elements," in IEEE Transactions on Audio, Speech and Language Processing, vol. 34, pp. 1815-1825, 2026, doi: 10.1109/TASLPRO.2026.3675799.

📝 The article is available in open access on IEEE Xplore: https://ieeexplore.ieee.org/document/11447415.

This repository implements the proposed framework to design steerable differential beamformers for arbitrary planar arrays of first-order elements. The codebase is compact and organized around two main entry-point scripts, with reusable helpers for beamforming, geometry generation, target beampattern construction, and plotting.

📁 Repository overview

The main workflow is organized around two entry-point scripts:

• main.m: runs a single experiment, designs the beamformer, and visualizes the resulting array geometry, beampatterns, WNG, and DF.
• main_montecarlo.m: runs repeated randomized experiments and plots mean and standard deviation statistics for the beampattern, WNG, and DF.

Supporting code is grouped as follows:

• beamforming/: beamformer design and performance evaluation routines.
• ideal_beampatterns/: target-pattern coefficient generators for omnidirectional, dipole, cardioid, hypercardioid, and supercardioid responses.
• plotting/: reusable visualization utilities for array geometry, beampatterns, WNG, and DF.
• utils/: helper functions for generating array geometries and related support routines.

⚙️ Requirements

• MATLAB
• Optimization Toolbox (fmincon is needed for hypercardioid and supercardioid target beampatterns).

The code is otherwise self-contained and uses standard MATLAB functionality.

🚀 Getting started

From MATLAB, open the repository folder and choose the script that matches the type of experiment you want to run. The two entry points, main.m and main_montecarlo.m, share the same overall workflow: they define the acoustic and beamforming parameters, generate the array configuration and first-order element characteristics, compute the coefficients of the desired beampattern, design the frequency-dependent beamformer weights, and evaluate the resulting response. The lower-level functionality is delegated to the helper routines in beamforming/, plotting/, utils/, and ideal_beampatterns/.

For a single experiment on one array realization, run:

main

For a statistical analysis over multiple randomized realizations, run:

main_montecarlo

🎛️ Main parameters

The following parameters can be adjusted directly in main.m and main_montecarlo.m:

• M: number of array elements.
• freq: frequency grid used for beamformer design and evaluation.
• N: differential beamforming order.
• shape: target beampattern ('omnidirectional', 'dipole', 'cardioid', 'hypercardioid', or 'supercardioid').
• theta_s: target beampattern steering direction.
• q_m, theta_m: element beam shape and orientations.
• n_exp: number of Monte Carlo experiments in main_montecarlo.m.

Geometry-related parameters depend on the specific array generation strategy adopted in the script. For example, max_rad and min_sep are relevant for the random circular-area geometry used in the article.

📚 Citation

If you use this repository, please cite the corresponding article:

F. Miotello, D. Albertini and A. Bernardini, "On the Extension of Differential Beamforming Theory to Arbitrary Planar Arrays of First-Order Elements," in IEEE Transactions on Audio, Speech and Language Processing, vol. 34, pp. 1815-1825, 2026, doi: 10.1109/TASLPRO.2026.3675799.

BibTeX:

@article{miotello2026extension,
  author={Miotello, Federico and Albertini, Davide and Bernardini, Alberto},
  journal={IEEE Transactions on Audio, Speech and Language Processing},
  title={On the Extension of Differential Beamforming Theory to Arbitrary Planar Arrays of First-Order Elements},
  year={2026},
  volume={34},
  pages={1815-1825},
  publisher={IEEE}
}