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 i-th sinusoidal-component
instantaneous magnitude and frequency, respectively. In practice, we consider
discrete-time 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 Short-Time 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