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LAB 2. ANGLE
MODULATION
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Objective
In this part, our objective is to study
ANGLE MODULATION in detail. We consider two different angle modulation methods:
frequency modulation (FM), and phase modulation (PM). The phase-locked loop is
provided to demodulate the angle-modulated signal.
Introduction
In angle modulation, the idea is
to transmit the message signal in the frequency or the phase of the carrier
signal. Angle modulation is more complex to implement and much more difficult
to analyze compared to the amplitude modulation schemes. In many cases only
approximate analysis can be performed. Also, significantly more bandwidth is
usually required with angle modulation. The major benefit of the
angle-modulated systems is high degree of noise immunity. That is, these
systems trade-off bandwidth for high-noise immunity.
2-1 FREQUENCY
MODULATION
In frequency modulation (FM), the
frequency of the carrier is varied depending on the message signal.
The frequency-modulated signal can
be written as
where 
with m(t) being the message signal and 
is the frequency
deviation constant.
The maximum-frequency deviation in
an FM system is given by
and the modulation index is
defined as
where W
denotes the bandwidth of the message signal m(t).
The detailed treatment of the
spectral characteristics of an angle modulated signal for a general
deterministic message signal m(t) is difficult. The bandwidth of the
frequency-modulated signal is infinity. However there exists an approximate
relation for the effective bandwidth of the modulated signal that is called
Carson’s rule. Carson’s rule states that 
is the approximate
bandwidth required to transmit the FM modulated signal where 
is the modulation
index and 
is the bandwidth of the message signal.
The demodulation of an FM
modulated signal involves finding the instantaneous frequency of the modulated signal
and then subtracting the carrier frequency to recover the message signal.
We can use the FM discriminator as a FM demodulator. The derivative FM modulated signal is given by
.
where
is always positive since typically
.
Therefore, the envelope of the
differentiated FM modulated signal is proportional to 
and can be detected
by using an envelope detector. The block diagram of the FM discriminator is
shown below.
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where the output signal can be easily used to recover m(t).
Let us consider the use of a PLL as a method of a FM demodulator.
The input to the PLL is the frequency-modulated signal. Here, we assume the
absence of noise in this discussion.
where 
for FM systems.
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The voltage-controlled oscillator (VCO) generates a
sinusoid of a fixed frequency, in this case the carrier frequency 
, in the absence of an input control voltage.
Suppose that the control voltage to the VCO is the
output of the loop filter, denoted as v(t). The instantaneous frequency of the
VCO is
The VCO output may be given as
where 
.
The phase comparator consists of a multiplier and a
filter that rejects the signal component centered at 2
. Therefore, the output is expressed as
where the
difference
constitutes the phase error.
Under this condition, we may express the phase error
with the Fourier transform as
The corresponding equation for the control voltage to
the VCO is
Suppose that we design G(f) such that
>> 1
in the frequency band |f| < W as the message
signal bandwidth.
We have
or, equivalently,
Because the control voltage of the VCO is the
proportional to the message signal, 
is the demodulated
signal.
2-2 PHASE
MODULATION
In phase modulation (PM) systems,
the phase of the carrier is changed according to the variations in the message
signal. Similar to the FM, the phase modulation generally expands the bandwidth
so that the effective bandwidth of the PM modulated signal is usually many
times the bandwidth of the message signal.
A PM modulated signal can be
written as
where 
, 
is phase deviation
constant, and m(t) is the message signal.
The maximum-phase deviation in a
PM system is given by
The modulation index is defined as
.
The demodulation of a PM signal is
performed by finding the phase of the signal and then recovering m(t).
Let us consider the use of a PLL as a method of a PM
demodulator. The input to the PLL is the phase-modulated signal. Here, we
assume the absence of noise in this discussion.
where 
, and 
is the phase
deviation constant.
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The VCO generates a sinusoid of a fixed frequency, in
this case the carrier frequency 
, in the absence of an input control voltage.
Suppose that the control voltage to the VCO is the
output of the loop filter, denoted as v(t). The instantaneous frequency of the
VCO is
The VCO output may be given as
where 
.
The phase comparator consists of a multiplier and a
filter that rejects the signal component centered at 2
. Therefore, the output is expressed as
where the
difference
constitutes the phase error.
Under this condition, we may express the phase error
with the Fourier transform as
The corresponding equation for the control voltage to
the VCO is
Suppose that we design G(f) such that
>> 1
in the frequency band |f| < W as the message
signal bandwidth.
We have
or, equivalently,
.
Because the control voltage of the VCO is the
proportional to the message signal, 
is the demodulated
signal.
2-3
NARROWBAND ANGLE
MODULATION
For a special case, suppose that 
. The angle-modulated signal can be represented by
where 
for the PM method,
and 
for the FM method.
It almost looks like a conventional amplitude
modulated signal. It is called as a narrowband angle modulated signal because
its required bandwidth is narrower than the bandwidth of the general angle
modulated signal. Its required
bandwidth is twice the bandwidth of the message signal. Narrowband angle
modulated signals do not provide better noise immunity compared to conventional
amplitude modulated signals. That is the reason why it is not used for
communication systems in practice. But the narrowband angle modulated signal
can be used in generating a wideband angle modulated signal.
First, let us see the block diagram of narrowband
angle modulator.
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[Narrowband angler modulator] |
When we generate a wideband angle modulated signal, a
narrowband angle modulated signal enters the frequency multiplier first. The output
of the frequency multiplier is already a wideband angle modulated signal. It is
represented as
.
However it may not have the desired carrier
frequency. Both a mixer and a bandpass filter are used to convert the output of
the frequency multiplier into the signal having the desired carrier frequency.
For example, assume that a down converter is used with the frequency, 
, of the local oscillator and a bandpass filter located at
the desired carrier frequency. The wideband angle modulated signal is written
by
.
Because we can control the values of 
and 
, the wideband angle modulated signal having the desired
carrier frequency can be generated.
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[Wideband angle modulator with using a narrowband
angle modulated signal] |
Problem 2-1. Frequency modulation with a sinusoidal
message signal.
Let 
.
Let 
, and Carrier signal is 
.
Assume that the modulation index 
is 6.
a) Find the
frequency deviation constant mathematically.
b) Find the
maximum frequency deviation mathematically.
c) Show the
modulated signal in time domain.
Show the distance among
peaks in the magnitude of the modulated signal.
(Hint. Show
the distance among peaks by zooming in graphs)
d)
Assume the modulation index is equal to 1.
Find the frequency deviation
constant mathematically.
Show the modulated signal in time
domain.
Show the distance among peaks in the magnitude of the modulated signal.
(Hint. Show
the distance among peaks by zooming in graphs)
e) Find the
power content of the carrier signal.
Problem 2-2. Frequency modulation with a sinc signal.
Let 
.
The carrier signal is 
.
The modulation index 
is assumed to be 6.
a)
Find u(t) mathematically.
b)
Find the frequency deviation constant mathematically.
c)
Find the maximum frequency deviation mathematically.
d)
Show the modulated signal in both frequency domain and time
domain.
Problem 2-3. Phase modulation with a sinusoidal
message signal.
Let 
.
Let the carrier signal be 
.
The modulation index 
is assumed to 6.
a) Find the
phase deviation constant mathematically.
b) Find the
maximum phase deviation mathematically.
c) Show the
modulated signal in time domain.
Show the distance among
peaks in the magnitude of the modulated signal.
(Hint. Show
the distance among peaks by zooming in graphs)
d)
Assume the modulation index is equal to 3,
Find the phase deviation constant
mathematically.
Show the modulated signal in time
domain.
Show the distance among peaks in the magnitude of the modulated signal.
(Hint. Show
the distance among peaks by zooming in graphs)
Problem 2-4. Phase modulation with a sinc signal.
Let 
.
The carrier signal is assumed to 
.
The modulation index 
is assumed to 6.
a)
Find u(t) mathematically.
b)
Find the phase deviation constant.
c)
Find the maximum phase deviation.
d)
Show the modulated signal in both frequency domain and time
domain.
Quiz.
1.
What is the major benefit of an FM system compared with an
amplitude modulation?
2.
In FM systems, let m(t) be
.
Find the
modulation index and u(t) mathematically.
3.
In quiz 2, show the relationship between the amplitude of the
message signal and the effective bandwidth.
4.
In FM systems, let the
message signal have a bandwidth of 600Hz. The modulation index is assumed to 6.
What is the effective bandwidth?
5.
As the frequency of the message signal, fm, increases in FM
systems, what is the following effects?
(Choose all the correct ones)
a)
Decrease the number of harmonics in the bandwidth of the
modulated signal
b)
Increase the number of harmonics in the bandwidth of the
modulated signal.
c)
Remain the number of harmonics in the bandwidth of the
modulated signal constant.
d)
Decrease the spacing between the harmonics.
e)
Increase slightly the
bandwidth.