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parameters of the model are the tribological properties of the contact condition
(stiffness, dynamic or static friction coef
cients, etc.) and the dynamic parameters
of the interaction (mainly the velocity and the normal force). Also, a synthesis
technique based on the modal analysis of physical objects (
(finite element modelling
of each object for precomputation of shapes and frequencies of the modes) was
proposed by (O
Brien et al.
2002
) in the context of interactive applications. Note
that this approach presents a limitation when the physical considerations involve
complex modelling and can less easily be taken into account for synthesis per-
spectives especially with interactive constraints.
Signal synthesis models that simulate the resulting vibration of the sound source
are based on a mathematical modelling of the signal. They are numerically easy to
implement and can be classi
'
ed in three groups as follows:
Additive synthesis: The sound is constructed as a superposition of elementary
sounds, generally sinusoidal signals modulated in amplitude and frequency
(Risset
1965
). For periodic or quasi-periodic sounds, these components have
average frequencies that are multiples of one fundamental frequency and are
called harmonics. The amplitude and frequency modulation (FM) laws should be
precise when one reproduces a real sound. The advantage of these methods is the
potential for intimate and dynamic modi
cations of the sound. Granular syn-
thesis can be considered as a special kind of additive synthesis, since it also
consists in summing elementary signals (grains) localized in both the time and
the frequency domains (Roads
1978
).
Subtractive synthesis: The sound is generated by removing undesired compo-
nents from a complex sound such as noise. This technique is linked to the theory
of digital
filtering (Rabiner and Gold
1975
) and can be related to some physical
sound generation systems such as speech (Flanagan et al.
1970
; Atal and
Hanauer
1971
). The advantage of this approach is the possibility of uncoupling
the excitation source and the resonance system. The sound transformations
related to these methods often use this property to make hybrid sounds or
crossed synthesis of two different sounds by combining the excitation source of a
sound and the resonant system of another (Makhoul
1975
; Kronland-Martinet
1989
).
Global (or non-linear) synthesis: The most well-known example of such meth-
ods is audio FM. This technique updated by Chowning (
1973
) revolutionized
commercial synthesizers. The advantages of this method are that it calls for very
few parameters, and that a small number of numerical operations can generate
complex spectra. They are, however, not adapted to precise signal control, since
slight parameter changes induce radical signal transformations. Other related
methods such as waveshaping techniques (Ar
b
1979
; Le Brun
1979
) have also
been developed.
In some cases, both approaches (physical and signal) can be combined to pro-
pose hybrid models, which have shown to be very useful when simulating certain
musical instruments (Ystad and Voinier
2001
; Bensa et al.
2004
).
