Jonathan Botts

Spectral and pseudospectral properties of finite difference models used in audio and room acoustics

Finite difference solutions to the wave equation are simple and flexible modeling tools for approximating physical systems in audio and room acoustics. Each model is characterized by a matrix operator and the time-stepping solution by a sequence of powers of the matrix. Spectral decomposition of representative matrices provide some practical insight into solution behavior and in some cases stability. In addition to computed eigenvalue spectra, pseudospectra provide a description of numerical amplification due to rounding errors in floating point arithmetic. The matrix analysis also shows that certain boundary implementations in non-cuboid geometries can be unstable despite satisfying conditions derived from von Neumann and normal mode analyses.

Finite volume time domain room acoustics simulation under general impedance boundary conditions

In room acoustics simulation and virtualization applications, accurate wall termination is a perceptually crucial feature. It is particularly important in the setting of wave-based modeling of 3D spaces, using methods such as the finite difference time domain method or finite volume time domain method. In this paper, general locally reactive impedance boundary conditions are incorporated into a 3D finite volume time domain formulation, which may be specialized to the various types of finite difference time domain method under fitted boundary termination. Energy methods are used to determine stability conditions for general room geometries, under a large family of nontrivial wall impedances, for finite volume methods over unstructured grids. Simulation results are presented, highlighting in particular the need for unstructured or fitted cells at the room boundary in the case of the accurate simulation of frequency-dependent room mode decay times.

Integrating finite difference schemes for scalar and vector wave equations

Room acoustic simulation is the process of generating approximate solutions to either the linearized Euler equations or the scalar wave equation. As for the continuous equations, the discrete approximations of both are equivalent. The vector formulation is less efficient, but it can inform several unexploited features of the scalar formulation. This paper first demonstrates the equivalence of the two schemes and explores how the vector formulation may be integrated into the more efficient scalar formulation to produce local velocity estimates and velocity sources on the pressure grid.

Effects of sources on time-domain finite difference models

Recent work on excitation mechanisms in acoustic finite difference models focuses primarily on physical interpretations of observed phenomena. This paper offers an alternative view by examining the properties of models from the perspectives of linear algebra and signal processing. Interpretation of a simulation as matrix exponentiation clarifies the separate roles of sources as boundaries and signals. Boundary conditions modify the matrix and thus its modal structure, and initial conditions or source signals shape the solution, but not the modal structure. Low-frequency artifacts are shown to follow from eigenvalues and eigenvectors of the matrix, and previously reported artifacts are predicted from eigenvalue estimates. The role of source signals is also briefly discussed.

Optimization of absorption placement using geometrical acoustic models and least squares

Given a geometrical model of a space, the problem of optimally placing absorption in a space to match a desired impulse response is in general nonlinear. This has led some to use costly optimization procedures. This letter reformulates absorption assignment as a constrained linear least-squares problem. Regularized solutions result in direct distribution of absorption in the room and can accommodate multiple frequency bands, multiple sources and receivers, and constraints on geometrical placement of absorption. The method is demonstrated using a beam tracing model, resulting in the optimal absorption placement on the walls and ceiling of a classroom.

Audibility of dispersion error in room acoustic finite-difference time-domain simulation as a function of simulation distance

Finite-difference time-domain (FDTD) simulation has been a popular area of research in room acoustics due to its capability to simulate wave phenomena in a wide bandwidth directly in the time-domain. A downside of the method is that it introduces a direction and frequency dependent error to the simulated sound field due to the non-linear dispersion relation of the discrete system. In this study, the perceptual threshold of the dispersion error is measured in three-dimensional FDTD schemes as a function of simulation distance. Dispersion error is evaluated for three different explicit, non-staggered FDTD schemes using the numerical wavenumber in the direction of the worst-case error of each scheme. It is found that the thresholds for the different schemes do not vary significantly when the phase velocity error level is fixed. The thresholds are found to vary significantly between the different sound samples. The measured threshold for the audibility of dispersion error at the probability level of 82% correct discrimination for three-alternative forced choice is found to be 9.1 m of propagation in a free field, that leads to a maximum group delay error of 1.8 ms at 20 kHz with the chosen phase velocity error level of 2%.

Design of IIR Filters With Bayesian Model Selection and Parameter Estimation

Bayesian model selection and parameter estimation are used to address the problem of choosing the most concise filter order for a given application while simultaneously determining the associated filter coefficients. This approach is validated against simulated data and used to generate pole-zero representations of head-related transfer functions.

Bayesian inference approach to room-acoustic modal analysis

Spectrum estimation is a problem common to many fields of physics, science, and engineering, and it has thus received a great deal of attention from the Bayesian data analysis community. In room acoustics, the modal or frequency response of a room is important for diagnosing and remedying acoustical defects. The physics of a sound field in a room dictates a model comprised of exponentially decaying sinusoids. Continuing in the tradition of the seminal work of Bretthorst and Jaynes, this work contributes an approach to analyzing the modal responses of rooms with a time-domain model. Room acoustic spectra are constructed of damped sinusoids, and the modelbased approach allows estimation of the number of sinusoids in the signal as well as their frequencies, amplitudes, damping constants, and phase delays. The frequency-amplitude spectrum may be most useful for characterizing a room, but in some settings the damping constants are of primary interest. This is the case for measuring the absorptive properties of...

Comment on “Optimum absorption and aperture parameters for realistic coupled volume spaces determined from computational analysis and subjective testing results” [J. Acoust. Soc. Am. 127, 223–232 (2010)]

A recent paper [D. T. Bradley and L. M. Wang, J. Acoust. Soc. Am. 127, 223-232 (2010)] has reported inconsistencies between the results of two different approaches for characterizing non-exponential decays in coupled-volume systems. This letter aims to expose the origin of these inconsistencies, which are due to a limitation in the methodology utilized for the analysis presented in the paper referenced above.

Bayesian inference for acoustic impedance boundaries in room-acoustic finite difference time-domain modeling

In room acoustics, the finite difference time-domain approach is increasingly being applied to model wave propagation in spaces with complex geometries. For realistic simulation, implementation of frequency-dependent, impedance boundary conditions, is necessary. This paper will demonstrate that the modeling and implementation of acoustic impedance boundaries within the finite difference time-domain approach represents tasks which can be solved by two levels of Bayesian inference. The impedance function is expressed as a parametric model with coefficients of finite order, and the order of the model is directly connected to the accuracy of the calculation, computational expense, and memory requirements. The implicit Occams razor for boundary impedance model selection, followed by coefficient estimation within the Bayesian framework, can be applied for optimizing computational expense and accuracy achieved in room-acoustic finite difference time-domain modeling.