Digital Signal Processing Reference
In-Depth Information
are different, the same information is given. If harmonics for both positive frequen-
cy and negative frequency are shown, the plots must be for the coefficients of the
complex exponentials. If harmonics for only positive frequency are shown, the plots
are of the coefficients of the sinusoids.
The usefulness of the frequency spectrum is evident from the square-wave
spectrum of Figure 4.5. It was mentioned in Example 4.2 that some electronic oscil-
lators generate a square wave as an intermediate step to producing a sinusoidal signal.
We now discuss this case as an example.
C
k
2C
k
Filtering for an electronic oscillator
EXAMPLE 4.3
From Figure 4.5, we see that a sinusoid of frequency is present in a square wave. Hence, in
the oscillator, we must remove (filter out) the frequencies at
v
0
3v
0
, 5v
0
,
and so on, to produce
a sinusoid of frequency of
The electronic oscillator can be depicted by the system of Figure 4.7. The input signal
to the filter is a square wave, and its output signal is a sinusoid of the same frequency. The en-
gineers designing the oscillator of Figure 4.7 have the two design tasks of (1) designing the
square-wave generator and (2) designing the filter. We introduce filter design in Chapter 6.
The system of Figure 4.7 is used in many oscillators, because the square wave is easy to gen-
erate and the higher frequencies are not difficult to filter out. A filter of the type required is
analyzed mathematically in Section 4.5.
v
0
.
■
Square-wave
generator
Filter
Figure 4.7
Electronic oscillator.
We next consider a second physical system that illustrates frequency spectra.
Filtering in a pendulum clock
EXAMPLE 4.4
We now consider a pendulum clock, with the pendulum depicted in Figure 4.8(a). The pen-
dulum was discussed in Section 1.2. From physics, we know that the motion of a simple pen-
dulum approximates a sinusoid, as shown in Figure 4.8(b). The mainspring of the clock
applies periodic pulses of force at one of the extreme points of each swing of the pendulum,
as indicated in Figure 4.8(a). We can approximate this force with the signal
f
(
t
) of Figure 4.8(c).
This force will have a Fourier series, with a spectrum as indicated in Figure 4.8(d).
We now consider the pendulum to be a system with the input
f
(
t
) and the output
as shown in Figure 4.9. It is evident that this system filters the higher harmonics of the force
signal to produce a sinusoid of frequency In the electronic oscillator of Example 4.3, a fil-
ter was added to remove the higher harmonics. For the pendulum clock, the pendulum itself
is a mechanical filter that removes the higher harmonics.
u(t)
u(t),
v
0
.
■
Note that in discussing the pendulum, we used the
concept
of frequency spec-
tra to
understand
the operation of the clock, without deriving a model or assigning
numbers to the system. We can see the importance of concepts in engineering. Of
