Digital Waveguides versus Finite Difference Structures: Equivalence and Mixed Modeling
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Digital Waveguides versus Finite Difference Structures: Equivalence and Mixed Modeling Matti Karjalainen Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology, 02150 Espoo, Finland Email: [email protected]
Cumhur Erkut Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology, 02150 Espoo, Finland Email: [email protected] Received 30 June 2003; Revised 4 December 2003 Digital waveguides and finite difference time domain schemes have been used in physical modeling of spatially distributed systems. Both of them are known to provide exact modeling of ideal one-dimensional (1D) band-limited wave propagation, and both of them can be composed to approximate two-dimensional (2D) and three-dimensional (3D) mesh structures. Their equal capabilities in physical modeling have been shown for special cases and have been assumed to cover generalized cases as well. The ability to form mixed models by joining substructures of both classes through converter elements has been proposed recently. In this paper, we formulate a general digital signal processing (DSP)-oriented framework where the functional equivalence of these two approaches is systematically elaborated and the conditions of building mixed models are studied. An example of mixed modeling of a 2D waveguide is presented. Keywords and phrases: acoustic signal processing, hybrid models, digital waveguides, scattering, FDTD model structures.
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INTRODUCTION
Discrete-time simulation of spatially distributed acoustic systems for sound and voice synthesis finds its roots both in modeling of speech production and musical instruments. The Kelly-Lochbaum vocal tract model [1] introduced a onedimensional transmission line simulation of speech production with two-directional delay lines and scattering junctions for nonhomogeneous vocal tract profiles. Delay sections discretize the d’Alembert solution of the wave equation [2] and the scattering junctions implement the acoustic continuity laws of pressure and volume velocity in a tube of varying diameter. Further simplification led to the synthesis models used as the basis for linear prediction of speech [3]. A similar modeling approach to musical instruments, such as string and wind instruments, was formulated later and named the technique of digital waveguides (DWGs) [4, 5]. For computational efficiency reasons, in DWGs twodirectional delay lines are often reduced to single delay loops [6]. DWGs have been further discussed in two-dimensional (2D) and three-dimensional (3D) modeling [5, 7, 8, 9, 10], combined sometimes with a finite difference approach into DWG meshes.
Finite difference schemes [11] were introduced to the simulation of vibrating string as a numerical integration solution of the wave equation [12, 13], and the approach has been developed further for example in [14] as a finite difference time domain (FDTD) simulation. The second-order finite difference scheme including propagation losses was formulated as a digital filter structure in [15], and its sta
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