DAFX’00, 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 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) 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 Outline ➤ Introduction ➤ 2-D Digital Waveguide Mesh Algorithms ➤ Frequency Warping Techniques ➤ Extending the Frequency Range ➤ Numerical Examples ➤ Conclusions Välimäki and Savioja 2000 2
HELSINKI UNIVERSITY OF TECHNOLOGY Introduction Introduction • 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) Välimäki and Savioja 2000 3
HELSINKI UNIVERSITY OF TECHNOLOGY Sophisticated 2-D Waveguide Structures 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 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 Välimäki and Savioja 2000 4
HELSINKI UNIVERSITY OF TECHNOLOGY Interpolated Rectangular Waveguide Mesh Interpolated Rectangular Waveguide Mesh Hypothetical Original WGM 8-directional Interpolated WGM WGM (Van Duyne & Smith, (Savioja & Välimäki, 1993) 1997, 2000) Välimäki and Savioja 2000 5
HELSINKI UNIVERSITY OF TECHNOLOGY Wave Propagation Speed Wave Propagation Speed Interpolated WGM Original WGM (Bilinear interpolation) Välimäki and Savioja 2000 6
HELSINKI UNIVERSITY OF TECHNOLOGY Wave Propagation Speed (2) Wave Propagation Speed (2) Interpolated WGM Original WGM (Bilinear interpolation) 0.2 0.2 0.1 0.1 2 c c 0 0 2 ξ ξ -0.1 -0.1 -0.2 -0.2 -0.2 0 0.2 -0.2 0 0.2 ξ 1 c ξ 1 c Välimäki and Savioja 2000 7
HELSINKI UNIVERSITY OF TECHNOLOGY Wave Propagation Speed (3) Wave Propagation Speed (3) Interpolated WGM Original WGM (Quadratic interpolation) 0.2 0.1 2 c 0 ξ -0.1 -0.2 -0.2 0 0.2 ξ 1 c Välimäki and Savioja 2000 8
HELSINKI UNIVERSITY OF TECHNOLOGY Wave Propagation Speed (4) Wave Propagation Speed (4) Interpolated WGM Original WGM (Optimal interpolation) 0.2 0.1 2 c 0 ξ -0.1 -0.2 -0.2 0 0.2 ξ 1 c (Savioja & Välimäki, 2000) Välimäki and Savioja 2000 9
HELSINKI UNIVERSITY OF TECHNOLOGY Relative Frequency Error (RFE) Relative Frequency Error (RFE) 5 RELATIVE FREQUENCY ERROR (%) (a) RFE in diagonal diagonal and 0 axial axial directions: -5 (a) original and -10 0 0.05 0.1 0.15 0.2 0.25 (b) bilinearly 5 (b) 0 interpolated -5 rectangular WGM -10 0 0.05 0.1 0.15 0.2 0.25 NORMALIZED FREQUENCY Välimäki and Savioja 2000 10
HELSINKI UNIVERSITY OF TECHNOLOGY Relative Frequency Error (RFE) (2) Relative Frequency Error (RFE) (2) RELATIVE FREQUENCY ERROR (%) RFE in diagonal diagonal and axial directions: axial Optimally interpolated rectangular WG mesh (up to 0.25 f s ) Välimäki and Savioja 2000 11
HELSINKI UNIVERSITY OF TECHNOLOGY 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 (Oppenheim et al. , 1971; Härmä et warped-FIR filter al. , JAES, Nov. 2000) Välimäki and Savioja 2000 12
HELSINKI UNIVERSITY OF TECHNOLOGY Frequency Warping: Warped-FIR Filter Frequency Warping: Warped-FIR Filter − + λ 1 z • Chain of first-order allpass filters ( ) = A z − + λ 1 1 z δ ( δ ( n n ) ) A ( ( z z ) ) A ( ( z z ) ) A ( ( z z ) ) A A A s (0) (0) s (1) (1) s (2) (2) s ( ( L L -1) -1) s s s s s w ( n n ) ) s w ( • s ( n ) is the signal to be warped • s w ( n ) is the warped signal • The extent of warping is determined by λ Välimäki and Savioja 2000 13
HELSINKI UNIVERSITY OF TECHNOLOGY λ Optimization of Warping Factor λ 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 Välimäki and Savioja 2000 14
HELSINKI UNIVERSITY OF TECHNOLOGY (a,b) Bilinear interpolation (c,d) Quadratic interpolation (e,f) Optimal interpolation (g,h) Triangular mesh Välimäki and Savioja 2000 15
HELSINKI UNIVERSITY OF TECHNOLOGY 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 ( λ 1 + λ 2 ) / (1 + λ 1 λ 2 ) • There is a way... Välimäki and Savioja 2000 16
HELSINKI UNIVERSITY OF TECHNOLOGY 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 operations can be cascaded – Many parameters to optimize: λ 1 , λ 2 , ... D 1 , D 2 ,... ( ) ( n ) x 1 n y M Frequency Sampling Frequency Sampling warping rate conv. warping rate conv. Välimäki and Savioja 2000 17
HELSINKI UNIVERSITY OF TECHNOLOGY Reduced Relative Frequency Error Reduced Relative Frequency Error (a) Warping with λ = –0.32 (b) Multiwarping with λ 1 = –0.92, D 1 = 0.998 λ 2 = –0.99, D 2 = 7.3 (c) Error in eigenmodes Välimäki and Savioja 2000 18
HELSINKI UNIVERSITY OF TECHNOLOGY Computational complexity 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 Välimäki and Savioja 2000 19
HELSINKI UNIVERSITY OF TECHNOLOGY 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... Välimäki and Savioja 2000 20
HELSINKI UNIVERSITY OF TECHNOLOGY Relative Frequency Error (RFE) (2) Relative Frequency Error (RFE) (2) RELATIVE FREQUENCY ERROR (%) RFE in diagonal diagonal and axial directions: axial Optimally interpolated rectangular WG mesh (up to 0.35 f s ) Välimäki and Savioja 2000 21
HELSINKI UNIVERSITY OF TECHNOLOGY Extending the Frequency Range (3) Extending the Frequency Range (3) • The mapping of frequencies for various WGMs MESH FREQUENCY MESH FREQUENCY Upper frequency (a) (b) 0.4 0.4 limit always 0.3536 0.3 0.3 0.2 0.2 (a) Original 0.1 0.1 0 0 0 0.5 0 0.5 (b) Optimally interp. NORMALIZED FREQUENCY NORMALIZED FREQUENCY WARPED FREQUENCY MESH FREQUENCY up to 0.25 (c) (d) 0.4 0.4 0.3 0.3 (c) Optimally interp. 0.2 0.2 up to 0.35 0.1 0.1 λ = -0.32736 0 0 0 0.5 0 0.5 (d) Warped case b NORMALIZED FREQUENCY NORMALIZED FREQUENCY Välimäki and Savioja 2000 22
HELSINKI UNIVERSITY OF TECHNOLOGY Simulation Result vs vs. Analytical Solution . Analytical Solution Simulation Result 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) Välimäki and Savioja 2000 23
HELSINKI UNIVERSITY OF TECHNOLOGY Error in Mode Frequencies Error in Mode Frequencies Error in eigenfrequencies RELATIVE FREQUENCY ERROR (%) of a square membrane Warped interpolated Warped interpolated WGM WGM Warped triangular Warped triangular WGM WGM Original WGM Original WGM Välimäki and Savioja 2000 24
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