A state-space representation of the boundary integral equation for room acoustic modelling

2026-04-18

AI Summary

This study addresses the lack of a unified theoretical framework in traditional room acoustic modeling, which hinders the integration of diverse models and impedes systematic control and inference. To overcome this limitation, the work extends state-space theory to the level of functions and operators, introducing a Boundary Integral Operator State-Space (BIOSS) model grounded in boundary integral equations. This formulation replaces conventional state vectors and matrices with state functions and integral operators. The approach yields equivalent representations across continuous and discrete domains as well as time and frequency domains, unifies feedback and feedforward transfer function structures, and establishes equivalence with existing acoustic models. By doing so, it offers a novel paradigm for model fusion and intelligent acoustic control.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation
• Cognitive Modeling & Cognitive Systems: Applications

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We introduce a new framework for room acoustics modelling based on a state-space model of the boundary integral equation representing the sound field in a room. Whereas state-space models of linear time-invariant systems are traditionally constructed by means of a state vector and a 4-tuple of system matrices, the state-space representation introduced in this work consists of a state function representing the pressure distribution at the room boundary, and a 4-tuple of integral operators. We refer to this representation as a boundary integral operator state-space (BIOSS) model and provide a physical interpretation for each of the integral operators. As many mathematical operations on vectors and matrices translate to functions and operators, the BIOSS representation can be manipulated to obtain two transfer function representations, having either a feedback or a parallel feedforward structure. Consequently, various equivalent representations for room acoustics are obtained in the BIOSS framework, in the time or frequency domain, and in continuous or discrete space. We discuss two future directions for how the proposed framework can be fertile for research on room acoustics modelling. Firstly, we identify equivalences between the BIOSS framework and various existing room acoustics models (boundary element models, delay networks, geometric models), which may be used to establish relations between existing models and to develop novel room acoustics models. Secondly, we postulate on how concepts from state-space theory, such as observability, controllability, and state realization, can be used for developing new inference and control methods for room acoustics.

Problem

• room acoustics
• state-space representation
• boundary integral equation
• acoustic modeling
• transfer function

Innovation

• state-space model
• boundary integral equation
• room acoustics
• integral operators
• transfer function