# Limitations and Performance Analysis of Spherical Sector Harmonics for Sound Field Processing
## 1 Introduction  
Spherical harmonics (SHs) are one of the fundamental approaches for spherical microphone array processing [1; 2; 3; 4]. They have been widely adopted in various acoustic tasks, including beamforming [5; 6], source localization [7; 8; 9; 10], noise cancellation [11; 12; 13; 14], and sound field recording [15]. However, in SH-based methods, implementing the required spherical microphone array can be challenging, especially in applications such as attaching these arrays to drones for whole-sphere sampling. To address the whole sphere sampling challenge, an orthonormal function set was developed for an accurate representation of the pressure over an arbitrary spherical sector region, named spherical sector harmonics (SSHs) [16]. Although SSHs have the potential to be implemented in various audio and acoustic problems, a thorough investigation of this needs to be included. In this paper, we conduct comprehensive investigations of SSHs and reveal three main limitations of them, offering valuable insights for their future implementation. SHs can decompose the sound field into angular-dependent orthonormal functions and corresponding coefficients, allowing for the analysis of a spatial sound field with a finite number of SH coefficients. The concept of using an open spherical microphone array to capture high-order sound fields was first introduced by [PERSON] and [PERSON] [1]. Subsequently, [PERSON] and [PERSON] developed a microphone array over a rigid sphere to implement this approach [2]. Further advancements in the design and analysis of spherical arrays were made by [PERSON] [3] and [PERSON] [17], with [PERSON] later proposing a dual-sphere microphone array [4]. Although SH-based spherical array processing hasgreatly benefited spatial audio and acoustics, the requirement for whole-sphere sampling poses challenges for implementation in certain applications. [PERSON] et al. [18] developed a method for sound field visualization based on sector design. To approximate the directivity pattern with data available only within a limited range of directions, [PERSON] and [PERSON] introduced the concept of spherical Slepian functions [19]. These functions form an orthogonal basis specifically designed for the restricted directional range of the sphere, particularly for rotationally symmetric regions such as spherical caps or segments [19; 20]. While spherical Slepian functions enable the decomposition and reconstruction of the sound field within a spherical cap or segment, they require frequency-dependent matrix inversion and may introduce significant errors for sources outside the restricted range [20]. By solving the Helmholtz equation, with sound-soft or sound-hard boundary conditions, the spherical cap and segment harmonics were proposed in [20; 21] and have been validated by microphone array prototypes [22; 23]. As the spherical cap and segment harmonics satisfy the Helmholtz equation, they can be implemented to interpolate and extrapolate the sound field. However, to apply the spherical cap and segment harmonics, the target region has to be bounded by a sound-soft or sound-hard boundary, limiting its broader implementation. Further addressing the need for microphone processing over arbitrary spherical sector regions, [PERSON] and [PERSON] developed SSHs that decompose and represent the sound pressure over a spherical sector region [16]. This advancement reduces the sampling requirements, simplifies microphone array construction, and expands its application to active noise control (ANC) [24], beamforming [25; 26], and sound source localization [25; 27]. In our previous work, we implemented SSHs in a spherical sector sound field extrapolation problem [28]. Compared to effectively representing sound pressure over a spherical sector region, SSHs do not require the sound-soft or sound-hard boundary to bind the spherical sector region; therefore, it can potentially be applied to broader applications.