Biomedical Engineering Reference
In-Depth Information
frequency. As we move out of the shaded region, just before getting to point
5, the oscillations stop [Gardner et al. 2001, Laje et al. 2002].
Now, it is ubiquitous in birdsong that certain syllables are repeated several
times. This, in terms of our path, can be reproduced if, from point 5 in the
parameter space, we return to point 1, and repeat the path as many times
as necessary. In Fig. 5.4b, we can see a sonogram of the syllables generated
(as explained in Chap. 4) using values of pressure and tension that change
along the path shown. To a piece of sound analysis software, the results
obtained with our equations are indistinguishable from a real recording in
digital format. Therefore, we can study the result as usual. We could, for
example, reproduce it in front of a bird and study the bird's behavior.
5.2.2 Paths in Parameter Space
In the previous example, the pressure started and ended at the same value
after a complete cycle, and so did the labial tension. However, there was a
delay between the cycles of pressure and tension. In our example, the tension
started to grow after the pressure reached its maximum value. We say that
the two cycles were out of phase, and we quantify this delay in terms of a
quantity that we call the phase difference
φ
0
[Gardner et al. 2001].
The simplest conceivable path in the parameter space reflecting our ideas
about a “cycle” in pressure and tension and a “delay” between them can be
written mathematically as
p
(
t
)=
p
0
+
p
1
cos(
φ
(
t
))
,
(5.8)
k
(
t
)=
k
0
+
k
1
cos(
φ
(
t
)+
φ
0
)
.
(5.9)
The parametrization of time through
φ
(
t
) allows us to control the speed at
which the path is traversed. According to our previous discussion,
dφ
(
t
)
/dt
must be very small compared with the frecuency of the vocalization. However,
the key parameter in (5.8) and (5.9) is the phase difference
φ
0
. Let us explore
the reason.
Notice that a delay in the tension with respect to the pressure for a
syllable such as the one displayed in Fig. 5.4 can be achieved with a value
of
φ
0
between
φ
and 2
φ
. How different would the syllable be if, instead of
a delay in the tension with respect to the pressure, both variables evolved
simultaneously? By “simultaneously” we mean
in phase
, that is,
φ
0
=0.In
this case, the path would look like that in Fig. 5.5. In this situation, as the
pressure increases, so does the tension. Therefore, at the moment at which
the oscillations start (between point 1 and point 2 in the parameter space),
the frequency of oscillation increases. The process continues until both the
pressure and the tension reach their maximum values. At this point the labia
are oscillating with a rich spectral content and a high frequency. From this
point, the pressure and tension begin to decrease (towards point 5 in the
