Room acoustic modal analysis using Bayesian inference

Acoustic modal analysis of room responses from the perspective of state-space balanced realization with application to field interpolation

Despite its importance in structural dynamics and vibration, modal analysis is rarely performed in acoustics due to the high modal density of sound fields. A novel acoustic modal analysis (AMA) approach is proposed in this paper for enclosed acoustic fields, such as a room, from the perspective of state-space formulation of control systems. A single-input-multiple-output (SIMO) state-space model is established in light of the balanced realization (BR), given impulse response measurements. The BR model is then converted to a modal form such that the modal parameters, including natural frequencies, damping ratios, and mode shapes, can be estimated. To reconstruct mode shapes, plane wave decomposition (PWD) and compressive sensing (CS) techniques are exploited to solve the underdetermined problem for a spatially sparse representation of mode shapes under the Schroeder frequency. As a result, a model of a continuous system can be "interpolated" for any arbitrary source-receiver positions on the basis of the estimated mode shapes. With the identified modal parameters, the low-frequency and early reflection part of room impulse responses (RIRs) can be synthesized for arbitrary source-sensor pairs. The proposed AMA acoustic field interpolation is validated by extensive simulations and experiments.

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.

Maa's equation for the number of normal modes of sound waves in rectangular rooms

The Reflections series takes a look back on historical articles from The Journal of the Acoustical Society of America that have had a significant impact on the science and practice of acoustics.

Estimation of locally reacting surface impedance at modal frequencies using an eigenvalue approximation technique

The accuracy of computational models for acoustics is often limited by a lack of reliable information concerning the frequency-dependent impedance of surface materials. This lack of information stems from the unavailability of reliable measurement methods for low frequencies. In this work, an approach is proposed, using eigenvalue analysis, for estimating the locally reacting, frequency-dependent impedance of a sound-absorbing sample. In particular, an eigenvalue approximation method is proposed and used in tandem with an optimization routine to obtain surface impedance estimates of an installed sample at modal frequencies. It is shown, using finite element simulations of an impedance tube and a small reverberation room, that the proposed method can provide reasonable estimates of the surface impedance of a sample placed on a boundary surface.

Multichannel deconvolution of vibrational signals: A state-space inverse filtering approach

Deconvolution of noisy measurements, especially when they are multichannel, has always been a challenging problem. The processing techniques developed range from simple Fourier methods to more sophisticated model-based parametric methodologies based on the underlying acoustics of the problem at hand. Methods relying on multichannel mean-squared error processors (Wiener filters) have evolved over long periods from the seminal efforts in seismic processing. However, when more is known about the acoustics, then model-based state-space techniques incorporating the underlying process physics can improve the processing significantly. The problems of interest are the vibrational response of tightly coupled acoustic test objects excited by an out-of-the-ordinary transient, potentially impairing their operational performance. Employing a multiple input/multiple output structural model of the test objects under investigation enables the development of an inverse filter by applying subspace identification techniques during initial calibration measurements. Feasibility applications based on a mass transport experiment and test object calibration test demonstrate the ability of the processor to extract the excitations successfully.

Transient recovery problem in acoustics: A multichannel model-based deconvolution approach

Critical acoustical systems operating in complex environments contaminated with disturbances and noise offer an extreme challenge when excited by out-of-the-ordinary, impulsive, transient events that can be undetected and seriously affect their overall performance. Transient impulse excitations must be detected, extracted, and evaluated to determine any potential system damage that could have been imposed; therefore, the problem of recovering the excitation in an uncertain measurement environment becomes one of multichannel deconvolution. Recovering a transient and its initial energy has not been solved satisfactorily, especially when the measurement has been truncated and only a small segment of response data is available. The development of multichannel deconvolution techniques for both complete and incomplete excitation data is discussed, employing a model-based approach based on the state-space representation of an identified acoustical system coupled to a forward modeling solution and a Kalman-type processor for enhancement and extraction. Synthesized data are utilized to assess the feasibility of the various approaches, demonstrating that reasonable performance can be achieved even in noisy environments.

Model-based Bayesian analysis in acoustics-A tutorial

Bayesian analysis has been increasingly applied in many acoustical applications. In these applications, prediction models are often involved to better understand the process under investigation by purposely learning from the experimental observations. When involving the model-based data analysis within a Bayesian framework, issues related to incorporating the experimental data and assigning probabilities into the inferential learning procedure need fundamental consideration. This paper introduces Bayesian probability theory on a tutorial level, including fundamental rules for manipulating the probabilities, and the principle of maximum entropy for assignment of necessary probabilities prior to the data analysis. This paper also employs a number of examples recently published in this journal to explain detailed steps on how to apply the model-based Bayesian inference to solving acoustical problems.

Multichannel spectral estimation in acoustics: A state-space approach

Spectral estimation is a necessary methodology to analyze the frequency content of noisy data sets especially in acoustic applications. Many spectral techniques have evolved starting with the classical Fourier transform methods based on the well-known Wiener-Khintchine relationship relating the covariance-to-spectral density as a transform pair culminating with more elegant model-based parametric techniques that apply prior knowledge of the data to produce a high-resolution spectral estimate. Multichannel spectral representations are a class of both nonparametric, as well as parametric, estimators that provide improved spectral estimates. In any case, classical nonparametric multichannel techniques can provide reasonable estimates when coupled with peak-peaking methods as long as the signal levels are reasonably high. Parametric multichannel methods can perform quite well in low signal level environments even when applying simple peak-picking techniques. In this paper, the performance of both nonparametric (periodogram) and parametric (state-space) multichannel spectral estimation methods are investigated when applied to both synthesized noisy structural vibration data as well as data obtained from a sounding rocket flight. It is demonstrated that for the multichannel problem, state-space techniques provide improved performance, offering a parametric alternative compared to classical methods.

Vibrational processing of a dynamic structural flight system: A multichannel spectral estimation approach

Dynamic testing of large flight vehicles (rockets) is not only complex, but also can be very costly. These flights are infrequent and can lead to disastrous effects if something were to fail during the flight. The development of sensors coupled to internal components offers a great challenge in reducing their size, yet still maintaining their precision. Sounding rockets provide both a viable and convenient alternative to the more costly vehicular flights. Some of the major objectives are to test various types of sensors for monitoring components of high interest as well as investigating real-time processing techniques. Signal processing presents an extreme challenge in this noisy multichannel environment. The estimation and tracking of modal frequencies from vibrating structures is an important set of features that can provide information about the components under test; therefore, high resolution multichannel spectral processing is required. The application of both single channel and multichannel techniques capable of producing reliable modal frequency estimates of a vibrating structure from uncertain accelerometer measurements is discussed.

Model-based Bayesian direction of arrival analysis for sound sources using a spherical microphone array

In many room acoustics and noise control applications, it is often challenging to determine the directions of arrival (DoAs) of incoming sound sources. This work seeks to solve this problem reliably by beamforming, or spatially filtering, incoming sound data with a spherical microphone array via a probabilistic method. When estimating the DoA, the signal under consideration may contain one or multiple concurrent sound sources originating from different directions. This leads to a two-tiered challenge of first identifying the correct number of sources, followed by determining the directional information of each source. To this end, a probabilistic method of model-based Bayesian analysis is leveraged. This entails generating analytic models of the experimental data, individually defined by a specific number of sound sources and their locations in physical space, and evaluating each model to fit the measured data. Through this process, the number of sources is first estimated, and then the DoA information of those sources is extracted from the model that is the most concise to fit the experimental data. This paper will present the analytic models, the Bayesian formulation, and preliminary results to demonstrate the potential usefulness of this model-based Bayesian analysis for complex noise environments with potentially multiple concurrent sources.

Multichannel processing of vibrational measurements: A constrained subspace application

Monitoring mechanical systems operating in uncertain environments contaminated with both environmental disturbances and noise lead directly to low signal-to-noise-ratios, creating an extremely challenging processing problem, especially in real-time. In order to estimate the performance of a particular system from uncertain vibrational data, it is necessary to identify its unique resonant (modal) frequency signature. The monitoring of structural modes to determine the condition of a device under investigation is essential, especially if it is a critical entity of an operational system. The development of a model-based scheme capable of the on-line tracking of the inherent structural modal frequencies by applying both constrained subspace identification techniques to extract the modal frequencies and state estimation methods to track the evolution is discussed. An application of this approach to a cylindrical structural device (pipe-in-air) is analyzed based on theoretical simulations along with controlled validation experiments, including injected anomalies illustrate the approach and performance. Statistics are gathered to bound potential processors for real-time performance employing these constrained techniques.

Bayesian Inference for Acoustic Direction of Arrival Analysis Using Spherical Harmonics

This work applies two levels of inference within a Bayesian framework to accomplish estimation of the directions of arrivals (DoAs) of sound sources. The sensing modality is a spherical microphone array based on spherical harmonics beamforming. When estimating the DoA, the acoustic signals may potentially contain one or multiple simultaneous sources. Using two levels of Bayesian inference, this work begins by estimating the correct number of sources via the higher level of inference, Bayesian model selection. It is followed by estimating the directional information of each source via the lower level of inference, Bayesian parameter estimation. This work formulates signal models using spherical harmonic beamforming that encodes the prior information on the sensor arrays in the form of analytical models with an unknown number of sound sources, and their locations. Available information on differences between the model and the sound signals as well as prior information on directions of arrivals are incorporated based on the principle of the maximum entropy. Two and three simultaneous sound sources have been experimentally tested without prior information on the number of sources. Bayesian inference provides unambiguous estimation on correct numbers of sources followed by the DoA estimations for each individual sound sources. This paper presents the Bayesian formulation, and analysis results to demonstrate the potential usefulness of the model-based Bayesian inference for complex acoustic environments with potentially multiple simultaneous sources.

Bayesian acoustic analysis of multilayer porous media

In many acoustical applications, porous materials may be stratified or physically anisotropic along their depth direction. In order to better understand the sound absorbing mechanisms of these porous media, the depth-dependent anisotropy can be approximated as a multilayer combination of finite-thickness porous materials with each layer being considered as isotropic. The uniqueness of this work is that it applies Bayesian probabilistic inference to determine the number of constituent layers in a multilayer porous specimen and macroscopic properties of their pores. This is achieved through measurement of the acoustic surface impedance and subsequent transfer-matrix analysis based on a valid theoretical model for the acoustical properties of porous media. The number of layers considered in the transfer-matrix analysis is varied, and Bayesian model selection is applied to identify individual layers present in the porous specimen and infer the parameters of their microstructure. Nested sampling is employed in this process to solve the computationally intensive inversion problem.