sinusoidal model
additive synthesis
stochastic part
deterministic part
Chapter 5
Sound Modelling
5.1
Spectral modelling
5.1.1
The sinusoidal model
A sound is expressed according to the sinusoidal model if it has the form
y(t) =
I
i=1
A
i
(t)e
j
i
(t)
,
(1)
where
i
(t) =
t

i
( )d , and A
i
(t) and
i
(t) are the ith sinusoidalcomponent
instantaneous magnitude and frequency, respectively. In practice, we consider
discretetime real signals. Therefore, we can write
y(n) =
I
i=1
A
i
(n) cos (
i
(n)) ,
(2)
with
i
(n) =
nT
0
i
( )d +
0,i
.
(3)
In principle, if I is arbitrarily high, any sound can be expressed according
to the sinusoidal model. This principle states the generality of the additive
synthesis approach. Actually, the noise components would require a multitude of
sinusoids, and it is therefore convenient to treat them separately by introduction
of a "stochastic" part e(n):
y(n) =
I
i=0
A
i
(n) cos (
i
(n))
Deterministic Part
+
e(n)
Stochastic Part
.
(4)
The separation of the stochastic part from the deterministic part can be done by
means of the ShortTime Fourier Transform using the scheme of figure 1. Here,
we rely on the fact that the STFT analysis retains the phases of the sinusoidal
components, thus allowing a reconstruction that preserves the wave shape [94].
In this way, the deterministic part can be subtracted from the original signal
91