S$^2$H Domain Processing for Acoustic Source Localization and Beamforming Using Microphone Array on Spherical Sector

Spherical microphone arrays (SMAs) are widely used for the localization of acoustic sources and beamforming. However, building an SMA over a rigid sphere is a challenging task, and it is uneconomic to build full SMAs when the sources are in some restricted regions of the environment. This paper addresses the issue of multiple source localization and beamforming using a microphone array on a spherical sector. The discontinuity over the boundary is handled by solving the Helmholtz equation over the sector. As the sound pressure is present only on the microphones over that sector, novel orthonormal spherical sector harmonics (S<inline-formula><tex-math notation="LaTeX">$^2$</tex-math></inline-formula>H) basis functions are developed. The basis functions defined over the sector are in generalized form, for which the usual spherical harmonics are a special case. To ensure orthonormality of S<inline-formula><tex-math notation="LaTeX">$^2$</tex-math></inline-formula>H basis functions, the orthogonality of the shifted associated Legendre polynomials and scaled exponential functions are established for the same sector. The new S<inline-formula><tex-math notation="LaTeX">$^2$</tex-math></inline-formula>H basis functions are utilized for far-field data model development. Subsequently, multiple signal classification and minimum variance distortionless response spectra are formulated in the S<inline-formula><tex-math notation="LaTeX">$^2$</tex-math></inline-formula>H domain for wideband and narrowband source localization. An ideal direction-invariant beampattern using S<inline-formula><tex-math notation="LaTeX">$^2$</tex-math></inline-formula>H basis functions is also derived. The performance of the proposed methods is analyzed using various source localization experiments. The acoustic image principle for beamforming is extended herein for source localization. Additionally, the proposed methods are compared with acoustic image principle-based source localization using spherical harmonics.

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