INTERPOLATED 3D DIGITAL WAVEGUIDE MESH WITH FREQUENCY WARPING

1
INTERPOLATED 3-D DIGITAL WAVEGUIDE MESH WITH
FREQUENCY WARPING
IEEE ICASSP 2001, Salt Lake City, May 2001

• Lauri Savioja1 and Vesa Välimäki2
• Helsinki University of Technology
• 1Telecommunications Software and Multimedia Lab.
• 2Lab. of Acoustics and Audio Signal Processing
• (Espoo, Finland)
• http//www.tml.hut.fi/, http//www.acoustics.hut.
  fi/

2
Outline

• Introduction
• 3-D Digital Waveguide Mesh
• Interpolated 3-D Digital Waveguide Mesh
• Optimization of Interpolation Coefficients
• Frequency Warping
• Simulation Example of A Cube
• Conclusions

3
Introduction

• Digital waveguides are useful in physical
  modeling of musical instruments and other
  acoustic systems 8
• 2-D digital waveguide mesh (WGM) for simulation
  of membranes, drums etc. 2
• 3-D digital waveguide mesh for simulation of
  acoustic spaces 3
• Numerous potential applications
• Acoustic design of concert halls, churches,
  auditoria, listening rooms, movie theaters,
  cabins of vehicles, or loudspeaker enclosures

4
Former Methods

• Ray-tracing
• A statistical method
• Inaccurate at low frequencies

5
Former Methods (2)

• Image-source method
• Based on mirror images of the sound source(s)
• Accurate modeling of low-order reflections
• Inaccurate at low frequencies

6
New Method

• Digital waveguide mesh
• Finite-difference method
• Sound propagates through the network from node to
  node
• Wide frequency range of good accuracy
• Requires much memory

7
Sophisticated Waveguide Mesh Structures

• In the original WGM, wave propagation speed
  depends on direction and frequency 3
• Rectangular mesh
• More advanced structures improve this problem,
  e.g.,
• Triangular and tetrahedral WGMs 3,
  7, (Fontana Rocchesso, 1995, 1998)
• Interpolated WGM 5, 6
• Direction-dependence is reduced but
  frequency-dependence remains
• Þ Dispersion !

8
Interpolated 2-D Waveguide Mesh
Hypothetical 8-directional 2-D WGM
Interpolated 2-D WGM 5, 6
Original 2-D WGM 2
9
Wave Propagation Speed
Interpolated 2-D WGM (bilinear interpolation)
Original 2-D WGM
0.2
0.2
0.1
0.1
c
c
0
2
0
2
x
x
-0.1
-0.1
-0.2
-0.2
-0.2
0
0.2
-0.2
0
0.2
c
x
c
x
1
1
10
Wave Propagation Speed (2)
Interpolated 2-D WGM (Optimal interpolation 6)
Original 2-D WGM
0.2
0.1
0
-0.1
-0.2
-0.2
0
0.2
11
Original 3-D Digital Waveguide Mesh

• Difference equation for pressure at each node
• p(n1, x, y, z) (1/3) p(n, x 1, y, z)
  p(n, x 1, y, z)
• p(n, x, y 1, z) p(n, x, y 1, z)
• p(n, x, y, z 1) p(n, x, y, z 1)
• p(n1, x, y, z)
• where p(n, x, y, z) is the sound pressure at
  time step n at position (x, y, z)

12
Interpolated 3-D Digital Waveguide Mesh

• Difference scheme for the interpolated 3-D WGM
• where coefficients h are

13
Interpolated 3-D Digital Waveguide Mesh (2)

2D-diagonal neighbors
Original WGM
All neighbors in interpolated WGM
3D-diagonal neighbors
14
Optimization of Coefficients

• Two constraints must be satisfied
• 1)
• 2) Wave travel speed at dc (0 Hz) must be unity
• From the first constraint, we may solve 1
  coefficient
• Another coefficient can be solved using the 2),
  for example

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Optimization of Coefficients (2)

• The remaining 2 coefficients can be optimized
• We searched for a solution where the difference
  between the min and max error curves is
  minimized
• ha 0.124867
• h2D 0.0387600
• h3D 0.0133567
• hc 0.678827

16
Relative Frequency Error (RFE)
(a) Original WGM (b) Interpolated WGM Line
types Axial blue 2D-diagonal cyan 3D-diagonal
red
17
Frequency Warping

• Dispersion error of the interpolated WGM can be
  reduced by frequency warping 11 because
• Difference between the max and min errors is
  small
• RFE curve is smooth
• Postprocessing of the response of the WGM
• 3 different approaches
• 1) Time-domain warping using a warped-FIR filter
  6, 12
• 2) Time-domain multiwarping
• 3) Frequency warping in the frequency domain

18
Frequency Warping Warped-FIR Filter

• Chain of first-order allpass filters

d(n)
s(0)
s(1)
s(2)
s(L-1)
sw(n)

• s(n) is the signal to be warped
• sw(n) is the warped signal
• The extent of warping is determined by l

19
Frequency Warping Resampling

• Every time-domain frequency-warping operation
  must be accompanied by a sampling rate conversion
• All frequencies are shifted by warping, including
  those that should not
• Resampling factor
• (Phase delay of the allpass filter at the zero
  frequency)
• With optimal warping and resampling, the maximal
  RFE is reduced to 3.8

20
Multiwarping

• How to add degrees of freedom to the time-domain
  frequency-warping to improve the accuracy?
• Frequency-warping and sampling-rate-conversion
  operations can be cascaded
• Many parameters to optimize l1, l2, ... D1,
  D2,...
• We call this multiwarping 12, 13
• Maximal RFE is reduced to 2.0

21
Frequency Warping in the Frequency Domain

• Non-uniform resampling of the Fourier transform
  14, 15
• Postprocessing of the output signal of the mesh
  in the frequency domain
• Warping function can be the average of the RFEs
  in 3 different directions
• Maximal RFE is reduced to 0.78

22
Improvement of Accuracy
Three versions of frequency warping applied to
the optimally interpolated WGM
(a) Single warping (b) Multiwarping (c) Warping
in the frequency domain
23
Simulation of a Cubic Space
Frequency response of a cubic space simulated
using the interpolated WGM
(a) Original WGM (b) Interpolated
Multiwarped (c) Interpolated warped in the
frequency domain (red lineanalytical)
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Conclusions

• Optimally interpolated 3-D digital waveguide mesh
• Interpolation yields nearly direction-dependent
  wave propagation characteristics
• Based on the rectangular mesh, which is easy to
  use
• The remaining dispersion can be reduced by using
  frequency warping
• In the time-domain or in the frequency-domain
• Future goal simulation of acoustic spaces using
  the interpolated 3-D waveguide mesh

25
References
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26
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