Optimal Space-Time Finite Difference Scheme for Binaural Room Impulse
Response Calculation
Yusuke Naka1,
Assad A. Oberai2, and
Barbara G. Shinn-Cunningham3
1 Department of Aerospace and Mechanical Engineering, Boston University
2 Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer
Polytechnic Institute
3 Boston University Hearing Research Center
Introduction
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Figure 1: The expanding wavefront (as an
isosurface) approaching the human model.
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Many acoustic events in our everyday lives occur in reverberant environments,
from small offices to large concert halls. In such an enclosure, echoes and
reverberation impact auditory perception in many ways, distorting auditory
spatial cues, rendering speech less intelligible, and providing cues for source
distance and room characteristics. In order to investigate the influence of
room acoustics on perception, computer simulations of sound that a listener
would hear in the room is invaluable. By using the binaural room impulse
responses (BRIRs), the impulse responses from a source position to the
listener's left and right ears, the effects of reverberation can be added to
any anechoic sound source.
A BRIR can be obtained by solving the wave equation in a room using a numerical
method. For the accuracy of the numerical method, the dispersion (phase) and
dissipation (magnitude) errors resulting from spatial and temporal
discretization are more important than the formal order of accuracy in a
Taylor series expansion of the differential operators in space and
time. Efforts have been made in various application fields, such as in
computational aero-acoustics (CAA), to reduce the dispersion and dissipation
errors by optimizing the discretization parameters in space and time [1,
2]. However, since the dispersion and dissipation errors accumulate over time,
more accurate numerical methods are required in room acoustics, where
reverberation makes it necessary to calculate BRIRs over long durations.
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Figure 2: The wavefront
passing the human model.
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Optimal Space-Time (OST) Finite Difference Scheme
In this study, a new approach for constructing low dispersion- and
dissipation-error finite difference schemes for the scalar wave equation in the
time domain is developed. The numerical parameters (spatial and temporal finite
difference coefficients) are determined by minimizing the total error (space
and time), unlike the previous approaches, in which the discretization schemes
are optimized only in either space or time, and spatial and temporal
discretizations are not coupled. The resulting scheme is referred to as the
optimal space-time (OST) finite difference scheme. Using the OST approach,
finite difference stencils and time integration schemes, with remarkable
accuracy, are developed in both two and three dimensions. For example, in three
dimensions, a 25-point stencil with a 10-stage time-integration scheme is
developed that incurs an error of less than 2 percent in propagating a high
wavenumber plane wave (6 points per wavelength) at moderate CFL (unity) through
a distance of 1,000 wavelengths.
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Figure 3: The wavefront
reflecting at the walls.
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Results
The OST scheme is applied to the wave equation in a 3m x 3m x 3m cubic room
with a simple model of a listener. The acoustic pressure at all finite
difference grid points are calculated at each time step, and the wave front
(isosurface) is visualized from these results. The visualized images show the
interesting wave phenomena such as diffraction around the listener and
reflection from the walls. The BRIRs (the acoustic pressure at listener's left
and right ears as a function of time) are also obtained. These BRIRs are
converted to audio files to be used for psychoacoustic experiments or for
virtual audio rendering, for example. Simulations were executed on 512 nodes
of Boston University's IBM Blue Gene. The visualization was accomplished
using RSI IDL, Alias Maya, SGI Performer and custom code.
Visualization Movie
This movie was originally shown in stereo on our
travelling
high-resolution stereo display at the
Supercomputing 2006 conference at a resolution of 2048 x 1536. For the web
version here, it has been downsampled to 720 x 540 mono and is available
in the Microsoft AVI and Apple Quicktime video formats.
The
visualization process is described in the
Results section above
and plays for 20 seconds.
(23MB AVI, 177MB QuickTime)
References
[1] C.K.W. Tam and J.C. Webb, "Dispersion-relation-preserving finite
difference schemes for computational acoustics," Journal of Computational
Physics, 107 (2), 262-281, 1993.
[2] F.Q. Fu, M.Y. Hussaini, and J.L. Manthey, "Low-dissipation and
low-dispersion Runge-Kutta schemes for computational acoustics," Journal of
Computational Physis, 124 (1), 177-191, 1996.
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The
visualization process is described in the
Results section above
and plays for 20 seconds.
(23MB AVI, 177MB QuickTime)
