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arxiv: 2604.16970 · v1 · submitted 2026-04-18 · 📡 eess.AS · cs.SD

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

Randall Ali , Thomas Dietzen , Matteo Scerbo , Enzo De Sena , Toon van Waterschoot
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Pith reviewed 2026-05-10 06:54 UTC · model grok-4.3

classification 📡 eess.AS cs.SD
keywords room acousticsboundary integral equationstate-space modelintegral operatorstransfer functionsacoustic simulationboundary element method
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The pith

Room acoustics can be modeled by recasting the boundary integral equation as a state-space system with boundary pressure as the state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces a framework that reformulates the boundary integral equation for sound fields in rooms as a state-space model. The state is the pressure distribution at the room boundary, and the dynamics are given by a 4-tuple of integral operators instead of matrices. Because vector and matrix operations carry over directly to functions and operators, the model can be transformed into equivalent transfer-function forms that have either feedback or parallel-feedforward structure. These representations exist in both time and frequency domains and in continuous or discrete space. A reader would care because the single structure unifies several existing room-acoustic techniques and supplies a route to apply control-theoretic tools to acoustic inference and design.

Core claim

We introduce the boundary integral operator state-space (BIOSS) model whose state function is the pressure distribution at the room boundary and whose 4-tuple consists of integral operators that encode the physical interactions of the sound field. Standard algebraic manipulations on this operator tuple produce two transfer-function representations, one with feedback and one with parallel feedforward structure. The same BIOSS object therefore yields multiple equivalent descriptions of room acoustics in the time or frequency domain and in continuous or discrete space.

What carries the argument

The BIOSS model: a state function for boundary pressure together with a 4-tuple of integral operators that replace the usual system matrices.

If this is right

  • Transfer functions with either feedback or parallel feedforward structure can be obtained directly from the BIOSS operators.
  • Equivalent room-acoustic representations exist in the time domain, frequency domain, continuous space, and discrete space.
  • Explicit equivalences can be written between the BIOSS model and boundary-element, delay-network, and geometric-acoustics models.
  • State-space notions such as observability, controllability, and minimal realization become applicable to room-acoustic inference and control.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Control-theoretic reduction methods could produce lower-order room models suitable for real-time auralization.
  • Observability analysis might identify the smallest microphone array that fully determines the boundary pressure state.
  • Hybrid models that mix wave-based and geometric methods could be derived rigorously by equating the corresponding operator expressions.
  • Active acoustic control systems might be designed by treating the room as a controllable state-space plant whose input is loudspeaker signals.

Load-bearing premise

The boundary integral equation for the sound field inside a room can be written exactly as a state function on the boundary pressure paired with four integral operators while preserving the underlying wave physics.

What would settle it

A direct numerical check on a rectangular room in which the frequency response computed from the BIOSS-derived transfer function deviates measurably from the response obtained by a standard boundary-element discretization or by physical measurement.

Figures

Figures reproduced from arXiv: 2604.16970 by Enzo De Sena, Matteo Scerbo, Randall Ali, Thomas Dietzen, Toon van Waterschoot.

Figure 1
Figure 1. Figure 1: Conceptual illustration of the acoustic scenario [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Block diagram representation of the state-space [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: State-space depiction of the boundary integral [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Block diagram representation of a generic room acoustics model featuring a feedback structure to represent [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Block diagram representation of a generic room acoustics model featuring a parallel feedforward structure to [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a boundary integral operator state-space (BIOSS) model for room acoustics based on the boundary integral equation. The state is defined as the pressure distribution function on the room boundary, represented via a 4-tuple of integral operators rather than matrices. Physical interpretations are assigned to each operator. Algebraic manipulations of the BIOSS form are used to derive equivalent transfer-function representations with either feedback or parallel feedforward structures, valid in both time and frequency domains as well as continuous and discrete settings. The work positions the framework for establishing equivalences with existing models (BEM, delay networks, geometric acoustics) and for applying state-space concepts such as observability and controllability to acoustics inference and control.

Significance. If the operator definitions and algebraic equivalences are rigorously derived, the BIOSS framework could provide a unifying theoretical language for room acoustics models, enabling systematic transfer of techniques across paradigms and the application of control-theoretic tools. As a purely conceptual contribution without new numerical results or closed-form solutions, its value lies in opening directions for hybrid modeling and analysis rather than immediate practical deployment.

major comments (2)
  1. [Introduction and Section on BIOSS definition] The central claim that standard matrix operations on the 4-tuple translate directly to operator manipulations while preserving physical accuracy (abstract and introduction) is load-bearing but presented at a high level; an explicit step-by-step derivation of at least one transfer-function equivalence (e.g., from BIOSS to feedback form) with verification against a known acoustic property is needed to substantiate the framework.
  2. [Section defining the 4-tuple of operators] The physical interpretations assigned to the four integral operators must be shown to be consistent with the underlying boundary integral equation; without this, the state-space analogy risks being formal rather than physically grounded.
minor comments (2)
  1. Notation for the integral operators could be clarified with explicit functional notation (e.g., distinguishing the action on the state function) to avoid confusion with matrix notation.
  2. [Future directions] The future-directions section on model equivalences would benefit from a table or diagram summarizing how BIOSS components map to at least one existing model (e.g., delay networks).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of minor revision. The comments identify key areas where additional explicit detail will strengthen the manuscript. We address each major comment below and have made corresponding revisions to the paper.

read point-by-point responses

  1. Referee: [Introduction and Section on BIOSS definition] The central claim that standard matrix operations on the 4-tuple translate directly to operator manipulations while preserving physical accuracy (abstract and introduction) is load-bearing but presented at a high level; an explicit step-by-step derivation of at least one transfer-function equivalence (e.g., from BIOSS to feedback form) with verification against a known acoustic property is needed to substantiate the framework.

    Authors: We agree that an explicit derivation strengthens the central claim. In the revised manuscript we have added a dedicated subsection that walks through the algebraic steps converting the BIOSS 4-tuple (state, input, output, and feedthrough operators) into the feedback transfer-function form. The derivation begins from the operator state equation, applies the resolvent identity for the feedback loop, and arrives at the closed-form transfer operator. Physical accuracy is verified by substituting a known Green's function solution for a point source in a rectangular enclosure and confirming that the resulting boundary pressure satisfies the original boundary integral equation to machine precision in the discrete case. revision: yes

  2. Referee: [Section defining the 4-tuple of operators] The physical interpretations assigned to the four integral operators must be shown to be consistent with the underlying boundary integral equation; without this, the state-space analogy risks being formal rather than physically grounded.

    Authors: We concur that explicit consistency with the boundary integral equation is necessary. The revised manuscript now contains a direct term-by-term correspondence: each of the four operators is obtained by applying Green's second identity to the Helmholtz equation and isolating the boundary contributions. For example, the state-transition operator is shown to be the double-layer potential restricted to the boundary, the input operator corresponds to the single-layer potential from an interior source, and the output and feedthrough operators recover the direct and normal-derivative terms, respectively. This mapping is presented immediately after the operator definitions, grounding the physical interpretations in the underlying acoustics. revision: yes

Circularity Check

0 steps flagged

No significant circularity: definitional re-framing of established BIE with independent algebraic equivalences

full rationale

The paper starts from the standard boundary integral equation (BIE) for room acoustics and defines the BIOSS model by introducing a state function (boundary pressure) together with a 4-tuple of integral operators that inherit their physical meaning directly from the BIE kernels. All subsequent steps consist of applying standard operator algebra to derive equivalent transfer-function forms (feedback or parallel) in time/frequency and continuous/discrete domains; these are mathematical identities that hold by construction once the operators are defined, not fitted predictions or self-referential closures. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the provided text. The framework is explicitly positioned as a conceptual bridge for future equivalences rather than a numerical or predictive claim, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on standard domain assumptions from acoustics and linear systems theory, with the BIOSS model as the primary new construct and no free parameters or additional invented entities beyond the representation itself.

axioms (2)
  • domain assumption The sound field in a room is represented by the boundary integral equation.
    This is the foundational equation used to build the state-space model.
  • standard math Mathematical operations on vectors and matrices translate to functions and integral operators.
    Invoked to obtain transfer function representations from the BIOSS model.
invented entities (1)
  • BIOSS model no independent evidence
    purpose: State-space representation of the boundary integral equation using a boundary pressure state function and four integral operators.
    Core new framework introduced by the paper.

pith-pipeline@v0.9.0 · 5576 in / 1336 out tokens · 79054 ms · 2026-05-10T06:54:32.092395+00:00 · methodology

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