Biomedical Engineering Reference
In-Depth Information
The second way is indirect: one needs to know the gain
G
ofthetubeas
a function of frequency, i.e., the mathematical expression
G
=
G
(
ω
) giving
rise to the central diagram in Fig. 2.9. In addition, the spectral content of the
input signal is needed. By this we mean the set of coe
cients
a
n
,
b
n
in (1.11),
represented by a function of (discrete) frequency
P
in
=
P
in
(
ω
n
) (recall that a
coe
cient labeled
n
stands for the amplitude of frequency
ω
n
). The spectral
content of the signal
after
the filter,
P
out
, is simply
P
out
(
ω
n
)=
G
(
ω
n
)
P
in
(
ω
n
)
.
(2.27)
This is known as
frequency-domain filtering
. Once we know
P
out
(
ω
n
), the
time-dependent output signal can be easily recovered by summing all the
spectral components, as in (1.11).
This is what is called a
passive filter
: the only thing it can do is enhance
or attenuate frequencies in the signal's spectrum. It cannot create frequencies
that are not present in the original signal. But it should be kept in mind that
it is not always possible to separate the effects of the source and the filter.
This is a delicate issue. So far, we have assumed that the dynamics of the
exciting source are independent of what is happening in the filter, and that
the source maintains a strict control of the situation [Laje et al. 2001]. This
is known as “source-filter separation”. The filter simply adds the injected
signal to the successive reflections, selecting some frequencies and suppressing
others. However, it is not impossible to think that the signal built up in the
tube could affect the dynamics of the exciting source. We shall run into this
interesting effect at a later point of our discussion.
2.3.3 The Emission from a Tube
The problem of the tube excited by a membrane at one end and open at the
other end is an important exercise in advancing towards our understanding
of birdsong: at one end of the tube perturbations are being induced, and the
result is the emission of sound at the other end. According to our discussion in
the previous sections, the problem reduces to solving the wave equation with
the proper boundary conditions. We can propose a solution for the pressure
perturbations of the form
p
=(
ae
ikx
+
be
−ikx
)
e
−iωt
,
(2.28)
with the boundary conditions
v
=
u
0
e
−iωt
membrane velocity at
x
=0
,
(2.29)
p
=0
at
x
=
L.
(2.30)
Since the velocity
v
of the wave satisfies (2.7), we obtain for the amplitudes
of the waves
