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 $^2$ 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 $^2$ H basis functions, the orthogonality of the shifted associated Legendre polynomials and scaled exponential functions are established for the same sector. The new S $^2$ 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 $^2$ H domain for wideband and narrowband source localization. An ideal direction-invariant beampattern using S $^2$ 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.

中文翻译：

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小号$ ^ 2 $H域处理，用于在球形扇区上使用麦克风阵列进行声源定位和波束成形

球形麦克风阵列（SMA）被广泛用于声源定位和波束形成。但是，在刚性球体上构建SMA是一项艰巨的任务，并且当源位于环境的某些受限区域时，构建完整的SMA是不经济的。本文解决了使用球形扇区上的麦克风阵列进行多源定位和波束成形的问题。边界上的不连续性是通过求解该扇区上的亥姆霍兹方程来处理的。由于声压仅存在于该扇区上的麦克风上，因此新颖的正交球面扇区谐波（S $ ^ 2 $ H）开发了基本功能。在扇区上定义的基本函数是广义形式，对此，通常的球谐函数是一种特殊情况。确保S的正交性 $ ^ 2 $ 对于同一扇区，建立了H基函数，移位关联的Legendre多项式的正交性和缩放的指数函数。新闻 $ ^ 2 $ H基础函数用于远场数据模型开发。随后，在S中制定了多个信号分类和最小方差无失真响应谱 $ ^ 2 $ H域用于宽带和窄带源定位。使用S的理想方向不变光束图案 $ ^ 2 $ 还推导了H基函数。使用各种源定位实验分析了所提出方法的性能。用于波束形成的声像原理在此扩展以用于源定位。另外，将所提出的方法与使用基于球谐函数的基于声像原理的源定位进行了比较。