Biomedical Engineering Reference
In-Depth Information
and the pressure wave arriving as a result of reflection at the end of the
tract. If these perturbations are
P
back
(
t
)attime
t
, the total perturbation is
the result of adding
s
(
t
)and
P
back
(
t
τ
), since it takes the pressure wave
traveling backwards a time
τ
to reach the opposite end of the tract. The source
of pressure perturbations
s
(
t
) is a function of all the variables describing
the dynamics of the labium. As discussed in Chaps. 1 and 2, the pressure
perturbations
s
(
t
) at the entrance of the vocal tract are generated by time
variations of the flow
U
(
t
) injected by the glottis, which in turn is a function
of the variables
x
and
x
.
The introduction of a time delay into the differential equation describing
the labial dynamics is responsible for a dramatic change in the kind of so-
lutions that can be expected. In order to gain an intuitive idea about this
observation, we can compare the simple dynamics obtained from the system
−
dN
(
t
)
dt
π
2
τ
N
(
t
)
=
−
(6.7)
with the solution of
dN
(
t
)
dt
π
2
τ
N
(
t
=
−
−
τ
)
.
(6.8)
In the first case, we obtain a linear decay to zero,
N
(
t
)=
A
exp(
πt/
2
τ
),
while in the second case the system evolves harmonically in time:
N
(
t
)=
A
cos(
πt/
2
τ
).
There are two parameters, then, that are important in order to obtain
complex dynamics within this framework: one (the
coupling parameter
)mea-
suring the amplitude of the perturbation injected by the glottis at the en-
trance of the vocal tract, and one describing the delay involved (which can be
taken simply as the time it takes a sound wave to travel the length
L
of the
vocal tract). For large enough values of the coupling parameter, the pressure
fluctuations that are established at the base of the trachea can be impor-
tant and affect the labial dynamics [Laje et al. 2001]. The vocal-tract input
pressure values found by the labia in two consecutive situations in which the
profile is divergent (i.e. opened to the trachea) are not going to be equal.
Whenever this is the case, the labial oscillations can be more complex than
expected, and the period of the signal (if it exists) can be very large. This is
a possible mechanism for the appearance of low frequencies (subharmonics)
in the spectral analysis of a signal.
−
6.2.3 Coupling Between Source and Vocal Tract
An expression for the coupling parameter is needed in order to continue the
analysis. In other words, we need to develop an expression for
s
(
t
) in (6.5) in
terms of the variables
x
and
x
. The coupling parameter (that is, the coe
cient
hopefully appearing in front of the variables in the expression for
s
(
t
)) will in
general be a combination of anatomical parameters of the syrinx and vocal
