Digital Signal Processing Reference
In-Depth Information
modulates the carrier signal
c
(
t
)
=
cos(2
π
f
c
t
) with the AM signal
s
(
t
)
given by Eq. (8.1).
(a) Determine the value of the modulation index
k
to ensure
s
(
t
)
≥
0 for
all
t
.
(b) Determine the ratio of the power lost because of the transmission of
the carrier in
s
(
t
) versus the total power of
s
(
t
).
(c) Sketch the spectrum of
x
(
t
) and
s
(
t
) for
f
1
=
10 kHz,
f
2
=
20 kHz,
and
f
c
=
50 kHz.
(d) Show how synchronous demodulation can be used to reconstruct
x
(
t
)
from
s
(
t
).
8.2
Repeat Problem 8.1 for the information signal
x
(
t
)
=
sinc(5
10
3
t
)
if the fundamental frequency of the carrier is given by
f
c
=
20 kHz.
8.3
An AM station uses a modulation index
k
of 0.75. What fraction of the
total power resides in the information signal? By repetition for different
values of
k
within the range 0
≤
k
≤
1, deduce whether low or high values
of modulation index are better for improved efficiency.
8.4
Synchronous demodulation requires both phase and frequency coherence
for perfect reconstruction of the information signal. Assume that the infor-
mation signal
x
(
t
)
=
2 sin(2
π
f
1
t
)
is used to modulate the carrier
c
(
t
)
=
cos(2
π
f
c
t
). However, the demod-
ulating carrier has a frequency offset given by
c
(
t
)
=
cos[2
π
f
c
+
f
)
t
].
Determine the spectrum of the demodulated signal. Can the information
signal be reconstructed in such situations?
8.5
A special case of amplitude modulation, referred to as the quadrature ampli-
tude modulation (QAM), modulates two information-bearing signals
x
1
(
t
)
and
x
2
(
t
) simultaneously using two different carriers
c
1
(
t
)
=
A
1
cos(2
π
f
c
t
)
and
c
2
(
t
)
=
A
2
sin(2
π
f
c
t
). The QAM signal is given by
s
(
t
)
=
A
1
[1
+
k
1
x
1
(
t
)] cos(2
π
f
c
t
)
+
A
2
[1
+
k
2
x
2
(
t
)] sin(2
π
f
c
t
)
,
where
k
1
and
k
2
are the two modulation indexes used for modulating
x
1
(
t
)
and
x
2
(
t
). Draw the block diagram of the demodulator that reconstructs
x
1
(
t
) and
x
2
(
t
) from the modulated signal.
8.6
Assume the frictional coefficient
r
of the spring damping system, shown in
Fig. 8.6, to equal zero. Determine the transfer function
H
(
s
) and impulse
response
h
(
t
) for the modified model. Based on the location of the poles,
comment on the stability of the spring damping system.
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