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
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