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Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms

DAFX’00, Verona, Italy, December 2000. Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms. Vesa Välimäki 1 and Lauri Savioja 2 Helsinki University of Technology 1 Laboratory of Acoustics and Audio Signal Processing 2 Telecommunications Software and Multimedia Lab.

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Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms

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  1. DAFX’00, Verona, Italy, December 2000 Interpolated and Warped 2-D Digital 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) Välimäki and Savioja 2000

  2. Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms Outline • Introduction • 2-D Digital Waveguide Mesh Algorithms • Frequency Warping Techniques • Extending the Frequency Range • Numerical Examples • Conclusions Välimäki and Savioja 2000

  3. Introduction • Digital waveguides for physical modeling of musical instruments and other acoustic systems (Smith, 1992) • 2-D digital waveguide mesh (WGM) for simulation of membranes, drums etc. (Van Duyne & Smith, 1993) • 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) Välimäki and Savioja 2000

  4. Sophisticated 2-D Waveguide Structures • 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 WGM (Fontana & Rocchesso, 1995, 1998; Van Duyne & Smith, 1995, 1996) • Interpolated rectangular WGM (Savioja & Välimäki, ICASSP’97, IEEE Trans. SAP 2000) • Direction-dependence is reduced but frequency-dependence remains ÞDispersion Välimäki and Savioja 2000

  5. Interpolated Rectangular Waveguide Mesh Hypothetical 8-directional WGM Original WGM Interpolated WGM (Van Duyne & Smith, 1993) (Savioja & Välimäki, 1997, 2000) Välimäki and Savioja 2000

  6. Wave Propagation Speed Interpolated WGM (Bilinear interpolation) Original WGM Välimäki and Savioja 2000

  7. Wave Propagation Speed (2) Interpolated WGM (Bilinear interpolation) Original WGM 0.2 0.2 0.1 0.1 c c 0 2 0 2 x x -0.1 -0.1 -0.2 -0.2 -0.2 0 0.2 -0.2 0 0.2 c x c x 1 1 Välimäki and Savioja 2000

  8. c 2 x c x 1 Wave Propagation Speed (3) Interpolated WGM (Quadratic interpolation) Original WGM 0.2 0.1 0 -0.1 -0.2 -0.2 0 0.2 Välimäki and Savioja 2000

  9. c 2 x c x 1 Wave Propagation Speed (4) Interpolated WGM (Optimal interpolation) Original WGM 0.2 0.1 0 -0.1 -0.2 -0.2 0 0.2 (Savioja & Välimäki, 2000) Välimäki and Savioja 2000

  10. Relative Frequency Error (RFE) 5 (a) RFE in diagonaland axial directions: (a) original and (b) bilinearly interpolated rectangular WGM 0 -5 -10 0 0.05 0.1 0.15 0.2 0.25 RELATIVE FREQUENCY ERROR (%) 5 (b) 0 -5 -10 0 0.05 0.1 0.15 0.2 0.25 NORMALIZED FREQUENCY Välimäki and Savioja 2000

  11. Relative Frequency Error (RFE) (2) RFE in diagonaland axial directions: Optimally interpolated rectangular WG mesh (up to 0.25fs) RELATIVE FREQUENCY ERROR (%) Välimäki and Savioja 2000

  12. 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 (Oppenheim et al., 1971; Härmä et al., JAES, Nov. 2000) Välimäki and Savioja 2000

  13. A(z) A(z) A(z) Frequency Warping: Warped-FIR Filter • Chain of first-order allpass filters d(n) s(0) s(1) s(2) s(L-1) sw(n) • s(n) is the signal to be warped • sw(n) is the warped signal • The extent of warping is determined by l Välimäki and Savioja 2000

  14. Optimization of Warping Factor l • 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 Välimäki and Savioja 2000

  15. (a,b) Bilinear interpolation (c,d) Quadratic interpolation (e,f) Optimal interpolation (g,h) Triangular mesh Välimäki and Savioja 2000

  16. 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 (l1 + l2) / (1 + l1l2) • There is a way... Välimäki and Savioja 2000

  17. 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 operations can be cascaded • Many parameters to optimize: l1, l2, ... D1, D2,... Välimäki and Savioja 2000

  18. Reduced Relative Frequency Error (a) Warping with l = –0.32 (b) Multiwarping with l1 = –0.92, D1 = 0.998 l2 = –0.99, D2 = 7.3 (c) Error in eigenmodes Välimäki and Savioja 2000

  19. 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 Välimäki and Savioja 2000

  20. 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... Välimäki and Savioja 2000

  21. Relative Frequency Error (RFE) (2) RFE in diagonaland axial directions: Optimally interpolated rectangular WG mesh (up to 0.35fs) RELATIVE FREQUENCY ERROR (%) Välimäki and Savioja 2000

  22. Extending the Frequency Range (3) • The mapping of frequencies for various WGMs 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 (a) (b) 0.4 0.4 0.3 0.3 MESH FREQUENCY MESH FREQUENCY 0.2 0.2 0.1 0.1 0 0 0 0.5 0 0.5 NORMALIZED FREQUENCY NORMALIZED FREQUENCY (c) (d) 0.4 0.4 0.3 0.3 MESH FREQUENCY WARPED FREQUENCY 0.2 0.2 0.1 0.1 l = -0.32736 0 0 0 0.5 0 0.5 NORMALIZED FREQUENCY NORMALIZED FREQUENCY Välimäki and Savioja 2000

  23. Simulation Result vs. Analytical Solution Magnitude spectrum of a square membrane (a) original (b) warped interpolated (l = –0.32736) (c) warped triangular (l = –0.10954) digital waveguide mesh (with ideal response in the background) Välimäki and Savioja 2000

  24. Error in Mode Frequencies Error in eigenfrequencies of a square membrane Warped interpolated WGM Warped triangular WGM RELATIVE FREQUENCY ERROR (%) Original WGM Välimäki and Savioja 2000

  25. Conclusions and Future Work • Accuracy of 2-D digital waveguide mesh simulations can be improved using 1) the interpolated or triangular WGM and 2) frequency warping or multiwarping • Dispersion can be reduced dramatically • In the future, the interpolation and warping techniques will be applied to 3-D WGM simulations • Modeling of boundary conditions and losses must be improved Välimäki and Savioja 2000

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