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HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20001 Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms Vesa Välimäki 1 and Lauri Savioja.

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Presentation on theme: "HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20001 Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms Vesa Välimäki 1 and Lauri Savioja."— Presentation transcript:

1 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20001 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. (Espoo, Finland) DAFX’00, Verona, Italy, December 2000

2 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20002 Outline ä Introduction ä 2-D Digital Waveguide Mesh Algorithms ä Frequency Warping Techniques ä Extending the Frequency Range ä Numerical Examples ä Conclusions Interpolated and Warped 2-D Digital Waveguide Mesh Algorithms

3 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20003 Digital waveguidesDigital waveguides for physical modeling of musical instruments and other acoustic systems (Smith, 1992) 2-D digital waveguide mesh2-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) Introduction

4 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20004 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 –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

5 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20005 Interpolated Rectangular Waveguide Mesh Original WGM Hypothetical 8-directional WGM Interpolated WGM (Savioja & Välimäki, 1997, 2000) (Van Duyne & Smith, 1993)

6 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20006 Wave Propagation Speed Original WGM Interpolated WGM (Bilinear interpolation)

7 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20007 Wave Propagation Speed (2) Original WGM Interpolated WGM (Bilinear interpolation) -0.200.2 -0.2 -0.1 0 0.1 0.2  1 c  2 c -0.200.2 -0.2 -0.1 0 0.1 0.2  1 c  2 c

8 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20008 Wave Propagation Speed (3) Original WGM -0.200.2 -0.2 -0.1 0 0.1 0.2  1 c  2 c Interpolated WGM (Quadratic interpolation)

9 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 20009 Wave Propagation Speed (4) Original WGM Interpolated WGM (Optimal interpolation) -0.200.2 -0.2 -0.1 0 0.1 0.2  1 c  2 c (Savioja & Välimäki, 2000)

10 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200010 diagonal RFE in diagonal and axial axial directions: (a) original and (b) bilinearly interpolated rectangular WGM 00.050.10.150.20.25 -10 -5 0 5 (a) 00.050.10.150.20.25 -10 -5 0 5 (b) NORMALIZED FREQUENCY RELATIVE FREQUENCY ERROR (%) Relative Frequency Error (RFE)

11 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200011 Relative Frequency Error (RFE) (2) diagonal axial RFE in diagonal and axial directions: Optimally interpolated rectangular WG mesh (up to 0.25f s ) RELATIVE FREQUENCY ERROR (%)

12 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200012 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 warped-FIR filterPostprocess the response of the WGM using a warped-FIR filter (Oppenheim et al., 1971; Härmä et al., JAES, Nov. 2000)

13 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200013 Frequency Warping: Warped-FIR Filter Chain of first-order allpass filters s(n) is the signal to be warped s w (n) is the warped signal The extent of warping is determined by A(z)A(z)A(z)A(z) A(z)A(z)A(z)A(z) A(z)A(z)A(z)A(z) s(0) s(1) s(2) s(L-1) (n)(n)(n)(n) sw(n)sw(n)sw(n)sw(n)

14 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200014 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

15 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200015 (a,b) Bilinear interpolation (c,d) Quadratic interpolation (e,f) Optimal interpolation (g,h) Triangular mesh

16 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200016 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 + 1 2 ) There is a way...

17 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200017 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: 1, 2,... D 1, D 2,...

18 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200018 Reduced Relative Frequency Error (a) Warping with  = –0.32 (b) Multiwarping with 1 = –0.92, D 1 = 0.998 2 = –0.99, D 2 = 7.3 (c) Error in eigenmodes

19 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200019 Computational complexity Original WGM: 1 binary shift & 4 additions Interpolated WGM: 3 MUL & 9 ADD Warped-FIR filter: O(L 2 ) 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

20 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200020 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...

21 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200021 Relative Frequency Error (RFE) (2) RELATIVE FREQUENCY ERROR (%) diagonal axial RFE in diagonal and axial directions: Optimally interpolated rectangular WG mesh (up to 0.35f s )

22 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200022 Extending the Frequency Range (3) The mapping of frequencies for various WGMs 00.5 0 0.1 0.2 0.3 0.4 (a) NORMALIZED FREQUENCY MESH FREQUENCY 00.5 0 0.1 0.2 0.3 0.4 (b) NORMALIZED FREQUENCY MESH FREQUENCY 00.5 0 0.1 0.2 0.3 0.4 (c) NORMALIZED FREQUENCY MESH FREQUENCY 00.5 0 0.1 0.2 0.3 0.4 (d) = -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

23 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200023 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)

24 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200024 Error in Mode Frequencies Warped triangular Warped triangular WGM WGM Warped interpolated Warped interpolated WGM WGM Original WGM Original WGM Error in eigenfrequencies of a square membrane RELATIVE FREQUENCY ERROR (%)

25 HELSINKI UNIVERSITY OF TECHNOLOGY Välimäki and Savioja 200025 Conclusions and Future Work Accuracy of 2-D digital waveguide mesh simulations can be improved using interpolated 1) the interpolated or triangular WGM and frequency warping multiwarping 2) frequency warping or multiwarping Dispersion can be reduced dramatically 3-DIn the future, the interpolation and warping techniques will be applied to 3-D WGM simulations Modeling of boundary conditions and losses must be improved

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