REDUCTION OF THE DISPERSION ERROR IN THE INTERPOLATED DIGITAL WAVEGUIDEMESH USING FREQUENCY WARPING Lauri Savioja and Vesa V¨ alim¨ aki Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology, P.O. Box 3000, FIN-02015 HUT, Espoo, Finland E-mail: Lauri.Savioja@hut.fi, Vesa.Valimaki@hut.fi http://www.acoustics.hut.fi/ ABSTRACT The digital waveguide mesh is an extension of the one-dimensional digital waveguide technique. The mesh is used for simulation of two- and three-dimensional wave propagation in musical in- struments and acoustic spaces. The rectangular digital waveguide mesh algorithm suffers from direction-dependent dispersion. By using the interpolated mesh, nearly uniform wave propagation char- acteristics are obtained in all directions. In this paper we show how the dispersion error of the interpolated mesh can be reduced by fre- quency warping. By using this technique the bandwidth where the frequency accuracy is within 1% tolerance is more than doubled. 1. INTRODUCTION One-dimensional digital waveguides are a discrete numerical meth- od used to model musical instruments, such as string and wind in- struments [1, 2]. Two-dimensional (2-D) and three-dimensional (3-D) extensions of digital waveguides have been proposed for simulation of plates, drums [3, 4], and acoustic spaces [5]. An- other method which is similar can be obtained by using multi- dimensional wave digital filters [6]. In the original multi-dimensional digital waveguide mesh, the wave propagation speed is a function of propagation direction [3, 4, 7]. By using more advanced structures, such as the triangular mesh [8, 9, 10], nearly uniform wave propagation characteristics can be obtained in all directions. Another way how this can be achieved is by using an interpolation technique with the rectangu- lar mesh as presented in an earlier study [7]. Although the dis- persion error of the triangular mesh is very small, the rectangular structure is still attractive for some applications. It is conceptually simple: the indexing of the mesh nodes is easy, and the tesselation of a given plane or space is straightforward, especially in the case of rectangular objects. Also in the interpolated rectangular mesh there still remains dispersion error which increases with frequency [7]. The error is not very harmful in music synthesis applications, but in high- accuracy numerical simulations the dispersion limits the valid band- width of simulations. In this paper we show that the dispersion er- ror in the interpolated rectangular digital waveguide mesh can be reduced by frequency warping. This is possible because the error is nearly independent of propagation direction. We examine the 2-D case, but the results may be extended to three dimensions. The interpolated rectangular digital waveguide mesh is briefly described and analyzed in Section 2. In Section 3, the frequency- warping technique is introduced. The performance of the proposed method is illustrated in Section 4 with simulated examples. Sec- tion 5 concludes the paper. 2. INTERPOLATED MULTI-DIMENSIONAL DIGITAL WAVEGUIDE MESH A multi-dimensional rectangular digital waveguide mesh is a reg- ular array of 1-D digital waveguides arranged along each perpen- dicular dimension, interconnected between all the unit delay ele- ments. A difference equation can be derived for the nodes of an -dimensional rectangular mesh [3, 4]: (1) where represents the displacement at a junction at time step , subscript denotes the junction to be calculated, and index rep- resents its axial neighbors. Ideally, waves should propagate at the same speed in all di- rections. In the original digital waveguide mesh, however, sample updates occur along the axial directions only. This approxima- tion causes inaccuracies in the wave propagation speed [3, 4, 7]. The ratio between the speed in the original 2-D digital waveg- uide mesh and the ideal speed is presented by the dispersion factor [10, 4]: (2) where and are the normalized spatial frequency coordinates, and , and is the ideal wave propagation speed, that is, spatial samples per sampling interval . The relative frequency error (RFE) is obtained from the dispersion factor as (3) where and is the normalized temporal fre- quency such that . The relative frequency error in axial and diagonal directions is shown in Fig. 1(a). There is no error in the diagonal direction but in the axial direction, where the maximal er- ror occurs, the wave propagation speed decreases with frequency so that 1% error is reached at , where is the sampling frequency. Note that we show the RFE as a function of tempo- ral frequency, since we want to compare our results against ideal results that are also available as a function of (see Section 4).