Biomedical Engineering Reference
In-Depth Information
6.2 Acoustic Feedback
6.2.1 Source-Filter Separation
One of the possible ways of obtaining oscillations of this sort, without aban-
doning the model that we introduced in Chap. 4, is through
acoustic feedback
.
When we analyzed the onset of oscillations for the syringeal labia, we men-
tioned that when the labia have a divergent profile, the average interlabial
pressure is low with respect to the bronchial pressure. The reason is that when
the profile is divergent, the average pressure between the labia is similar to
the tracheal pressure (and similar in turn to atmospheric pressure). We also
discussed the existence of pressure perturbations at the input of the vocal
tract as a result of adding the flow perturbations injected by the sources to
the perturbations induced by reflected pressure waves. In that discussion, we
assumed implicitly that the existence of pressure perturbations in the vocal
tract did not affect the labial dynamics. The theory that assumes that the
labial dynamics are independent of whatever happens in the filter is known
as the
source-filter theory
. In many cases, as during normal human speech
or the song of most bird species, this hypothesis is a most sensible one. But
source-filter separation does not hold in every case, or at least we cannot a
priori a
rm that it holds.
6.2.2 A Time-Delayed System
Formally, the dynamics of a labium are described in terms of its midpoint
position
x
by
M x
+
B x
+
Kx
=
P
g
,
(6.3)
where
P
g
is the average interlabial pressure, given by
P
g
=
P
i
+(
P
s
− P
i
)
f
(
x, x
) ;
(6.4)
f
(
x, x
) is a generic function, which, in the flapping model, is
f
(
x, x
)=
1
a
2
/a
1
(see Chap. 4). The source-filter approximation assumes that the
pressure at the entrance of the tract
P
i
is zero (that is, atmospheric pressure).
A more realistic approximation takes into account the fact that this pressure
is the result of adding perturbations due to the modulation of the airflow and
the perturbations arriving back after being reflected, mostly at the beak:
−
P
i
(
t
)=
s
(
t
)+
P
back
(
t − τ
)
,
(6.5)
P
back
(
t
)=
−γP
i
(
t − τ
)
,
(6.6)
where
τ
is the time it takes for a sound wave to travel the length of the
tract, and
γ
is the reflection coe
cient at the beak. This equation expresses
the fact that the pressure perturbations at the input of the tract are the
result of adding the perturbations induced by the labial fluctuations (
s
(
t
))
