Biomedical Engineering Reference
In-Depth Information
Fig. 2.8.
Superposition of two standing waves in a tube open at one end and closed
at the other. The resulting wave is not a standing wave, but a traveling wave at
the average frequency
wherewehavedefined
k
=(
k
1
+
k
2
)
/
2, ∆
k
=(
k
2
−
k
1
)
/
2,
ω
=(
ω
1
+
ω
2
)
/
2
and ∆
ω
=(
ω
2
−
ω
1
)
/
2. Notice that the resulting wave is a traveling wave
(plus its reflection) at the average frequency
ω
, with an amplitude modulated
at a lower frequency ∆
ω
.
2.3 Filtering a Signal
2.3.1 Conceptual Filtering
It is time to examine what happens when a signal with a rich spectral con-
tent meets a filter. Our discussion of resonances was performed in terms of
harmonic excitations produced by a membrane. When did we use the fact
that our excitations were harmonic? We assumed this in order to claim that
a signal was equal to itself but inverted if we waited for half a period. This
is not necessarily true if the signal is not harmonic. Does this mean that for
forcings more realistic than simply harmonic ones, we have to forget about
our discussion? No, since we saw in Chap. 1 that a periodic signal of arbi-
trary form can be thought of as a sum of harmonic signals. In this way, we
can analyze what happens to a spectrally rich signal in terms of what hap-
pens to its harmonic components. Let us take the case, analyzed in detail
in Chap. 1, of the triangular time-periodic function. It is spectrally rich (we
could approximate it with a sum of properly weighted harmonic functions),
and we can describe such richness by showing in a diagram the frequency of
each component versus its relative weight. Such a diagram is illustrated in
Fig. 2.9. Each peak in the “source” part of the figure represents one of the
harmonic components that it is necessary to add in order to reproduce the
signal, as discussed in Chap. 1. The frequency at which a peak is located rep-
resents the frequency of the component, and its height indicates the relative
weight of the component in the sum.
