Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformat
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Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformation Method L. Trautmann Multimedia Communications and Signal Processing, University of Erlangen-Nuremberg, Cauerstrasse 7, 91058 Erlangen, Germany Email: [email protected] Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology, P.O. Box 3000, 02015 Espoo, Finland Email: [email protected]
R. Rabenstein Multimedia Communications and Signal Processing, University of Erlangen-Nuremberg, Cauerstrasse 7, 91058 Erlangen, Germany Email: [email protected] Received 30 June 2003; Revised 14 November 2003 The functional transformation method (FTM) is a well-established mathematical method for accurate simulations of multidimensional physical systems from various fields of science, including optics, heat and mass transfer, electrical engineering, and acoustics. This paper applies the FTM to real-time simulations of transversal vibrating strings. First, a physical model of a transversal vibrating lossy and dispersive string is derived. Afterwards, this model is solved with the FTM for two cases: the ideally linearly vibrating string and the string interacting nonlinearly with the frets. It is shown that accurate and stable simulations can be achieved with the discretization of the continuous solution at audio rate. Both simulations can also be performed with a multirate approach with only minor degradations of the simulation accuracy but with preservation of stability. This saves almost 80% of the computational cost for the simulation of a six-string guitar and therefore it is in the range of the computational cost for digital waveguide simulations. Keywords and phrases: multidimensional system, vibrating string, partial differential equation, functional transformation, nonlinear, multirate approach.
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INTRODUCTION
Digital sound synthesis methods can mainly be categorized into classical direct synthesis methods and physics-based methods [1]. The first category includes all kinds of sound processing algorithms like wavetable, granular and subtractive synthesis, as well as abstract mathematical models, like additive or frequency modulation synthesis. What is common to all these methods is that they are based on the sound to be (re)produced. The physics-based methods, also called physical modeling methods, start at the physics of the sound production mechanism rather than at the resulting sound. This approach has several advantages over the sound-based methods. (i) The resulting sound and especially transitions between successive notes always sound acoustically realistic as far as the underlying model is sufficiently accurate. (ii) Sound variations of acoustical instruments due to dif-
ferent playing techniques or different instruments within one instrument family are described in the physics-based methods with only a few parameters. These parameters can be adjusted in advance to simulate a distinct acoustical instrument or they can be controlled by the musician to morph between
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