a basic form of digital modulation known as binary frequency-shift-keying (BFSK), discussed in Chapter 7.
186
CHAPTER 4 ! ANGLE MODULATION
ADDITIONAL
PROBLEMS
4.10 Consider an interval ¢t of an FM wave s1t2 ! Ac cos3u1t24 such that u1t2 satisfies the condition
4.8 Sketch the PM and FM waves produced by the sawtooth wave shown in Fig. 4.17 as the source
u1t " ¢t2 # u1t2 ! p
of modulation.
Hence, show that if ¢t is sufficiently small, the instantaneous frequency of the FM wave inside
this interval is approximately
given by
m(t)
fi !
1
2¢t
A
4.11 The sinusoidal modulating
wave
m1t2 ! Am cos12pfm t2
t
Normalized bandwidth, BT /!f
T0 phase sensitivity
2T0
3T0 unmodulated carrier wave has
kp . The
is applied to a phase modulator
with
0
Ac . Determine the spectrum of the resulting phase-modulated wave,
frequency
186
CHAPTERfc 4and
! amplitude
ANGLE MODULATION
IGURE 4.17 Problem 4.8
assuming that theFmaximum
phase deviation b ! kp Am does not exceed 0.3 radian.
s1t2
!
A
u1t2
4.10
Consider
an
interval
¢t
of
an
FM
wave
cos3u1t24
such
that
satisfies
the
condition
4.12 A carrier wave is frequency-modulated using a sinusoidal
signal of frequency fm and amplitude Am .
c
(a)
Determine
the
values
of
the
modulation
index
for
which
component of
the FM
wave
b
u1t
"
¢t2
#
u1t2
! the
p ofcarrier
4.9 In a frequency-modulated radar the instantaneous frequency
the transmitted
carrier
is varis
reduced
to
zero.
For
this
calculation
you
may
use
the
values
of
given
in
Appendix
3.
1b2
J
0
ied Hence,
as in Fig.
4.18.
is generated
by frequency
modulation
with
a periodic
trianshow
thatSuch
if ¢taissignal
sufficiently
small, the
instantaneous
frequency
of the
FM wave
inside
(b)
In
a
certain
experiment
conducted
with
and
increasing
(starting
from
zero
A
f
!
1
kHz
m
m
gular
modulating
The instantaneous
this
interval is wave.
approximately
given by frequency of the received echo signal is shown dashed
volt),
is found
carrier component
of the
FM wave and
is reduced
to echo
zero signals
for the are
first
in Fig.
4.18itwhere
the the
round-trip
delay time. The
received
t isthat
1 transmitted
time
when
What
is
the
frequency
sensitivity
of
the
modulator?
What
is
A
!
2
volts.
m
f
!
applied to a mixer, and the difference frequencyi component is retained. Assuming that f0t V the
1
2¢t
value
of Am forthe
which
the carrier
is reduced
to zeroaveraged
for the second
time?
for all
number
of beatcomponent
cycles at the
mixer output,
over one
second, t, determine
4.11
The
sinusoidal
wave
4.13
carrier
ofmodulating
frequency
frequency-modulated
by a t,
sinusoidal
of ampliinAterms
ofwave
the peak
deviation100
of theiscarrier
frequency, the delay
and the wave
repetition
fre¢f MHz
tude 20f0Vofand
100
kHz. (The
The
frequency
sensitivity
of
the
modulator
is
25
kHz/V.
m1t2
!A
cos12pf
t2
quency
thefrequency
transmitted
signal.
beat
refers
to
a
signal
whose
frequency
is
the
differm
m
(a)is Determine
approximate
bandwidth
ofsignals.)
the FM wave,
using
Carson’s rule.
ence
between
thethe
of the
two input
kp . The
applied to
a frequencies
phase
modulator
with
phase
sensitivity
unmodulated
carrier wave has
(b)frequency
Determine
the
bandwidth
obtained
by
transmitting
only
those
side-frequencies
with amplifc and amplitude Ac . Determine the spectrum of the resulting phase-modulated
wave,
tudes
that
exceed
one
percent
of
the
unmodulated
carrier
amplitude.
Use
the
universal
curve
assuming that fthe
maximum
phase
deviation
b
!
k
A
does
not
exceed
0.3
radian.
p
m
i(t)
Transmitted
Fig. 4.9
this calculation.
4.12 Aof
carrier
wavefor
is frequency-modulated
signal of frequency fm and amplitude Am .
signal using a sinusoidal
Echo
(c)(a)Repeat
your the
calculations,
assuming
thatindex
the amplitude
ofthe
the
modulating
wave
is doubled.
Determine
values
of
the
modulation
for
which
carrier
component
of
the
FM wave
b
4.6 Transmission
Bandwidth
of
FM
Waves
171
fc + #f
(d) Repeat
your
calculations,
assuming
that
the
modulation
frequency
is
doubled.
is reduced to zero. For this calculation you may use the values of J01b2 given in Appendix 3. fc PM wave
tAm cos12pf
4.14 Consider
wide-band
produced
by fthe
sinusoidal
modulating
t2,
(b) In aacertain
experiment
conducted
with
increasingwave
frommzero
Am (starting
1 kHz and
m !
40
using avolt),
modulator
withthat
a phase
sensitivity
equal to
radians
perisvolt.
kthe
it isf found
the
carrier
component
of
FM
wave
reduced
to
zero
for
the
first
p
c – #f
timethat
when
the frequency
of the modulator? What is the
2 volts. What
(a) Show
if A
the
maximum
phase isdeviation
of thesensitivity
m !
1PM wave is large compared with one
!
––
value the
of Abandwidth
component
is reduced
zero
for the second
time? fm .
radian,
ofthe
thecarrier
PM
wave
varies linearly
modulation
frequency
m for which
f0 withtothe
20
4.13(b)ACompare
carrier wave
frequency 100
is frequency-modulated
by of
a sinusoidal
wave
amplithisof
characteristic
of aMHz
wide-band
PM wave with that
a wide-band
FMofwave.
FIGURE
4.18100
Problem
4.9
tude 4.19
20 Vshows
and
frequency
kHz. of
The
frequency sensitivity
the modulator
is 25 kHz/V.
4.15 Figure
the block
diagram
a closed-loop
feedback of
system
for the carrier-frequency
(a) Determine
approximate
bandwidth
of the The
FM voltage-controlled
wave, using Carson’s
rule. shown in
stabilization
of a the
wide-band
frequency
modulator.
oscillator
10
the(b)
figure
constitutes
the frequency
modulator.
Using the only
ideasthose
of mixing
(i.e., frequency
Determine
the bandwidth
obtained
by transmitting
side-frequencies
with transampli8
that exceed
one percent
offrequency
the unmodulated
carrier amplitude.
the universal
lation)tudes
(described
in Chapter
3) and
discrimination
(describedUse
in this
chapter),curve
dis6 cuss how
of Fig.
for this
calculation.
the 4.9
feedback
system
of Fig. 4.19 is capable of exploiting the frequency accuracy of the
crystal
oscillator
stabilize theassuming
voltage-controlled
oscillator.of the modulating wave is doubled.
(c) Repeat
yourtocalculations,
that the amplitude
4
(d) Repeat your calculations, assuming that the modulation frequency is doubled.
4.14 Consider a wide-band PM wave produced by the sinusoidal modulating wave Am cos12pfm t2,
VoltageusingMessage
a modulator with
a phase sensitivity equal to kp radians per volt.Frequency-stabilized
controlled
2
signal
m(t)
(a) Show that if the maximum
phase deviation of the PM wave is large compared
oscillator
FM wave with one
radian, the bandwidth of the PM wave varies linearly with the modulation frequency fm .
1 (b) Compare this characteristic of a wide-band PM wave with that of a wide-band FM wave.
0.1 Figure0.2
0.4 the0.6
0.8 1.0
2 of a closed-loop
4
6
8 10
20
4.15
4.19 shows
block
diagram
feedback
system
for the40
carrier-frequency
Low-pass
Frequency
Crystal
Mixer
Modulation modulator.
index, !
stabilization of a wide-band
frequency
The
voltage-controlled
oscillator
filter
discriminator
oscillator shown in
the figure constitutes the frequency modulator. Using the ideas of mixing (i.e., frequency transFIGURE 4.9 Universal curve for evaluating the one percent bandwidth of an FM wave.
lation) (described in Chapter 3) and frequency discrimination (described in this chapter), disFIGURE 4.19 Problem 4.15
cuss how the feedback system of Fig. 4.19 is capable of exploiting the frequency accuracy of the
stabilize the
voltage-controlled
oscillator.
value over crystal
which oscillator
the carriertofrequency
actually
deviates. This
means that the small values
of modulation index b are relatively more extravagant in transmission bandwidth than
the larger values of b.
VoltageMessage
controlled
signal m(t)
oscillator
! ARBITRARY MODULATING WAVE
Frequency-stabilized
FM wave
Consider next the more general case of an arbitrary modulating wave m1t2 with its highest frequency component denoted by W; that is, W denotes the message bandwidth. We now
Low-pass
Frequency
Crystal
Mixer
have a more difficult situation to deal
way of tackling it is
to seek a worst-case
filterwith. One discriminator
oscillator
evaluation of the transmission bandwidth. Specifically, the bandwidth required to transmit
an FM wave generated
an arbitrary
modulating wave is based on a worst-case toneFIGURE by
4.19
Problem 4.15
modulation analysis. We first determine the so-called deviation ratio D, defined as the ratio
of the frequency deviation ¢f, which corresponds to the maximum possible amplitude of
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Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes
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