Biomedical Engineering Reference
In-Depth Information
40
time
time
0
x
=
-120
0.0
frequency (Hz)
200
0
frequency (Hz)
200
0.0
frequency (Hz)
200
source
x
filter
=
output
Fig. 2.9.
Filtering a signal.
Left
: a source generates a triangular wave (time series
at
top left
, spectral content at
bottom left
). This triangular signal is used to excite
a tube open at one end and closed at the other (the filter), whose response function
is shown in the middle. In accordance with the response function of the tube, some
frequencies entering the tube will resonate (the frequencies close to a peak), while
others will be attenuated. The result is shown on the
right
: the spectral content of
the output signal (
bottom right
) has changed. The time series of the output signal
is shown at the
top right
Now it is time to filter the signal. By this we mean that we excite a tube
of air by means of a signal that is no longer harmonic (as in our discussion of
resonances and modes), but instead is something like our triangular function.
According to our analysis of resonances in a tube, some components of the
excitation are capable of establishing important oscillations in the tube, while
others will eventually be damped. The gain diagrams discussed in Sect. 2.2.4
gave an idea of precisely those effects. In this way, it is possible to know what
will happen to the signal after it has been filtered by the tube: those spectral
components falling in the regions of resonant frequencies will survive, while
the others will see their relative weight diminished. In Fig. 2.9, we show in
addition the gain of a tube open at one end and closed at the other, and the
result of filtering our triangular signal. The frequency of the resulting signal
is the same as the frequency of the original triangular signal, but the timbre
has been changed by the effects of the tube [Titze 1994].
2.3.2 Actual Filtering
There are two ways of actually applying a filter to a signal. The direct way is,
obviously, forcing the tube to vibrate by means of a loudspeaker emitting the
signal at one end of the tube. This can be accomplished either experimentally
(with a real tube) or numerically with the help of a computer, by keeping
track of the propagation and reflection of the waves in the tube, which is
known as
time-domain filtering
. The details of this procedure are described
in Sect. 6.2.2.
