Biomedical Engineering Reference
In-Depth Information
the spectrum reveals the existence of a physical phenomenon that we cannot
interpret with the elements presented so far. In summary, we have two sam-
ples of complex sounds, for which the origins of the complexity are quite
different. It is natural, then, to wonder if the complexity of a vocalized sound
is necessarily a signature of the complexity of the neural instructions sent to
the vocal organ by the brain. The first researchers who became interested in
this problem described the possible physical nontrivial behaviors of syringeal
labia subjected to a simple constant airflow [Fee et al. 1998]. The study was
carried out in vitro on syringes surgically extracted from zebra finches (
Tae-
niopygia guttata
). The degree of complexity of the sound signal produced in
these experiments was the same as that found in the natural vocalizations of
the bird. In this way, it was established that there is a degree of complexity
associated with the mechanics of the vocal organ, beyond the richness of its
neural control.
6.1.2 Subharmonics
We presented the sonogram in Fig. 6.1b as an example of a complex sound
signal. Why? Remember (Chap. 1) that a signal of a given fundamental fre-
quency
F
repeats itself every
T
time units (
T
is the period associated with
the fundamental frequency;
T
=1
/F
). It is possible for a signal to undergo
a modification of its spectral content and acquire or lose harmonics (for ex-
ample, after being filtered by the vocal tract), but it will always repeat itself
after
T
time units. The harmonics, which are multiples of the fundamental fre-
quency, make an important contribution to the timbre of the sound, but they
do not modify its periodicity, since they repeat themselves every
T/n
units
of time, where
n
=1
,
2
,
3
,...
. In contrast, the appearance of
subharmon-
ics
(that is, frequencies that are submultiples of the fundamental frequency)
corresponds to signals that repeat themselves not after
T
but after
nT
time
units. In the language of Chap. 1, a time function such as
y
1
=
A
cos(
ωt
)+
supra
cos(2
ωt
)
(6.1)
has a period
T
=2
π/ω
, no matter the value of
supra
. Its sonogram will
consist of a fundamental frequency
F
1
=1
/T
with amplitude
A
plus its first
harmonic
F
2
=2
F
1
=2
/T
with amplitude
supra
. In contrast, a function such
as
y
1
=
A
cos(
ωt
)+
sub
cos
1
2
ωt
(6.2)
has a period
T
new
=2
T
(twice as large) as soon as
sub
is different from zero,
since 2
π/T
new
=
ω/
2=(1
/
2)(2
π/T
). What mechanisms can give rise to such
a behavior?
