INTERPOLATED 3-D DIGITAL WAVEGUIDE MESH WITH FREQUENCY WARPING

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



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



• 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



Former MethodsFormer Methods• Ray-tracing

–A statistical method

–Inaccurate at low frequencies



• 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)



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



• 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



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]



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



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



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)



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



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



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



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



(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.



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



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))



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



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



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%



(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



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



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



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.



[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.

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INTERPOLATED 3-D DIGITAL WAVEGUIDE MESH WITH FREQUENCY WARPING