{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:28:37Z","timestamp":1760146117505,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T00:00:00Z","timestamp":1727395200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Research Project of Anhui Educational Committee","award":["2023AH050422","APDI202304"],"award-info":[{"award-number":["2023AH050422","APDI202304"]}]},{"name":"Open Project of Anhui Digital Intelligent Engineering Research Center for Agricultural Products Quality Safety under the Peak Cultivation Discipline of Electronic Information","award":["2023AH050422","APDI202304"],"award-info":[{"award-number":["2023AH050422","APDI202304"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>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\u20134 kHz.<\/jats:p>","DOI":"10.3390\/s24196277","type":"journal-article","created":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T07:19:37Z","timestamp":1727680777000},"page":"6277","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Design of Differential Loudspeaker Line Array for Steerable Frequency-Invariant Beamforming"],"prefix":"10.3390","volume":"24","author":[{"given":"Yankai","family":"Zhang","sequence":"first","affiliation":[{"name":"Anhui Digital Intelligent Engineering Research Center for Agricultural Products Quality Safety, Fuyang Normal University, Fuyang 236037, China"},{"name":"Anhui Province Photovoltaic Industry Common Technology Research Center, Fuyang Normal University, Fuyang 236037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-4834-2945","authenticated-orcid":false,"given":"Qian","family":"Xiang","sequence":"additional","affiliation":[{"name":"Anhui Province Photovoltaic Industry Common Technology Research Center, Fuyang Normal University, Fuyang 236037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiaoxi","family":"Zhu","sequence":"additional","affiliation":[{"name":"Faculty of Engineering and IT, University of Technology Sydney, Sydney, NSW 2007, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,27]]},"reference":[{"key":"ref_1","unstructured":"Don, H., and Johnson, D.E.D. 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