106
D. Rocchesso: Sound Processing
direct manipulation
gestural controllers
mass-spring-damper system
Chebyshev polynomial is defined by the recursive relation:
T
0
(x) = 1
T
1
(x) = x
T
n
(x) = 2xT
n-1
(x) - T
n-2
(x) ,
(30)
and it has the property
T
n
(cos ) = cos n .
(31)
In virtue of property (31), if the nonlinear distorting function is a degree-m
Chebyshev polynomial, the output y, obtained by using a sinusoidal oscillator
x(n) = cos
0
n, is y(n) = cos (m
0
n), i.e., the m-th harmonic of x.
In order to produce the spectrum
y(n) =
k
h
k
cos (k
0
n) ,
(32)
it is sufficient to use the linear composition of Chebyshev functions
F (x) =
k
h
k
T
k
(x)
(33)
as a nonlinear distorting function.
Varying the oscillator amplitude A, the amount of distortion and the spec-
trum of the output sound are varied as well. However, the overall output ampli-
tude does also vary as a side effect, and some form of compensation has to be
introduced if a constant amplitude is desired. This is a clear drawback of NLD
as compared to FM. Time-varying spectral variations can also be introduced by
adding a control signal to the oscillator output x, so that the nonlinear function
is dynamically shifted.
5.4
Physical models
Instead of trying to model the air pressure signal as it appears at the entrance of
the ear canal, we can simulate the physical behavior of mechanical systems that
produce sound as a side effect. If the simulation is accurate enough, we would
obtain veridical sound dynamics and a detailed control in terms of physical
variables. This allows direct manipulation of the sound synthesis model and
direct coupling with gestural controllers.
5.4.1
A physical oscillator
Let us consider a simple mechanical mass-spring-damper system, as depicted in
figure 11. Let f be an exogenous force that drives the system. It is a mechanical
series connection, as the components share the same x position and the forces
sum up to zero:
f
m
= f
R
+ f
k
+ f m¨
x = -R x - kx + f .
(34)
By taking the Laplace transform of (34) (with null initial conditions) we get
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