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Välimäki and Savioja 2000 1
HELSINKI UNIVERSITY OF TECHNOLOGY
Interpolated and Warped 2-D Di gitalInterpolated and Warped 2-D Di gitalWaveguide Mesh Al gorithmsWaveguide Mesh Al gorithms
Vesa Välimäki 1 and Lauri Savioja 2
Helsinki University of Technology1Laboratory of Acoustics and Audio Signal Processing
2Telecommunications Software and Multimedia Lab.
(Espoo, Finland)
DAFX’00, Verona, Italy, December 2000
Välimäki and Savioja 2000 2
HELSINKI UNIVERSITY OF TECHNOLOGY
Outline➤ Introduction
➤ 2-D Digital Waveguide Mesh Algorithms
➤ Frequency Warping Techniques
➤ Extending the Frequency Range
➤ Numerical Examples
➤ Conclusions
Interpolated and Warped 2-D Di gitalInterpolated and Warped 2-D Di gitalWaveguide Mesh Al gorithmsWaveguide Mesh Al gorithms
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•• Digital waveguidesDigital waveguides for physical modeling of musicalinstruments and other acoustic systems (Smith, 1992)
•• 22-D digital waveguide mesh-D digital waveguide mesh (WGM) for simulation ofmembranes, drums etc. (Van Duyne & Smith, 1993)
•• 3-D digital waveguide mesh3-D digital waveguide mesh for simulation of acousticspaces (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 dependson 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 Wave guide StructuresSophisticated 2-D Wave guide Structures
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Interpolated Rectan gular Wave guide MeshInterpolated Rectan gular Wave guide Mesh
Original WGMHypothetical8-directional
WGMInterpolated WGM
(Savioja & Välimäki,1997, 2000)
(Van Duyne & Smith,1993)
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Wave Propa gation SpeedWave Propa gation Speed
Original WGMInterpolated WGM
(Bilinear interpolation)
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Wave Propa gation Speed Wave Propa gation Speed (2)(2)
Original WGMInterpolated WGM
(Bilinear interpolation)
-0.2 0 0.2
-0.2
-0.1
0
0.1
0.2
ξ1c
ξ2c
-0.2 0 0.2
-0.2
-0.1
0
0.1
0.2
ξ1c
ξ2c
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Wave Propa gation Speed Wave Propa gation Speed (3)(3)
Original WGM
-0.2 0 0.2
-0.2
-0.1
0
0.1
0.2
ξ1c
ξ2c
Interpolated WGM(Quadratic interpolation)
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Wave Propa gation Speed Wave Propa gation Speed (4)(4)
Original WGMInterpolated WGM
(Optimal interpolation)
-0.2 0 0.2
-0.2
-0.1
0
0.1
0.2
ξ1c
ξ2c
(Savioja & Välimäki, 2000)
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HELSINKI UNIVERSITY OF TECHNOLOGY
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 interpolatedrectangular WG mesh(up to 0.25fs)
RE
LAT
IVE
FR
EQ
UE
NC
Y E
RR
OR
(%
)
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Frequency Warpin gFrequency Warpin g
• Dispersion error of the interpolated WGM can bereduced using frequency warping because
– The difference between the max and min errorsis small
– The RFE curve is smooth
• Postprocess the response of the WGM using awarped-FIR filter warped-FIR filter (Oppenheim et al., 1971; Härmä etal., JAES, Nov. 2000)
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Frequency Warpin g: Warped-FIR FilterFrequency Warpin g: 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 λ
AA((zz))AA((zz))AA((zz))
ss(0)(0) ss(1)(1) ss(2)(2) ss((LL-1)-1)
A zz
z( ) = +
+
−
−
1
11
λλ
δδ((nn))
ssww((nn))
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Optimization of Warpin g Factor Optimization of Warpin g Factor λλ• 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) Optimalinterpolation
(g,h) Triangularmesh
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Higher-Order Frequency Warpin g?Higher-Order Frequency Warpin g?
• How to add degrees of freedom to the warping toimprove 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 warpingusing (λ1 + λ2) / (1 + λ1λ2)
• There is a way...
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Multiwarpin gMultiwarpin g• 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-conversionoperations can be cascaded
– Many parameters to optimize: λ1, λ2, ... D1, D2,...
Frequencywarp ing
Samp l i ngrate conv.
F requencywarp ing
Samp l i ngrate conv.
)(1 nx )(nyM
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Reduced Relative Frequency ErrorReduced Relative Frequency Error
(a) Warping with
λ = –0.32
(b) Multiwarping with
λ1 = –0.92, D1 = 0.998
λ2 = –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 0.25instead of 0.1 or so
– If no need to extend bandwidth, smaller mesh sizemay be used
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Extendin g the Frequency Ran geExtendin g the Frequency Ran ge• It is known that the limiting frequency of the original
waveguide mesh is 0.25
– The point-to-point transfer functions on the meshare functions of z–2 , i.e., oversampling by 2
• Fontana and Rocchesso (1998): triangular WG meshhas 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 interpolatedrectangular WG mesh
(up to 0.35fs)
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HELSINKI UNIVERSITY OF TECHNOLOGY
Extendin g the Frequency Ran ge Extendin g the Frequency Ran ge (3)(3)
• The mapping of frequencies for various WGMs
0 0.50
0.1
0.2
0.3
0.4 (a)
NORMALIZED FREQUENCYM
ES
H F
RE
QU
EN
CY
0 0.50
0.1
0.2
0.3
0.4 (b)
NORMALIZED FREQUENCY
ME
SH
FR
EQ
UE
NC
Y
0 0.50
0.1
0.2
0.3
0.4 (c)
NORMALIZED FREQUENCY
ME
SH
FR
EQ
UE
NC
Y
0 0.50
0.1
0.2
0.3
0.4 (d)
λ = -0.32736
NORMALIZED FREQUENCY
WA
RP
ED
FR
EQ
UE
NC
Y
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 ResultSimulation Result vs vs . Analytical Solution. Analytical SolutionMagnitude spectrum ofa square membrane(a) original(b) warped interpolated
(λ = –0.32736)(c) warped triangular
(λ = –0.10954)digital waveguide mesh
(with ideal response inthe background)
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Error in Mode FrequenciesError in Mode Frequencies
Warped triangularWarped triangular
WGM WGM
Warped interpolatedWarped interpolated
WGM WGM
Original WGMOriginal WGM
Error in eigenfrequenciesof 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 simulationscan 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 warpingtechniques will be applied to 3-D3-D WGM simulations
• Modeling of boundary conditions and losses mustbe improved