INTERPOLATED AND WARPED 2-D DIGITAL WAVEGUIDE MESH ALGORITHMS

Transcription

1 Proeedings of the COST G-6 Conferene on Digital Audio Effets (DAFX-), Verona, Italy, Deember 7-9, INTERPOLATED AND WARPED -D DIGITAL WAVEGUIDE MESH ALGORITHMS Vesa Välimäki Lab. of Aoustis and Audio Signal Proessing Helsinki University of Tehnology Lauri Savioja Teleommun. Software and Multimedia Lab. Helsinki University of Tehnology ABSTRACT Interpolated and warped digital waveguide mesh algorithms have been developed to overome the problem aused by diretion and frequeny-dependene of wave travel speed in digital waveguide mesh simulations. This paper reviews the interpolation methods appliable in the two-dimensional ase. The bilinear interpolation tehnique and two other approahes are briefly reapitulated. The use of -D quadrati interpolation results in a mesh struture that is effiiently implemented using only additions and binary shifts. The optimized interpolated -D mesh is obtained by hoosing the free parameters of the interpolation method so that the differene in wave travel speed in the axial and diagonal diretions is minimized. The output signal of any digital waveguide mesh an be proessed with a warped FIR filter to redue the dispersion error at all frequenies. The frequeny-warping fator must be optimized individually for eah mesh struture. We introdue an extended method alled multiwarping, whih an be used to further derease the dispersion error. The frequeny range of digital waveguide mesh simulations is also disussed.. INTRODUCTION The digital waveguide mesh was introdued in 993 by Van Duyne and Smith [, ]. They showed that the method is suitable for sound synthesis of perussion instruments although it suffers from diretion-dependent dispersion. In 994, Savioja et al. extended the use of the digital waveguide mesh to three dimensions, and presented simulation results of wave propagation in aousti spaes [3]. The method has turned out to be useful in aousti design of listening rooms, for example. Reently, the digital waveguide mesh has been used in the modeling of drums [4] and the violin body [5]. The interpolated mesh was developed to overome the diretion-dependent wave propagation harateristis and dispersion error [6]. However, it was only possible to redue the diretiondependene while the dispersion was not muh affeted it was merely rendered almost independent of diretion. Lukily, the remaining dispersion error an be made onsiderably smaller using frequeny warping [6 9]. It is implemented by postproessing the output signal of the mesh using a warped FIR filter. Fontana and Rohesso have also shown that the frequenywarping an be inorporated in the mesh struture []. The best results so far have been obtained by using the triangular waveguide mesh [ 3] together with a warped FIR filter [4, 6]. Also, the interpolated retangular mesh an be improved onsiderably using frequeny warping [6]. In Setion of this paper, we review the interpolation methods for the retangular -D digital waveguide mesh. In Setion 3 we disuss frequeny-warping methods and introdue a novel approah, multiwarping. Setion 4 fouses on the frequeny range of digital waveguide simulations. It is suggested that the available frequeny range an be extended with the interpolation and warping methods. In Setion 5, it is shown by numerial examples how muh the different mesh algorithms an gain from the interpolation and frequeny-warping tehniques, when the warping fator is optimized individually for eah of them. Setion 6 onludes the paper and presents diretions for future work.. DIGITAL WAVEGUIDE MESH ALGORITHMS.. Original Digital Waveguide Mesh The original digital waveguide mesh [, ] onsists of a retangular grid where the signal value at every node is updated at eah sampling interval aording to the following formula: 4 p( n) = pk ( n ) p( n ) k = where n is the disrete time index, p (n) is the node to be updated, p k (n ) are its four nearest neighbors, and p (n ) is the value of the enter node two sample intervals ago. Thus, it is neessary to store two omplete meshes. The first part of the update step () an also be formulated using the following matrix (see also [6], Eq. (5)) h orig = This interpretation is equivalent to the notion of point-spreading funtion used in image proessing. Figures and show two different views of the wave propagation speed in the original mesh as a funtion of two frequeny variables. The wave travel speed is onstant in diagonal diretions, while in other diretions the speed dereases with frequeny, that is, the dispersion error inreases. This is also seen in Fig. that shows the minimal (diagonal) and maximal (axial) relative frequeny error (RFE) aused by dispersion. For details on omputing the RFE urves, see [6]. () ()

2 Proeedings of the COST G-6 Conferene on Digital Audio Effets (DAFX-), Verona, Italy, Deember 7-9, ξ ξ ξ ξ ξ ξ () Figure. Equal wave travel speed ontours for the original, bilinearly interpolated, and () optimally interpolated digital waveguide mesh where ontours desend from the enter point in inrements of %. The distane from the enter gives the normalized temporal frequeny f = ξ, and (ξ, ξ,) determines the propagation diretion. The irle (thik line) indiates the highest normalized temporal frequeny of the retangular waveguide mesh, whih is.5. RFE (%) NORMALIZED SPEED ξ.. ξ Figure. Normalized wave travel speed in the original digital waveguide mesh as a funtion of two normalized frequeny variables ξ and ξ, and the minimal (dashed line) and maximal (solid line) RFE... Bilinearly Interpolated Waveguide Mesh The aim of the interpolated waveguide mesh was to suppress the diretion-dependene and dispersion of the original waveguide mesh [6]. The motivation was that sample updates should our in more diretions than just four as in the original mesh. The interpolation tehnique effetively inserts new nodes in the mesh the ontribution of the hypothetial nodes is then spread on the existing neighboring nodes to obtain a realizable struture. This is a -D appliation of frational delay filters [5]. The method of superimposing signal samples onto a non-integer point of a digital delay line was introdued in [6], where this operation was named as deinterpolation. See also [7, pp. 8 35], where deinterpolation is disussed in detail. In the initial version of the interpolated mesh algorithm [6], four hypothetial nodes were inserted in diagonal diretions, and we hose to use bilinear interpolation, whih is a two-dimensional extension of linear interpolation. In this ase, the point-spreading funtion is h = bilin = (3) In the bilinearly interpolated mesh, the wave travel speed is not onstant in any diretion, but the behavior is more homogeneous as a funtion of diretion than in the original mesh, as seen by omparing Figs. and. The remaining dispersion auses the mode frequenies of the aousti system that is being modeled to be lower than they should, as illustrated by the RFE urve in Fig. 3. Fortunately, the error is pratially independent of diretion, and hene a single orretion funtion an be used to redue the frequeny error..3. Quadrati -D Interpolation It is also possible to use higher-order interpolation to implement the interpolated mesh. Here we present the use of seond-order Lagrange interpolation, or quadrati interpolation, whih leads to the following point-spreading funtion: h quad = 3 3 = (4) Note that onvolution with this kernel an be realized without multipliations, sine all the elements in (4) are easily implemented as binary shifts and additions.

3 Proeedings of the COST G-6 Conferene on Digital Audio Effets (DAFX-), Verona, Italy, Deember 7-9, RELATIVE FREQUENCY ERROR (%) () (e) (g) λ = (d) λ = (f) λ = (h) λ = Figure 3. Maximal and minimal (dashed and solid lines) relative frequeny error in the bilinearly, () quadratially, and (e) optimally interpolated, and (g) the triangular mesh. The orresponding RFEs after frequeny warping are displayed on the right. The dotted line indiates the zero error. Figure 3() shows the RFE of the interpolated mesh when quadrati interpolation is used. Perhaps surprisingly, the quadrati interpolation does not yield as good results as the bilinear interpolation. The reason is that the quadrati interpolation has to work outside its optimal range, whih is.5 d.5 in eah diretion, where d is the frational delay from the node to be updated (see [7, pp. 9 94] for a disussion on approximation errors of Lagrange interpolation filters). The delay values required in this interpolation problem are d =.77 and d =.77. Note that it is impossible to use higher-order polynomial interpolation with the 3-by-3 point-spreading funtion. For example, the use of third-order Lagrange interpolation would lead to a 5-by-5 matrix..4. Optimized -D Interpolation There are only three free parameters in the 3-by-3 point-spreading funtions: the diagonal, axial, and entral value. It is thus relatively easy to optimize the values of these oeffiients so that the best possible harateristis are ahieved. We have optimized the oeffiients of the 3-by-3 matrix so that the differene between the minimal and maximal frequeny error below the normalized frequeny.5 is minimized while the error at the zero frequeny is fored to zero [6]. The iteration yields the following pointspreading funtion: h opt = The resulting wave travel speed is shown in Fig. (), and the orresponding RFE urves in axial and diagonal diretions are in Fig. 3(e). It an be seen that the differene between the two extreme ases of RFE urves is smaller than for the other interpolated retangular meshes. (5) (n) A (z) A (z) A(z) x m () x m ( ) x m ( ) x m ( L ).5. Triangular Digital Waveguide Mesh Also other sampling latties than the retangular one have been onsidered for digital waveguide mesh models. The triangular mesh has been found to be a good hoie [ 3]. Figure 3(g) indiates that the RFE in the triangular mesh is smaller than in any of the retangular mesh strutures disussed above. In addition, the differene between the smallest and largest error is small. The main drawbak of the triangular mesh is the triangular tessellation of the plane, whih may be more ompliated than the retangular one in some ases. 3. FREQUENCY-WARPING TECHNIQUES The dispersion error appearing in digital waveguide mesh simulations an be redued using frequeny warping, when the error is almost idential in all diretions and the error funtion is monotoni and relatively smooth [4, 6]. Then it is possible to design a frequeny-warping filter, whih shifts the frequenies and redues the error when the output signal of the digital waveguide mesh x(n) is proessed with it. 3.. How to Use Frequeny Warping Frequeny warping an be implemented with a warped FIR filter, whih onsists of a hain of first-order allpass filters A(z), as shown in Fig. 4 [7, 6, 9]. The tap oeffiients are set equal to the signal samples x m (n) to be warped, i.e., the output signal of the digital waveguide mesh in this ase. When a unit impulse is fed into the filter, the warped signal y m (n) is obtained at the output. The extent of warping is ontrolled by the allpass filter oeffiient λ that we all the warping fator. In addition, a sampling rate onversion is required, beause all frequenies not only the high frequenies are shifted as a onsequene of frequeny warping (see [6] for details). 3.. Redution of Dispersion Error y m (n) We have optimized λ for eah ase so that the maximal RFE in the frequeny band from up to 5% of the sampling rate is minimized. Figure 3 presents the RFE of the bilinearly interpolated mesh after its output signal has been frequeny warped using warping fator λ = Note that the RFE has been x ( n ) Frequeny Sampling Frequeny Sampling y M (n) warping rate onv. warping rate onv. Figure 4. Frequeny warping an be implemented using a hain of idential first-order allpass filters. Multiwarping is realized by alternating first-order frequeny-warping operations and sample rate onversions. 3

4 Proeedings of the COST G-6 Conferene on Digital Audio Effets (DAFX-), Verona, Italy, Deember 7-9, onsiderably redued at high frequenies, enabling aurate wideband simulations. Figure 3(d) orresponds to the warped quadratially interpolated mesh, whih is not so suessful beause the differene between the minimal and maximal error urves gets large at middle and high frequenies. The RFE of the warped optimally interpolated mesh shown in Fig. 3(f) is superior over other interpolated retangular mesh algorithms. Its maximum RFE is.%. However, the warped triangular mesh [4, 6] produes the smallest RFE (.6%) of all possibilities, see Fig. 3(h). MESH FREQUENCY MESH FREQUENCY Multiwarping For some time, our aim has been to find a new frequeny-warping method to augment the degrees of freedom. It would be desirable to have a tehnique in whih the auray ould be enhaned at will by seleting a higher order warping operation. Formerly, suh a tehnique has not been available. In theory, it would be possible to use higher order allpass transfer funtions A(z) instead of the first-order one. However, replaing unit delays with seond or higher order allpass filters leads to both folding and warping in the delay-line implementation [8], whih is undesirable. Another attrative solution would be to apply the first-order warping many times suessively, but this is a waste of effort sine there will not be more degrees of freedom in this proedure than in a single frequeny warping. This an be shown by noting that the frequeny warping obtained with two suessive warping operations with λ and λ is equivalent to warping with (λ + λ ) / ( + λ λ ). We propose a new approah that we all multiwarping [9], where an arbitrary number of suessive first-order frequeny warping and resampling operations are performed alternately as illustrated in Fig. 4. The use of sampling rate onversions failitates the design of nonlinear frequeny shifts that are impossible to ahieve with a single warping, sine at a new sampling rate, the frequeny shifts our at different frequenies. The sampling rate onversions are done by fator D, suh that if < D < the proess is upsampling (interpolation) and if D > it is downsampling (deimation). There are various strategies to find the optimal values for the warping and resampling oeffiients. It is a multivariable optimization task, whih should be restrited so that the downsampling operations do not ut the effetive frequeny band too muh. Another riterion to take into aount is that if λ is lose to. the output signal of frequeny warping beomes long, sine the pole of the allpass filter is loated lose to the unit irle. 4. EXTENDING THE FREQUENCY RANGE In this setion, we disuss the question of the upper frequeny limit of the interpolated and warped waveguide mesh simulations. It is known that the limiting frequeny in the ase of the original waveguide mesh is.5, and above that frequeny only mirror images of lower frequenies our []. However, when interpolation and frequeny-warping methods are used, it is no longer obvious what the highest frequeny is. In the following, we demonstrate how it is possible to extend the frequeny range of digital waveguide mesh simulations. Fontana and Rohesso have disussed earlier that the triangular mesh is superior over retangular or hexagonal tilings in terms of valid frequeny band []. Figure 5 shows the mapping of the original normalized frequenies to the frequenies ourring on the mesh in the ase of the original (Fig. 5) and optimally interpolated meshes, when MESH FREQUENCY ()..4 the frequeny limit has been set to.5 (in Fig. 5) and.35 (in Fig 5()). The maximum value of all axial mapping urves is.5, and it ours at normalized frequeny This appears to be the highest frequeny that an be simulated using the retangular digital waveguide meshes. Frequeny warping ould at its best shift the mesh frequenies so that.3536 would again our at the right frequeny. Assuming ideal frequeny warping, the remaining error is aused by the differenes between different diretions. As an example, we display in Fig. 5(d) the frequeny mapping of the warped interpolated mesh optimized up to the normalized frequeny.5. The warping fator used is λ = It an be seen that both the axial and diagonal frequeny mappings follow the ideal mapping urve (dotted line in Fig. 5(d)) well until about.5. Above this frequeny, also the differene between the diagonal and axial properties of the mesh begins to inrease substantially, and it is impossible to extend the bandwidth muh higher. However, it would be possible to warp the interpolated mesh that has been optimized up to.35 (see Fig. 5()), and then it would be feasible to obtain a good auray up to NUMERICAL EXAMPLES. λ = Figure 5. Mapping of frequenies in the original, optimally interpolated (up to.5), and () another optimally interpolated (up to.35) waveguide mesh (d) together with warped mapping for ase b. The axial diretion is shown with the solid and the diagonal diretion with the dashed line. The dotted line indiates the desired ideal mapping funtion (y = x). In the following we present numerial examples to illustrate the properties of the methods disussed above. We have simulated a square membrane with rigid boundaries (refletion oeffiient ) using different digital waveguide mesh algorithms, and ompared the results. The mesh size was 6 6 nodes. For details of the simulation, see [6]. Figure 6 shows the magnitude spetra of the simulated membrane in three ases: the original, the warped optimized interpolated, and () the warped triangular digital waveguide mesh. The ideal magnitude spetrum is given for omparison in eah ase. WARPED FREQUENCY (d) 4

5 Proeedings of the COST G-6 Conferene on Digital Audio Effets (DAFX-), Verona, Italy, Deember 7-9, MAGNITUDE (db) RELATIVE FREQUENCY ERROR (%) () Figure 6. Magnitude spetrum of an ideal retangular membrane (dashed lines) together with the simulation results (solid lines) obtained with the original, the warped optimally interpolated, and () the warped triangular mesh Figure 7. Error in eigenmode frequenies of the simulated retangular membrane (squares = original mesh, hexagrams = warped interpolated mesh, triangles = warped triangular mesh). The errors in the frequenies of eigenmodes of the membrane, as a differene from the ideal eigenfrequenies alulated by the Rayleigh equation, are given in Fig. 7. The error behavior of the original mesh reveals that some eigenmodes our at nearly the orret frequeny while others are too low by several perent. This is explained by the non-homogeneous nature of the original mesh (see Figs. and ). The errors in the warped interpolated mesh are smaller they are within ±.% in the frequeny band from to 5% of the sampling rate (see Fig. 7). However, the warped triangular mesh is still better: the maximum error on the same frequeny band is only.6%. These results are in good agreement with the theoretial RFE urves presented in Figs. 3(f) and 3(h), sine the mode frequeny errors are found between the theoretial minimum and RELATIVE FREQUENCY ERROR (%) () Figure 8. RFE in the optimally interpolated warped and multiwarped waveguide mesh with () a omparison of the orresponding eigenmode frequeny errors obtained from a numerial simulation of a membrane (squares = warped, irles = multiwarped). maximum errors in all ases. It must be understood that if the RFE is positive, the mode frequenies given by a simulation are shifted up, and if the RFE is negative, the frequenies are shifted downwards. To redue the error further, multiwarping onsisting of a hain of two frequeny-warping and resampling operations an be performed, as illustrated in Fig. 4. Finding the optimal warping and resampling fators is not trivial. In the ase of performing both warping and sampling rate onversion twie, there are four variables. Of these only three are free, sine the last resampling must be performed suh that there will be no shifts at the lowest frequenies. In this ase, we applied the Fmins funtion in Matlab to searh for the optimal values for the two warping fators and one resampling oeffiient. The resulting values λ =.995, D = , λ = , and D = 7.3 give the maximal RFE of.6% for the optimally interpolated retangular mesh, whih is about one half of the error obtained with a single warping (.%). The orresponding RFE urves are presented in Fig. 8 and 8. Figure 8() represents the RFE obtained by simulating the square membrane with the warped and multiwarped interpolated digital waveguide meshes. The line with squares in Fig. 8() stands for the result with one frequeny warping and is in good agreement with the urves shown in Fig. 8. The urve with irles in Fig. 8() orresponds to the multiwarped ase shown in Fig. 8. This example demonstrates that the error in mode frequenies an be halved using multiwarping. Similar results have been obtained with the triangular digital waveguide mesh, where the error an be redued from.6% to.35% by using a two-stage multiwarping instead of one warping. Adding more frequeny-warping and resampling operations to the multiwarping an further redue the error. The differene between the maximal and minimal error, i.e., the diretion-dependene, will restrit the improvement that an be ahieved using multiwarping. As the number of asaded warping and resampling operations is inreased, the omputational load inreases almost linearly. Sine multiwarping must be exeuted as an off-line 5

6 Proeedings of the COST G-6 Conferene on Digital Audio Effets (DAFX-), Verona, Italy, Deember 7-9, method, suh as postproessing after a simulation run in our appliation, the inrease in omputation time is not very ritial. 6. CONCLUSIONS AND FUTURE WORK This paper reviewed various interpolated and warped digital waveguide mesh algorithms and disussed the frequeny range of digital waveguide mesh simulations. The retangular mesh using quadrati interpolation an be implemented without multipliations. The optimized interpolated mesh is better than other interpolated retangular mesh strutures having a 3-by-3 matrix as the point-spreading funtion. The warped triangular mesh produes simulation results with the smallest dispersion error. The multiwarping tehnique was also disussed. The method onsists of suessive frequeny-warping and resampling operations. The maximal RFE of the interpolated or triangular digital waveguide mesh is redued by about 5% by using two asaded frequeny-warping proedures and sampling rate onversions instead of a single warping and resampling. 6.. Diretions for Future Researh Digital waveguide mesh methods are still not well understood. There are many issues related to wave propagation that annot be implemented aurately yet. These inlude modeling of internal losses of the medium and refletions at the boundaries. In the ase of wall refletions, it would be neessary to be able to model the aoustial properties of materials as a funtion of frequeny but also as a funtion of inident angle. A straightforward implementation of a refletion that onsists of inserting a filter behind the wall, next to the last layer of nodes, yields a diretion-dependent behavior, whih only works as planned for a plane wave arriving along the diagonal axis of a retangular mesh. While this paper onentrated on the two-dimensional digital waveguide mesh methods, the interpolation and frequeny-warping methods are appliable to three-dimensional simulations as well. Promising results have been published in [], where the interpolation tehnique was adapted to the 3-D retangular digital waveguide mesh. 7. ACKNOWLEDGMENTS The work of V. Välimäki has been supported by a postdotoral researh grant from the Aademy of Finland. 8. REFERENCES [] S. Van Duyne and J. O. Smith, Physial modeling with the -D digital waveguide mesh, in Pro. Int. Computer Musi Conf., Tokyo, Japan, pp. 4 47, Sept. 5, 993. [] S. Van Duyne and J. O. Smith, The -D digital waveguide mesh, in Pro. IEEE WASPAA, New Paltz, NY, Ot [3] L. Savioja, T. Rinne, and T. Takala, Simulation of room aoustis with a 3-D finite differene mesh, in Pro. Int. Computer Musi Conf., Aarhus, Denmark, pp , Sept [4] M. Aird, J. Laird, and J. Fith, Modelling a drum by interfaing -D and 3-D waveguide meshes, in Pro. Int. Computer Musi Conf., Berlin, Germany, pp. 8 85, Aug.. [5] P. Huang, S. Serafin, and J. O. Smith, A waveguide mesh model of high-frequeny violin body resonanes, in Pro. Int. Computer Musi Conf., Berlin, Germany, pp , Aug.. [6] L. Savioja and V. Välimäki, Reduing the dispersion error in the digital waveguide mesh using interpolation and frequeny-warping tehniques, IEEE Trans. Speeh and Audio Proessing, vol. 8, no., pp , Mar.. [7] A. Oppenheim, D. Johnson, and K. Steiglitz, Computation of spetra with unequal resolution using the fast Fourier transform, Pro. IEEE, vol. 59, pp. 99 3, Feb. 97. [8] M. Karjalainen, A. Härmä, U. K. Laine, and J. Huopaniemi, Warped filters and their audio appliations, in Pro. IEEE WASPAA, New Paltz, NY, Ot [9] A. Härmä, M. Karjalainen, L. Savioja, V. Välimäki, U. K. Laine, and J. Huopaniemi, Frequeny-warped signal proessing for audio appliations, J. Audio Eng. So., vol. 48,. In press. [] F. Fontana and D. Rohesso, Online orretion of dispersion error in D waveguide meshes, in Pro. Int. Computer Musi Conf., Berlin, Germany, pp. 78 8, Aug.. [] F. Fontana and D. Rohesso, A new formulation of the D waveguide mesh for perussion instruments, in Pro. Colloquio di Informatia Musiale, Bologna, Italy, pp. 7 3, Nov [] S. Van Duyne and J. O. Smith, The 3D tetrahedral digital waveguide mesh with musial appliations, in Pro. Int. Computer Musi Conf., Hong Kong, pp. 9 6, Aug [3] F. Fontana and D. Rohesso, Physial modeling of membranes for perussion instruments, Austia united with Ata Austia, vol. 84, no. 3, pp , May/June 998. [4] L. Savioja and V. Välimäki, Redution of the dispersion error in the triangular digital waveguide mesh using frequeny warping, IEEE Signal Proessing Letters, vol. 6, no. 3, pp. 58 6, Mar [5] T. I. Laakso, V. Välimäki, M. Karjalainen, and U. K. Laine, Splitting the unit delay Tools for frational delay filter design, IEEE Signal Proessing Mag., vol. 3, no., pp. 3 6, Jan [6] V. Välimäki, M. Karjalainen, and T. I. Laakso, Frational delay digital filters, in Pro. IEEE Int. Symp. on Ciruits and Systems, Chiago, IL, vol., pp , May 993. [7] V. Välimäki, Disrete-Time Modeling of Aousti Tubes Using Frational Delay Filters. Dotoral thesis. Helsinki Univ. of Teh., Lab. of Aoustis and Audio Signal Proess., Espoo, Finland, De Available online at the following URL: [8] J. A. Moorer, The manifold joys of onformal mapping: appliations to digital filtering in the studio, J. Audio Eng. So., vol. 3, no., pp , Nov [9] L. Savioja and V. Välimäki, Multiwarping for enhaning the frequeny auray of digital waveguide mesh simulations, unpublished manusript, submitted for publiation,. [] F. Fontana and D. Rohesso, Signal-theoreti haraterization of waveguide mesh geometries for membrane simulation, in Pro. Int. Computer Musi Conf., Ann Arbor, MI, pp. 6 63, Ot [] L. Savioja, Improving the three-dimensional digital wave guide mesh by interpolation, in Pro. Nordi Aoustial Meeting, Stokholm, Sweden, pp , Sept