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DAFX00, Verona, Italy, December 2000 HELSINKI UNIVERSITY OF TECHNOLOGY Interpolated and Warped 2-D Digital Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms Waveguide Mesh Algorithms Vesa Vlimki 1 and Lauri Savioja 2

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Välimäki and Savioja 2000 1

HELSINKI UNIVERSITY OF TECHNOLOGY

Interpolated and Warped 2-D Digital Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms Waveguide Mesh Algorithms

Vesa Välimäki1 and Lauri Savioja2

Helsinki University of Technology

1Laboratory of Acoustics and Audio Signal Processing 2Telecommunications Software and Multimedia Lab.

(Espoo, Finland)

DAFX’00, Verona, Italy, December 2000

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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 Digital Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms Waveguide Mesh Algorithms

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  • Digital waveguides

Digital waveguides for physical modeling of musical instruments and other acoustic systems (Smith, 1992)

  • 2

2-D digital waveguide mesh

  • D digital waveguide mesh (WGM) for simulation of

membranes, drums etc. (Van Duyne & Smith, 1993)

  • 3-D digital waveguide mesh

3-D digital waveguide mesh for simulation of acoustic spaces (Savioja et al., 1994)

  • Violin body (Huang et al., 2000)
  • Drums (Aird et al., 2000)

Introduction Introduction

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  • In the original WGM, wave propagation speed depends
  • n direction and frequency (Van Duyne & Smith, 1993)
  • More advanced structures ease this problem, e.g.,

– –Triangular WGM Triangular WGM (Fontana & Rocchesso, 1995, 1998; Van Duyne & Smith, 1995, 1996) – –Interpolated rectangular WGM Interpolated rectangular WGM (Savioja & Välimäki, ICASSP’97, IEEE Trans. SAP 2000)

  • Direction-dependence is reduced but frequency-

dependence remains

⇒ Dispersion

Dispersion

Sophisticated 2-D Waveguide Structures Sophisticated 2-D Waveguide Structures

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Interpolated Rectangular Waveguide Mesh Interpolated Rectangular Waveguide Mesh

Original WGM Hypothetical 8-directional WGM Interpolated WGM (Savioja & Välimäki, 1997, 2000) (Van Duyne & Smith, 1993)

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Wave Propagation Speed Wave Propagation Speed

Original WGM Interpolated WGM (Bilinear interpolation)

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HELSINKI UNIVERSITY OF TECHNOLOGY

Wave Propagation Speed Wave Propagation Speed (2)

(2)

Original WGM Interpolated WGM (Bilinear interpolation)

  • 0.2

0.2

  • 0.2
  • 0.1

0.1 0.2

ξ1c ξ

2c

  • 0.2

0.2

  • 0.2
  • 0.1

0.1 0.2

ξ1c ξ

2

c

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HELSINKI UNIVERSITY OF TECHNOLOGY

Wave Propagation Speed Wave Propagation Speed (3)

(3)

Original WGM

  • 0.2

0.2

  • 0.2
  • 0.1

0.1 0.2

ξ1c ξ

2c

Interpolated WGM (Quadratic interpolation)

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Wave Propagation Speed Wave Propagation Speed (4)

(4)

Original WGM Interpolated WGM (Optimal interpolation)

  • 0.2

0.2

  • 0.2
  • 0.1

0.1 0.2

ξ1c ξ

2c

(Savioja & Välimäki, 2000)

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RFE in diagonal diagonal and axial axial directions: (a) original and (b) bilinearly interpolated rectangular WGM

0.05 0.1 0.15 0.2 0.25

  • 10
  • 5

5 (a) 0.05 0.1 0.15 0.2 0.25

  • 10
  • 5

5 (b) NORMALIZED FREQUENCY

RELATIVE FREQUENCY ERROR (%)

Relative Frequency Error (RFE) Relative Frequency Error (RFE)

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Relative Frequency Error (RFE) Relative Frequency Error (RFE) (2)

(2)

RFE in diagonal diagonal and axial axial directions: Optimally interpolated rectangular WG mesh (up to 0.25fs)

RELATIVE FREQUENCY ERROR (%)

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Frequency Warping Frequency 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 λ

A A( (z z) ) A A( (z z) ) A A( (z z) ) s s(0) (0) s s(1) (1) s s(2) (2) s s( (L L-1)

  • 1)

A z z z ( ) = + +

− − 1 1

1 λ λ

δ δ( (n n) ) s sw

w(

(n n) )

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Optimization of Warping Factor Optimization of Warping Factor λ

λ

  • 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) Bilinear interpolation (c,d) Quadratic interpolation (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 (λ1 + λ2) / (1 + λ1λ2)

  • There is a way...
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Multiwarping Multiwarping

  • 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
  • perations can be cascaded

– Many parameters to optimize: λ1, λ2, ... D1, D2,...

Frequency warping Sampling rate conv. Frequency warping Sampling rate conv.

) (

1 n

x ) (n yM

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Reduced Relative Frequency Error Reduced 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 complexity Computational 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 Range Extending 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)

RELATIVE FREQUENCY ERROR (%)

RFE in diagonal diagonal and axial axial 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

0.5 0.1 0.2 0.3 0.4 (a) NORMALIZED FREQUENCY MESH FREQUENCY 0.5 0.1 0.2 0.3 0.4 (b) NORMALIZED FREQUENCY MESH FREQUENCY 0.5 0.1 0.2 0.3 0.4 (c) NORMALIZED FREQUENCY MESH FREQUENCY 0.5 0.1 0.2 0.3 0.4 (d) λ = -0.32736 NORMALIZED FREQUENCY WARPED FREQUENCY

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 Simulation Result vs

  • vs. Analytical Solution

. Analytical Solution

Magnitude spectrum of a square membrane (a) original (b) warped interpolated (λ = –0.32736) (c) warped triangular (λ = –0.10954) digital waveguide mesh (with ideal response in the background)

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Error in Mode Frequencies Error in Mode Frequencies

Warped triangular Warped triangular WGM WGM Warped interpolated Warped interpolated WGM WGM Original WGM Original WGM Error in eigenfrequencies

  • f a square membrane

RELATIVE FREQUENCY ERROR (%)

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Conclusions and Future Work Conclusions and Future Work

  • Accuracy of 2-D digital waveguide mesh simulations

can be improved using 1) the interpolated interpolated or triangular WGM and 2) frequency warping frequency warping or multiwarping multiwarping

  • Dispersion can be reduced dramatically
  • In the future, the interpolation and warping

techniques will be applied to 3-D 3-D WGM simulations

  • Modeling of boundary conditions and losses must

be improved