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HELSINKI UNIVERSITY OF TECHNOLOGY Savioja and Välimäki 20011 INTERPOLATED 3-D DIGITAL WAVEGUIDE...
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Savioja and Välimäki 2001 1
HELSINKI UNIVERSITY OF TECHNOLOGY
INTERPOLATED 3-D DIGITAL INTERPOLATED 3-D DIGITAL WAVEGUIDE MESH WITH WAVEGUIDE MESH WITH FREQUENCY WARPINGFREQUENCY WARPINGLauri Savioja1 and Vesa Välimäki2
Helsinki University of Technology1Telecommunications Software and Multimedia Lab.
2Lab. of Acoustics and Audio Signal Processing
(Espoo, Finland)
http://www.tml.hut.fi/, http://www.acoustics.hut.fi/
IEEE ICASSP 2001, Salt Lake City, May 2001IEEE ICASSP 2001, Salt Lake City, May 2001
Savioja and Välimäki 2001 2
HELSINKI UNIVERSITY OF TECHNOLOGY
Introduction 3-D Digital Waveguide Mesh Interpolated 3-D Digital Waveguide Mesh Optimization of Interpolation Coefficients Frequency Warping Simulation Example of A Cube Conclusions
OutlineOutline
Savioja and Välimäki 2001 3
HELSINKI UNIVERSITY OF TECHNOLOGY
• Digital waveguidesDigital waveguides are useful in physical modeling of musical instruments and other acoustic systems [8]
• 2-D digital waveguide mesh2-D digital waveguide mesh (WGM) for simulation of membranes, drums etc. [2]
• 3-D digital waveguide mesh3-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
IntroductionIntroduction
Savioja and Välimäki 2001 4
HELSINKI UNIVERSITY OF TECHNOLOGY
Former MethodsFormer Methods• Ray-tracing
–A statistical method
–Inaccurate at low frequencies
Savioja and Välimäki 2001 5
HELSINKI UNIVERSITY OF TECHNOLOGY
• Image-source method –Based on mirror images
of the sound source(s)–Accurate modeling of
low-order reflections–Inaccurate at low
frequencies
Former MethodsFormer Methods (2) (2)
Savioja and Välimäki 2001 6
HELSINKI UNIVERSITY OF TECHNOLOGY
New MethodNew 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
Savioja and Välimäki 2001 7
HELSINKI UNIVERSITY OF TECHNOLOGY
• 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 WGMInterpolated WGM [5], [6]• Direction-dependence is reduced but frequency-
dependence remains
Dispersion !Dispersion !
Sophisticated Waveguide Mesh StructuresSophisticated Waveguide Mesh Structures
Savioja and Välimäki 2001 8
HELSINKI UNIVERSITY OF TECHNOLOGY
Interpolated 2-D Waveguide MeshInterpolated 2-D Waveguide Mesh
Original 2-D WGM [2]Hypothetical8-directional 2-D WGM
Interpolated 2-D WGM
[5], [6]
Savioja and Välimäki 2001 9
HELSINKI UNIVERSITY OF TECHNOLOGY
Wave Propagation SpeedWave Propagation Speed
Original 2-D WGM Interpolated 2-D WGM(bilinear interpolation)
-0.2
0 0.2
-0.2
-0.1
0
0.1
0.2
1c
2c
-0.2
0 0.2
-0.2
-0.1
0
0.1
0.2
1c
2c
Savioja and Välimäki 2001 10
HELSINKI UNIVERSITY OF TECHNOLOGY
Wave Propagation Speed Wave Propagation Speed (2)(2)
Original 2-D WGMInterpolated 2-D WGM
(Optimal interpolation [6])
-0.2
0 0.2
-0.2
-0.1
0
0.1
0.2
1c
2c
Savioja and Välimäki 2001 11
HELSINKI UNIVERSITY OF TECHNOLOGY
Original 3-D Digital Waveguide MeshOriginal 3-D Digital Waveguide Mesh
• Difference equation for pressure at each node
p(n+1, 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(n–1, x, y, z)
where p(n, x, y, z) is the sound pressure at time step n at position (x, y, z)
Savioja and Välimäki 2001 12
HELSINKI UNIVERSITY OF TECHNOLOGY
Interpolated 3-D Digital Waveguide MeshInterpolated 3-D Digital Waveguide Mesh
• Difference scheme for the interpolated 3-D WGM
where coefficients h are
),,,1(
),,,(),,(),,,1(1
1
1
1
1
1
zyxnp
mzlykxnpmlkhzyxnpk l m
0 if
3 if
2 if
1 if
,
,
,
,
),,(3
2
mlk
mlk
mlk
mlk
h
h
h
h
mlkh
c
D
D
a
Savioja and Välimäki 2001 13
HELSINKI UNIVERSITY OF TECHNOLOGY
Interpolated 3-D Digital Waveguide MeshInterpolated 3-D Digital Waveguide Mesh (2) (2)
(a ) (b )
(c ) (d )
Original WGM
All neighbors
in interpolated
WGM
3D-diagonal
neighbors
2D-diagonal
neighbors
Savioja and Välimäki 2001 14
HELSINKI UNIVERSITY OF TECHNOLOGY
Optimization of CoefficientsOptimization 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:
2]2
463[2 32max cDDa
hhhhb
DDac hhhh 32 81262
)3121(12
123 aDD hhh
Savioja and Välimäki 2001 15
HELSINKI UNIVERSITY OF TECHNOLOGY
Optimization of CoefficientsOptimization of Coefficients (2) (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
Savioja and Välimäki 2001 16
HELSINKI UNIVERSITY OF TECHNOLOGY
(a) Original WGM
(b) Interpolated WGM
Line typesAxial — blueAxial — blue2D-diagonal — cyan2D-diagonal — cyan3D-diagonal — red3D-diagonal — red
Relative Frequency Error (RFE)Relative Frequency Error (RFE)Title:3Drfes_col.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.
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HELSINKI UNIVERSITY OF TECHNOLOGY
Frequency WarpingFrequency 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 warped-FIR filter [6], [12]
2) Time-domain multiwarping
3) Frequency warping in the frequency domain
Savioja and Välimäki 2001 18
HELSINKI UNIVERSITY OF TECHNOLOGY
Frequency Warping: Warped-FIR Filter Frequency Warping: Warped-FIR Filter
• Chain of first-order allpass filters
• s(n) is the signal to be warped
• sw(n) is the warped signal
• The extent of warping is determined by
AA((zz))AA((zz))AA((zz))
ss((00)) ss((11)) ss((22)) ss((L-1L-1))
A zz
z( )
1
11
((nn))
ssww((nn))
Savioja and Välimäki 2001 19
HELSINKI UNIVERSITY OF TECHNOLOGY
Frequency Warping: ResamplingFrequency 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%
1
1D
Savioja and Välimäki 2001 20
HELSINKI UNIVERSITY OF TECHNOLOGY
MultiwarpingMultiwarping• 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: 1, 2, ... D1, D2,...
• We call this multiwarping [12], [13]• Maximal RFE is reduced to 2.0%
Frequencywarping
Samplingrate conv.
Frequencywarping
Samplingrate conv.
)(1 nx )(nyM
Savioja and Välimäki 2001 21
HELSINKI UNIVERSITY OF TECHNOLOGY
Frequency Warping in the Frequency Warping in the Frequency Domain Frequency Domain
• Non-uniform resampling of the Fourier transformNon-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%
Savioja and Välimäki 2001 22
HELSINKI UNIVERSITY OF TECHNOLOGY
(a) Single warping
(b) Multiwarping
(c) Warping in the frequency domain
Improvement of Accuracy Improvement of Accuracy Title:warpoptint3Drfes_col.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.
Three versions of frequency warping applied to the optimally interpolated WGM
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HELSINKI UNIVERSITY OF TECHNOLOGY
Title:simres3D_col.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.
(a) Original WGMOriginal WGM
(b) Interpolated & Interpolated & MultiwarpedMultiwarped
(c) Interpolated & Interpolated & warped in the warped in the frequency domainfrequency domain
(red line—analyticalred line—analytical)
Simulation of a Cubic Space Simulation of a Cubic Space Frequency response of a cubic space simulated using the interpolated WGM
Savioja and Välimäki 2001 24
HELSINKI UNIVERSITY OF TECHNOLOGY
ConclusionsConclusions• Optimally interpolatedOptimally interpolated 3-D digital waveguide mesh
– Interpolation yields nearly direction-dependent wave propagation characteristics
– Based on the rectangular meshrectangular mesh, which is easy to use
• The remaining dispersion can be reduced by using frequency warpingfrequency warping
– In the time-domain or in the frequency-domain
• Future goal: simulation of acoustic spaces using the interpolated 3-D waveguide mesh
Savioja and Välimäki 2001 25
HELSINKI UNIVERSITY OF TECHNOLOGY
ReferencesReferences[1] L. Savioja, T. Rinne, and T. Takala, “Simulation of room acoustics with a 3-D finite
difference mesh,” in Proc. Int. Computer Music Conf., Aarhus, Denmark, Sept. 1994.
[2] S. Van Duyne and J. O. Smith, “The 2-D digital waveguide mesh,” in Proc. IEEE WASPAA’93, New Paltz, NY, Oct. 1993.
[3] S. Van Duyne and J. O. Smith, “The tetrahedral digital waveguide mesh,” in Proc. IEEE WASPAA’95, New Paltz, NY, Oct. 1995.
[4] L. Savioja, “Improving the 3-D digital waveguide mesh by interpolation,” in Proc. Nordic Acoustical Meeting, Stockholm, Sweden, Sept. 1998, pp. 265–268.
[5] L. Savioja and V. Välimäki, “Improved discrete-time modeling of multi-dimensional wave propagation using the interpolated digital waveguide mesh,” in Proc. IEEE ICASSP’97, Munich, Germany, April 1997.
[6] L. Savioja and V. Välimäki, “Reducing the dispersion error in the digital waveguide mesh using interpolation and frequency-warping techniques,” IEEE Trans. Speech and Audio Process., March 2000.
[7] S. Van Duyne and J. O. Smith, “The 3D tetrahedral digital waveguide mesh with musical applications,” in Proc. Int. Computer Music Conf., Hong Kong, Aug. 1996.
Savioja and Välimäki 2001 26
HELSINKI UNIVERSITY OF TECHNOLOGY
[8] J. O. Smith, “Principles of digital waveguide models of musical instruments,” in Applications of Digital Signal Processing to Audio and Acoustics, M. Kahrs and K. Brandenburg, Eds., chapter 10, pp. 417–466. Kluwer Academic, Boston, MA, 1997.
[9] J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Chapman & Hall, New York, NY, 1989.
[10] L. Savioja and V. Välimäki, “Reduction of the dispersion error in the triangular digital waveguide mesh using frequency warping,” IEEE Signal Process. Letters, March 1999.
[11] A. Oppenheim, D. Johnson, and K. Steiglitz, “Computation of spectra with unequal resolution using the Fast Fourier Transform,” Proc. IEEE, Feb. 1971.
[12] V. Välimäki and L. Savioja, “Interpolated and warped 2-D digital waveguide mesh algorithms,” in Proc. DAFX-00, Verona, Italy, Dec. 2000.
[13] L. Savioja and V. Välimäki, “Multiwarping for enhancing the frequency accuracy of digital waveguide mesh simulations,” IEEE Signal Processing Letters, May 2001.
[14] J. O. Smith, Techniques for Digital Filter Design and System Identification with Application to the Violin, Ph.D. thesis, Stanford University, June 1983.
[15] J.-M. Jot, V. Larcher, and O. Warusfel, “Digital signal processing issues in the context of binaural and transaural stereophony,” in 98th AES Convention, Paris, Feb. 1995.