Digital Signal Processing Reference
In-Depth Information
Fig. 2.5. Amplitude modulation
(AM) system.
km
(
t
)
(1 +
km
(
t
))
+
m
(
t
)
modulating
signal
attenuator
modulator
s
(
t
)
modulated
signal
offset for
modulation
A
cos(2
p
f
c
t
)
called the modulated signal. The steps involved in an amplitude modulator
are illustrated in Fig. 2.5, where the modulating signal
m
(
t
) is first processed
by attenuating it by a factor
k
and adding a dc offset such that the resulting
signal (1
+
km
(
t
)) is positive for all time
t
. The modulated signal is produced
by multiplying the processed input signal (1
+
km
(
t
)) with a high-frequency
carrier
c
(
t
)
=
A
cos(2
π
f
c
t
). Multiplication by a sinusoidal wave of frequency
f
c
shifts the frequency content of the modulating signal
m
(
t
) by an additive
factor of
f
c
. Mathematically, the amplitude modulated signal
s
(
t
) is expressed as
follows:
s
(
t
)
=
A
[1
+
km
(
t
)] cos(2
π
f
c
t
)
,
(2.9)
where
A
and
f
c
are, respectively, the amplitude and the fundamental frequency
of the sinusoidal carrier.
It may be noted that the amplitude
A
and frequency
f
c
of the carrier signal,
along with the attenuation factor
k
used in the modulator, are fixed; therefore,
Eq. (2.9) provides a direct relationship between the input and the output signals
of an amplitude modulator. For example, if we set the attenuation factor
k
to
0.2 and use the carrier signal
c
(
t
)
=
cos(2
π
10
8
t
), Eq. (2.9) simplifies to
10
8
t
)
.
s
(
t
)
=
[1
+
0
.
2
m
(
t
)] cos(2
π
(2.10)
Amplitude modulation is covered in more detail in Chapter 7.
2.1.4 Mechanical water pump
The mechanical pump shown in Fig. 2.6 is another example of a linear CT
system. Water flows into the pump through a valve V1 controlled by an electrical
circuit. A second valve V2 works mechanically as the outlet. The rate of the
outlet flow depends on the height of the water in the mechanical pump. A
higher level of water exerts more pressure on the mechanical valve V2, creating
a wider opening in the valve, thus releasing water at a faster rate. As the level
of water drops, the opening of the valve narrows, and the outlet flow of water is
reduced.
A mathematical model for the mechanical pump is derived by assuming that
the rate of flow
F
in
of water at the input of the pump is a function of the input
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