Biomedical Engineering Reference
In-Depth Information
7 Synthesizing Birdsong
Understanding the mechanisms behind birdsong production should enable us
to generate realistic synthetic song. A synthesizer could be built in such a
way that the acoustic properties of the song were reproduced by mechanisms
that were qualitatively different from those found in the actual birds, but
this is not our goal. We have focused on the generation of synthetic song
by reproducing the mechanisms found in the real system, as a way to build
confidence in our modeling. Other applications of our synthesizers will be
discussed at the end of the chapter. Two different strategies were followed.
First, we have produced synthetic song by generating audio files that can
be read by PC sound player software. Those audio files contain synthetically
generated sound pressure time series data, obtained by numerical integration
of the equations used to model the mechanisms involved in birdsong. Our
second strategy consisted of building an analog integrator of those equations,
using commercial integrated circuits. The output can be listened to by simply
connecting a loudspeaker to the electronic circuit.
7.1 Numerical Integration and Sound
In previous chapters, we derived systems of differential equations that would
allow us to compute the dynamical evolution of the variables involved. In
particular, the midpoint of a labium
x
and its velocity
y
were found to obey
(in a very simple model) a pair of first-order differential equations
x
=
y,
y
=
cx
2
y,
−
(
t
)
x
+(
β
(
t
)
−
b
)
y
−
(7.1)
where
(
t
)and
β
(
t
) are slowly varying temporal functions, periodic on the
timescale of the syllable, for phrases consisting of repeated syllables. The time
evolution of
x
and
y
, that is, the rule
t →
(
x
(
t
)
,y
(
t
)), is in principle com-
pletely determined by the differential equations above, once we have chosen
an initial condition
x
(
t
0
)=
x
0
,
y
(
t
0
)=
y
0
. But how can we actually compute
the solutions
x
(
t
)=(
x
(
t
)
,y
(
t
))? Analytical formulas are out of the question,
but good approximations are possible.
