Model-based Bayesian analysis in acoustics-A tutorial

Bayesian estimation of dissipation and sound speed in tube measurements using a transfer-function model

This study discusses acoustic dissipation, which contributes to inaccuracies in impedance tube measurements. To improve the accuracy of these measurements, this paper introduces a transfer function model that integrates diverse dissipation prediction models. Bayesian inference is used to estimate the important parameters included in these models, describing dissipation originating from various mechanisms, sound speed, and microphone positions. By using experimental measurements and considering a hypothetical air layer in front of a rigid termination as the material under test, Bayesian parameter estimation allows a substantial enhancement in characterization accuracy by incorporating the dissipation and sound speed estimates. This approach effectively minimizes residual absorption coefficients attributed to both boundary-layer effects and air medium relaxation. The incorporation of dissipation models leads to a substantial reduction (to 1%) in residual absorption coefficients. Moreover, the use of accurately estimated parameters further enhances the accuracy of actual tube measurements of materials using the two-microphone transfer function method.

Microphone array analysis of the first non-axisymmetric mode for the detection of pipe conditions

This paper reports on the use of a circular microphone array to analyze the reflections from a pipe defect with enhanced resolution. A Bayesian maximum a posteriori algorithm is combined with the mode decomposition approach to localize pipe defects with six or fewer microphones. Unlike all previous acoustic reflectometry techniques, which only estimate the location of a pipe defect along the pipe, the proposed method uses the phase information about the wave propagated in the form of the first non-axisymmetric mode to estimate its circumferential position as well as axial location. The method is validated against data obtained from a laboratory measurement in a 150 mm diameter polyvinyl chloride pipe with a 20% in-pipe blockage and 100 mm lateral connection. The accuracy of localization of the lateral connection and blockage attained in this measurement was better than 2% of the axial sensing distance and 9° error in terms of the circumferential position. The practical significance of this approach is that it can be implemented remotely on an autonomous inspection robot so that accurate axial location and circumferential position of lateral connections and small blockages can be estimated with a computationally efficient algorithm.

Validation of Bayesian design for broadband microslit panel absorbers using causal inference

This paper discusses experimental validations of multilayer microslit panels (MSPs) designed via Bayesian inference to obtain both high sound absorption and wide bandwidth simultaneously. Microslit perforation in thin panels is similar to microperforated panels [Xiang, Fackler, Hou, and Schmitt (2022). J. Acoust. Soc. Am. 151(5), 3094-3103]. MSP absorbers in single-layer configurations are functioning in a limited frequency range. By stacking the MSPs in multiple layered structures, absorbing performance may be widened in frequency ranges while retaining high absorption coefficients. Besides design challenges of multiple MSPs in layered structures to fulfill a practical requirement and minimize fabrication complexity, this paper further discusses challenges in experimental validations when experimental results undesirably deviate from the initial Bayesian design. Causation analysis is applied to the validation efforts where a causal model-based inference effectively provides causal reasoning of fabrication inaccuracies. Along with the causal inference, a causal reasoning conducted in this work can guide corrections due to fabrication inaccuracies during the iterative validation process.

Uncertainty quantification for direction-of-arrival estimation with conformal prediction

Uncertainty quantification (UQ) of deep learning (DL)-based acoustic estimation methods is useful for establishing confidence in the predictions. This is crucial to enable the real-world applicability of DL-based systems for acoustic tasks. Specifically, it is proposed to use conformal prediction (CP) for UQ in direction-of-arrival (DOA) estimation. CP is a statistically rigorous method to provide confidence intervals for an estimated quantity without making distributional assumptions. With CP, confidence intervals are computed via quantiles of user-defined scores. This easy-to-use method can be applied to any trained classification/regression model if an appropriate score function is chosen. The proposed approach shows the potential to enhance the real-time applicability of DL methods for DOA estimation. The advantages of CP are illustrated for different DL methods for DOA estimation in the presence of commonly occurring environmental uncertainty. Codes are available online (https://github.com/NoiseLabUCSD/ConformalPrediction).

Bayesian selection of plane-wave decomposition models

Plane-wave decompositions, whereby a measured sound field is described as a superposition of plane waves, are central to many applications in acoustics and audio engineering. This letter applies a Bayesian probabilistic inference framework to the plane wave decomposition problem and examines the Deviance Information Criterion (DIC) for selecting the optimum number of waves in the decomposition. The framework learns the model directly from the data and, as such, adapts to the wavefield under study. The DIC is applied to data measured in two reverberant sound fields (highly-reverberant and lightly-damped) to determine the simplest models providing the preferred fit to the data.

Graph-based sequential beamforming

This paper presents a Bayesian estimation method for sequential direction finding. The proposed method estimates the number of directions of arrivals (DOAs) and their DOAs performing operations on the factor graph. The graph represents a statistical model for sequential beamforming. At each time step, belief propagation predicts the number of DOAs and their DOAs using posterior probability density functions (pdfs) from the previous time and a different Bernoulli-von Mises state transition model. Variational Bayesian inference then updates the number of DOAs and their DOAs. The method promotes sparse solutions through a Bernoulli-Gaussian amplitude model, is gridless, and provides marginal posterior pdfs from which DOA estimates and their uncertainties can be extracted. Compared to nonsequential approaches, the method can reduce DOA estimation errors in scenarios involving multiple time steps and time-varying DOAs. Simulation results demonstrate performance improvements compared to state-of-the-art methods. The proposed method is evaluated using ocean acoustic experimental data.

Boundary admittance estimation for wave-based acoustic simulations using Bayesian inference

Acoustic surface admittance/impedance at domain boundaries is essential for wave-based acoustic simulations. This work applies two levels of Bayesian inference to estimate the order and the parameter values of the multipole admittance model. The frequency-dependent acoustic admittance is experimentally measured. Incorporating the maximum entropy strategy, the unified Bayesian framework is applied to the multipole approximation. Analysis results demonstrate that multipole model-based Bayesian inference is well suited to estimating the arbitrary frequency-dependent boundary condition within a wave-based simulation framework.

Bayesian estimations of dissipation, sound speed, and microphone positions in impedance tubes

Sound speed, microphone positions, and tube wall dissipation are critical parameters for absorption and impedance measurements using the transfer-function method in an impedance tube. This work applies a Bayesian method, based on a reflection coefficient model of an air layer and a boundary layer dissipation model, to estimate the values of these parameters for tube measurements. This estimation is based on experimental measurements obtained in the empty impedance tube with a rigid termination. Analysis results demonstrate that this method is able to accurately estimate the dissipation coefficient, the sound speed, and the microphone positions for highly accurate tube measurements.

Neural network for multi-exponential sound energy decay analysis

An established model for sound energy decay functions (EDFs) is the superposition of multiple exponentials and a noise term. This work proposes a neural-network-based approach for estimating the model parameters from EDFs. The network is trained on synthetic EDFs and evaluated on two large datasets of over 20 000 EDF measurements conducted in various acoustic environments. The evaluation shows that the proposed neural network architecture robustly estimates the model parameters from large datasets of measured EDFs while being lightweight and computationally efficient. An implementation of the proposed neural network is publicly available.

Bayesian design of broadband multilayered microperforated panel absorbers

In some noise control and architectural acoustics applications, nonfibrous, hygienic materials are desirable or even strictly required. In meeting such restrictive requirements, microperforated panel (MPP) sound absorbers represent a potential solution. Yet, they typically possess limited absorption bandwidth. Combining multiple MPPs into a multilayer system may broaden the absorption frequency ranges while maintaining high absorption. When increasing the overall absorption bandwidth, each additional MPP layer also increases the complexity of the design process because the design parameters are correspondingly increased by four per each additional layer. This paper applies a Bayesian inferential framework to the design of multilayer MPP absorbers with a parsimonious structural configuration, which penalizes the overlayered configurations. This Bayesian framework demonstrates that the practical design of multilayer MPP absorbers may be accomplished with two levels of model-based inference: model selection and parameter estimation. The design process proceeds inversely from a design target to design parameters, including the required number of MPP layers and their corresponding MPP parameters. This paper discusses the Bayesian design formulation, unified implementation of two levels of Bayesian inference, and experimental validation of a Bayesian design for a multilayered MPP absorber, which is able to meet the design target arising from practice.

Spatial reconstruction of the sound field in a room in the modal frequency range using Bayesian inference

Spatial characterization of the sound field in a room is a challenging task, as it usually requires a large number of measurement points. This paper presents a probabilistic approach for sound field reconstruction in the modal frequency range for small and medium-sized rooms based on Bayesian inference. A plane wave expansion model is used to decompose the sound field in the examined domain. The posterior distribution for the amplitude of each plane wave is inferred based on a uniform prior distribution with limits based on the maximum sound pressure observed in the measurements. Two different application cases are studied, namely a numerically computed sound field in a non-rectangular two-dimensional (2D) domain and a measured sound field in a horizontal evaluation area of a lightly damped room. The proposed reconstruction method provides an accurate reconstruction for both examined cases. Further, the results of Bayesian inference are compared to the reconstruction with a deterministic compressive sensing framework. The most significant advantage of the Bayesian method over deterministic reconstruction approaches is that it provides a probability distribution of the sound pressure at every reconstruction point, and thus, allows quantifying the uncertainty of the recovered sound field.

Gaussian processes for sound field reconstruction

This study examines the use of Gaussian process (GP) regression for sound field reconstruction. GPs enable the reconstruction of a sound field from a limited set of observations based on the use of a covariance function (a kernel) that models the spatial correlation between points in the sound field. Significantly, the approach makes it possible to quantify the uncertainty on the reconstruction in a closed form. In this study, the relation between reconstruction based on GPs and classical reconstruction methods based on linear regression is examined from an acoustical perspective. Several kernels are analyzed for their potential in sound field reconstruction, and a hierarchical Bayesian parameterization is introduced, which enables the construction of a plane wave kernel of variable sparsity. The performance of the kernels is numerically studied and compared to classical reconstruction methods based on linear regression. The results demonstrate the benefits of using GPs in sound field analysis. The hierarchical parameterization shows the overall best performance, adequately reconstructing fundamentally different sound fields. The approach appears to be particularly powerful when prior knowledge of the sound field would not be available.