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