Biomedical Engineering Reference
In-Depth Information
which that there is no reflection at the end of the vocal tract),
p
i
can be
approximated by
p
i
=
β
(
y
1
+
y
2
)
.
(6.25)
The coupling parameter
β
is analogous to the acoustic inertance
I
in (6.11),
up to a coe
cient made up of a combination of anatomical parameters. Notice
that the pressure at the input of the trachea
p
i
keeps track of the oscillations
at both of the sources, and therefore couples the two oscillations.
A simulation using this model is shown in Fig. 6.10. The syllable was
generated by driving (6.22)-(6.25) with a time-dependent
p
s
, in much the
same way as the syllables of Chap. 5 were. The remaining parameters were
symmetric (except the frequency-controlling parameters
k
1
and
k
2
) and con-
stant in time. The isolated-source spectra are apparent, with fundamental
frequencies
F
1
and
F
2
around 2000 Hz. In addition, the signature of coupling
is readily seen: the heterodyne frequencies
f
mn
appearing as light strokes in
the sonogram, which are sums and differences of multiples of
F
1
and
F
2
:
f
mn
=
mF
1
+
nF
2
,
m, n∈ Z.
(6.26)
1.0
10000
-1.0
0.0
0.04
time (s)
.16
0.04
time (s)
.16
Fig. 6.10.
Synthetic syllable generated with the theoretical model for coupled
sources (6.22)-(6.25). The model parameters are all symmetric except for
k
1
=
k
2
.
In addition to the spectral components of the isolated sources (the two dark strokes
around 2000 Hz and their corresponding harmonics), a series of lighter strokes ap-
pear. These new frequencies are sums and differences of multiples of the two fun-
damental frequencies
6.4.3 Interact, Don't Interact
But now the question arises of why the two sources do not couple in general.
In principle, the bird has the ability to control the bilateral structure. For
example, control of the dorsal muscles (gating muscles) is lateralized. Con-
sequently, unilateral silencing is possible by active closing of one side of the
syrinx.
