Välimäki and Savioja 2000 1
HELSINKI UNIVERSITY OF TECHNOLOGY
Interpolated and Warped 2-D Digital Interpolated and Warped 2-D Digital Waveguide Mesh AlgorithmsWaveguide Mesh Algorithms
Vesa Välimäki1 and Lauri Savioja2
Helsinki University of Technology1Laboratory of Acoustics and Audio Signal Processing2Telecommunications Software and Multimedia Lab.
(Espoo, Finland)
DAFX’00, Verona, Italy, December 2000
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Outline Introduction 2-D Digital Waveguide Mesh Algorithms Frequency Warping Techniques Extending the Frequency Range Numerical Examples Conclusions
Interpolated and Warped 2-D Digital Interpolated and Warped 2-D Digital Waveguide Mesh AlgorithmsWaveguide Mesh Algorithms
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• Digital waveguidesDigital waveguides for physical modeling of musical instruments and other acoustic systems (Smith, 1992)
• 22-D digital waveguide mesh-D digital waveguide mesh (WGM) for simulation of membranes, drums etc. (Van Duyne & Smith, 1993)
• 3-D digital waveguide mesh3-D digital waveguide mesh for simulation of acoustic spaces (Savioja et al., 1994)
- Violin body (Huang et al., 2000)
- Drums (Aird et al., 2000)
IntroductionIntroduction
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• In the original WGM, wave propagation speed depends on direction and frequency (Van Duyne & Smith, 1993)
• More advanced structures ease this problem, e.g.,–Triangular WGMTriangular WGM (Fontana & Rocchesso, 1995,
1998; Van Duyne & Smith, 1995, 1996)– Interpolated rectangular WGMInterpolated rectangular WGM (Savioja & Välimäki,
ICASSP’97, IEEE Trans. SAP 2000)• Direction-dependence is reduced but frequency-
dependence remains
DispersionDispersion
Sophisticated 2-D Waveguide StructuresSophisticated 2-D Waveguide Structures
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Interpolated Rectangular Waveguide MeshInterpolated Rectangular Waveguide Mesh
Original WGMHypothetical8-directional
WGMInterpolated WGM
(Savioja & Välimäki, 1997, 2000)
(Van Duyne & Smith, 1993)
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Wave Propagation SpeedWave Propagation Speed
Original WGMInterpolated WGM
(Bilinear interpolation)
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Wave Propagation Speed Wave Propagation Speed (2)(2)
Original WGMInterpolated 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
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Wave Propagation Speed Wave Propagation Speed (3)(3)
Original WGM
-0.2
0 0.2
-0.2
-0.1
0
0.1
0.2
1c
2c
Interpolated WGM(Quadratic interpolation)
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Wave Propagation Speed Wave Propagation Speed (4)(4)
Original WGMInterpolated WGM
(Optimal interpolation)
-0.2
0 0.2
-0.2
-0.1
0
0.1
0.2
1c
2c
(Savioja & Välimäki, 2000)
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RFE in diagonaldiagonal and
axialaxial directions:
(a) original and
(b) bilinearly
interpolated
rectangular WGM
0 0.05
0.1
0.15
0.2
0.25
-10
-5
0
5(a)
0 0.05
0.1
0.15
0.2
0.25
-10
-5
0
5(b)
NORMALIZED FREQUENCY
RE
LAT
IVE
FR
EQ
UE
NC
Y E
RR
OR
(%
)
Relative Frequency Error (RFE)Relative Frequency Error (RFE)
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Relative Frequency Error (RFE)Relative Frequency Error (RFE) (2) (2)
RFE in diagonaldiagonal and
axialaxial directions:
Optimally interpolated
rectangular WG mesh
(up to 0.25fs)
RE
LAT
IVE
FR
EQ
UE
NC
Y E
RR
OR
(%
)
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Frequency WarpingFrequency Warping
• Dispersion error of the interpolated WGM can be reduced using frequency warping because
– The difference between the max and min errors is small
– The RFE curve is smooth• Postprocess the response of the WGM using a
warped-FIR filter warped-FIR filter (Oppenheim et al., 1971; Härmä et al., JAES, Nov. 2000)
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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(0)(0) ss(1)(1) ss(2)(2) ss((LL-1)-1)
A zz
z( )
1
11
((nn))
ssww((nn))
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Optimization of Warping Factor Optimization of Warping Factor • Different optimization strategies can be used, such as
- least squares
- minimize maximal error (minimax)
- maximize the bandwidth of X% error tolerance• We present results for minimax optimization
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(a,b) Bilinearinterpolation
(c,d) Quadraticinterpolation
(e,f) Optimal interpolation
(g,h) Triangular mesh
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Higher-Order Frequency Warping?Higher-Order Frequency Warping?
• How to add degrees of freedom to the warping to improve the accuracy?
– Use a chain of higher-order allpass filters?
Perhaps, but aliasing will occur... No.– Many 1st-order warpings in cascade?
No, because it’s equivalent to a single warping using (1 + 2) / (1 + 12)
• There is a way...
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MultiwarpingMultiwarping• Every frequency warping operation must be
accompanied by sampling rate conversion– All frequencies are shifted by warping, including
those that should not• Frequency-warping and sampling-rate-conversion
operations can be cascaded
– Many parameters to optimize: 1, 2, ... D1, D2,...
Frequencywarping
Samplingrate conv.
Frequencywarping
Samplingrate conv.
)(1 nx )(nyM
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Reduced Relative Frequency ErrorReduced Relative Frequency Error
(a) Warping with
= –0.32
(b) Multiwarping with
1 = –0.92, D1 = 0.998
2 = –0.99, D2 = 7.3
(c) Error in eigenmodes
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Computational complexityComputational complexity
• Original WGM: 1 binary shift & 4 additions
• Interpolated WGM: 3 MUL & 9 ADD
• Warped-FIR filter: O(L2) where L is the signal length
• Advantages of interpolation & warping
– Wider bandwidth with small error: up to 0.25 instead of 0.1 or so
– If no need to extend bandwidth, smaller mesh size may be used
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Extending the Frequency RangeExtending the Frequency Range
• It is known that the limiting frequency of the original waveguide mesh is 0.25
– The point-to-point transfer functions on the mesh are functions of z–2 , i.e., oversampling by 2
• Fontana and Rocchesso (1998): triangular WG mesh has a wider frequency range, up to about 0.3
• How about the interpolated WG mesh?
– The interpolation changes everything
– Maybe also the upper frequency changes...
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Relative Frequency Error (RFE) Relative Frequency Error (RFE) (2)(2)
RE
LAT
IVE
FR
EQ
UE
NC
Y E
RR
OR
(%
)
RFE in diagonaldiagonal and
axialaxial directions:
Optimally interpolated
rectangular WG mesh
(up to 0.35fs)
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Extending the Frequency Range Extending the Frequency Range (3)(3)
• The mapping of frequencies for various WGMs
0 0.5
0
0.1
0.2
0.3
0.4
(a)
NORMALIZED FREQUENCY
ME
SH
F
RE
QU
EN
CY
0 0.5
0
0.1
0.2
0.3
0.4
(b)
NORMALIZED FREQUENCY
ME
SH
F
RE
QU
EN
CY
0 0.5
0
0.1
0.2
0.3
0.4
(c)
NORMALIZED FREQUENCY
ME
SH
F
RE
QU
EN
CY
0 0.5
0
0.1
0.2
0.3
0.4
(d)
= -0.32736
NORMALIZED FREQUENCY
WA
RP
ED
F
RE
QU
EN
CY
Upper frequency
limit always 0.3536
(a) Original
(b) Optimally interp.
up to 0.25
(c) Optimally interp.
up to 0.35
(d) Warped case b
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Simulation Result vs. Analytical Solution Simulation Result vs. Analytical Solution
Magnitude spectrum of a square membrane
(a) original
(b) warped interpolated
( = –0.32736)
(c) warped triangular
( = –0.10954)
digital waveguide mesh
(with ideal response in the background)
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Error in Mode Frequencies Error in Mode Frequencies
Warped Warped triangulartriangular
WGMWGM
Warped Warped interpolatedinterpolated
WGMWGM
Original Original WGMWGM
Error in eigenfrequencies of a square membrane
RE
LAT
IVE
FR
EQ
UE
NC
Y E
RR
OR
(%
)
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Conclusions and Future WorkConclusions and Future Work
• Accuracy of 2-D digital waveguide mesh simulations can be improved using
1) the interpolatedinterpolated or triangular WGM and
2) frequency warping frequency warping or multiwarpingmultiwarping• Dispersion can be reduced dramatically• In the future, the interpolation and warping
techniques will be applied to 3-D3-D WGM simulations• Modeling of boundary conditions and losses must
be improved
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