Three-dimensional beamforming with the scattering of spherical shell for inclined small aperture array of vector sensors

An efficient three-dimensional beamforming method applied to an inclined small aperture planar array of vector sensors (SAPAVS) is proposed to solve the problems of direction-dependent gain-phase error caused by the scattering of a spherical shell and fixed gain-phase error. Firstly, the expressions of received signals for each channel are unified with the assistance of the spherical harmonic expansion method. Subsequently, the coefficients of spherical harmonic expansions can be determined by employing several auxiliary sources based on these expressions. Finally, the three-dimensional beamforming method is obtained by combining the SAPAVS expression and the ideal beam response, enabling the realization of direction of arrival (DOA) estimation. The agreement between the theoretical and simulated results demonstrates the feasibility of the proposed method. Under the same array geometry, compared to the pressure sensor array, the proposed method based on the vector sensor array will have higher robustness. Moreover, as the auxiliary sound sources are more dispersed, the robustness of the algorithm will be further enhanced.