Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformation Method

EURASIP Journal on Advances in Signal Processing, Dec 2004

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.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1155%2FS1110865704312059.pdf

Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformation Method

EURASIP Journal on Applied Signal Processing Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformation Method L. Trautmann 0 1 R. Rabenstein 0 1 0 Multimedia Communications and Signal Processing, University of Erlangen-Nuremberg , Cauerstrasse 7, 91058 Erlangen , Germany 1 Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology , P.O. Box 3000, 02015 Espoo , Finland 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. and phrases; multidimensional system; vibrating string; partial differential equation; functional transformation; nonlinear; multirate approach 1. 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 different 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 real world instruments to obtain more degrees of freedom in the expressiveness and variability. The second item makes physical modeling methods quite useful for multimedia applications where only a very limited bandwidth is available for the transmission of music as, for example, in mobile phones. In these applications, the physical model has to be transferred only once and afterwards it is sufficient to transfer only the musical score while keeping the variability of the resulting sound. The starting points for the various existing physical modeling methods are always physical models varying for a certain vibrating object only in the model accuracies. The application of the basic laws of physics to an existing or imaginary vibrating object results in continuous-time, continuousspace models. These models are called initial-boundaryvalue problems and they contain a partial differential equation (PDE) and some initial and boundary conditions. The discretization approaches to the continuous models and the digital realizations are different for the single physical modeling methods. One of the first physical modeling algorithm for the simulation of musical instruments was made by Hiller and Ruiz 1971 in [2] with the finite difference method. It directly discretizes the temporal and spatial differential operators of the PDE to finite difference terms. On the one hand, this approach is computationally very demanding; since temporal and spatial sampling intervals have to be chosen small for accurate simulations. Furthermore, stability problems occur especially in dispersive vibrational objects if the relationship between temporal and spatial sampling intervals is not chosen properly [3]. On the other hand, the finite difference method is quite suitable for studies in which the vibration has to be evaluated in a dense spatial grid. Therefore, the finite difference method has mainly been used for academic studies rather than for real-time applications (see, e.g., [4, 5]). However, the finite difference method has recently become more popular also for real-time applications in conjunction wit (...truncated)


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1155%2FS1110865704312059.pdf

L Trautmann, R Rabenstein. Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformation Method, EURASIP Journal on Advances in Signal Processing, 2004, pp. 745924, Volume 2004, Issue 7, DOI: 10.1155/S1110865704312059