用空间微分重构核域波束成形，可生成任意非轴对称波束。
Beamforming in the Reproducing Kernel Domain Based on Spatial Differentiation
- 将方向响应视为声场空间微分，用多项式微分算子表示
- 基于霍布森定理推导出内场再生核，实现简洁解析表达
- 统一解释传统球谐域波束成形，揭示其理论结构与扩展性
本文提出一种基于再生核域的新型波束成形框架，其核心思想是将方向响应统一解释为声场的空间微分。通过多项式微分算子表示方向响应，该方法可构造任意波束模式，包括非轴对称类型。内场相关再生核的推导得到霍布森定理的数学支持，从而获得简洁的解析表达式。此外，该框架重新诠释了传统的球谐域波束成形器，将其视为空间微分算子，清晰揭示了其理论结构与可扩展性。在二维空间中进行了三项数值仿真，验证了方法的有效性。
原文摘要 · Abstract (English)
This paper proposes a novel beamforming framework in the reproducing kernel domain, derived from a unified interpretation of directional response as spatial differentiation of the sound field. By representing directional response using polynomial differential operators, the proposed method enables the formulation of arbitrary beam patterns including non-axisymmetric. The derivation of the reproducing kernel associated with the interior fields is mathematically supported by Hobson's theorem, which allows concise analytical expressions. Furthermore, the proposed framework generalizes conventional spherical harmonic domain beamformers by reinterpreting them as spatial differential operators, thereby clarifying their theoretical structure and extensibility. Three numerical simulations conducted in two-dimensional space confirm the validity of the method.
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