EC312 Lesson 12: Intro to Modulation
Objectives:
(a) Define baseband signal and describe associated considerations with regards to a communications system.
(b) Define modulation.
(c) Discuss the role of modulation in a signal transmission and the methods of modulating a sinusoidal carrier.
(d) Determine the sidebands, bandwidth, modulation index, and percentage modulation of an AM signal.
(e) Determine when overmodulation exists for an AM signal.
(f) Describe the relationship of sideband power to total AM signal power.
In Lesson 11, we defined a communications system and learned that the Wireless section of EC312 will focus
on communications in which the channel is free space.
We also learned that in order to send any information through a communications systems it must be in the form
of a signal (which is the name given to the function that conveys our information), and that if our channel is free
space, it means we’re dealing with the Electromagnetic Spectrum (EM). Finally, we learned that signals can be
represented as a function of either time or frequency.
1. Baseband Signals
If you’re sitting in your EC312 classroom in the basement of Rickover Hall and you speak quietly to the student
next to you, will that person be able to hear you? Will you be heard across the room? How about at the end of
that loooong Rickover passageway? Does anyone in Bancroft have a chance of hearing what you’re grumbling
about in your EC312 classroom? Why not? Recall from our last lesson that the sound of your voice is
comprised of frequencies in the EM spectrum, and the Voice Frequency band of the spectrum ranges from 300
Hz to 3 kHz. So why can’t your voice be heard down the hallway in Rickover or in Bancroft Hall? Your voice
frequencies don’t travel that far – they attenuate over a distance. Your voice is your signal which is conveying
the information you want to say, and is what is known as a baseband signal.
Baseband signals are unmodified signals at their original frequencies. Before signals can be transmitted, they
must be converted to a format that is compatible with the medium.
 Microphones convert acoustic pressure waves (sound) into voltages.
 This is explained a bit more in the first two minutes of the “Cool Edit” tutorial, located in the
Resources Section of the EC312 webpage
 Video cameras convert light into analog or digital voltage signals.
 Computer inputs (keyboard or mouse) are converted to binary electrical signals.
You can’t hear your baseband voice or music played in Rickover in Bancroft Hall, yet you talk to people or
listen to music in the Hall where the source of the voice or music is physically located elsewhere all the time.
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How does that really work? Is it as simple as converting the pressure waves of your voice to an electrical signal
and plugging it into an antenna to transmit? (Easy, right? Cell towers and radio towers are just antennas, after
all.) Let’s try an example and see…
Example
To transmit a signal with an antenna, the size of the antenna must be at least a tenth of a wavelength.
What is the approximate size of the antenna required to transmit the sound of a tuning fork (A = 440 Hz)?
Solution:
Wait! To transmit that lousy tuning fork my antenna
needs to be at least 68 km? That’s over 42 miles! We’d
need an antenna that extends into the Mesosphere for
that. Clearly, that’s not going to work.
Well, I know that if I want to listen to the Navy game on
the radio (because for some reason I avoided the
mandatory fun), I can tune in to 1430 WNAV.
Since I was paying close attention to our discussion of
the EM spectrum, I know that the 1430 actually means
1430 kHz. So what size antenna does WNAV use?
Example
What is the wavelength of an AM radio station whose frequency is 1430 kHz?
Solution:
What is the approximate antenna size if the station uses an antenna that is half the wavelength?
Solution:
105 meters? Okay, that’s still big –
it’s about 115 yards - almost an
entire football field… But at least I
don’t need an antenna that reaches
into outer space now. In reality,
WNAV’s antenna, pictured at right,
is 117 meters. We’re close!
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The purpose of those two examples was to demonstrate that we need to somehow get our baseband information
to a higher frequency in order to be able to transmit it across our channel. The higher frequency gives us a
reasonable antenna size, with the added benefit that signals attenuate less quickly at higher frequencies. We can
transmit baseband information at higher frequencies using a process called modulation.
2. Modulation
To overcome limitations of the communications channel and permit multiple access, information signals are
impressed upon a higher-frequency carrier signal for transmission. This process is called modulation. Now
we’re dealing with two signals:
1. Original (“Baseband” Signal)
- frequency is too low to transmit
2. Higher frequency (“Carrier”) Signal
- we can transmit this, so we use it to carry our
information
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2
vm = Vm sin (2 fm t + θ)
vc = Vc sin (2 fc t + θ)
1.5
1.5
1
1
0.5
Voltage (V)
Voltage (V)
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
0
Information signal vm
0.001
0.002
(for tuning
fork f0.006
= 440
Hz 0.008
)
0.004
0.005
0.007
m
0.003
0.009
Carrier signal vc (carrier frequency fc = 500 kHz)
-2
0
0.001
0.01
0.002
0.003
0.004
0.005 0.006
T ime (sec)
0.007
0.008
0.009
0.01
T ime (sec)
Mathematically, the sine wave representing the higher-frequency carrier is given by:
vc = Vc sin (2 fc t + θ)
We can vary any of these three variables in accordance with the low-frequency information signal to achieve
modulation.
 varying Vc (amplitude)  amplitude modulation (AM)

varying fc (frequency)  frequency modulation (FM)

varying  (phase angle)  phase modulation (PM)
A modulator is a component within the transmitter that mixes the baseband intelligence signal with a higherfrequency carrier.
The basic types of modulation are, in fact:
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
Amplitude Modulation (AM) – Varying the voltage Vc of the carrier with the info signal.

Frequency Modulation (FM) – Varying the frequency fc of the carrier with the info signal.
1
Voltage (V)
0.5
0
-0.5
-1
0
0.5
1
1.5
2
T ime (msec)
2.5
2
T ime (msec)
2.5
3
3.5
4
-3
x 10
1
Voltage (V)
0.5
0
-0.5
-1
0

0.5
1
1.5
3
3.5
4
-3
x 10
Phase Modulation (PM) – Varying phase angle of the carrier with the info signal.
Since the intention of the “Wireless” section of EC312 is to give you a broad understanding of wireless
communications techniques rather than to make you all communications engineers, we’re only going to go into
more detail with Amplitude Modulation (AM). This is NOT to say that Frequency Modulation and Phase
Modulation are unimportant – they’re very important and very widely used, and we will see their use when we
deal with digital communication, but AM is the easiest modulation to analyze,visualize and demonstrate.
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3. Amplitude modulation (AM)
In amplitude modulation, the information signal varies the amplitude of the carrier sine wave. For simplicity,
consider a sine wave information signal, vm (a 440 Hz tuning fork).
vm  Vm sin 2 f mt
2
1.5
1
Voltage (V)
0.5
0
-0.5
-1
-1.5
-2
0
0.001
0.002
0.003
0.004
0.005
0.006
T ime (sec)
0.007
0.008
0.009
0.01
Information signal vm (for tuning fork fm= 440 Hz )
The AM wave (vAM) is the product of the carrier and the intelligence signal (with an added offset, Vc) and is
given by:
vAM  (Vc  Vm sin 2 f mt )sin 2 f ct
The “envelope” (shown below in orange) of the modulating signal varies above and below the peak carrier
amplitude, Vc. In order for AM to convey the original signal accurately and prevent distortion, the modulating
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signal amplitude (Vm) must be less that the carrier signal amplitude (Vc). This relationship between these two
values is called the Modulation Index (m).
Modulation Index
The relationship between the modulating signal amplitude, Vm , and the carrier amplitude, Vc , is expressed as a
ratio called the modulation index, m, defined as:
m
Vm
VC
Sometimes m is expressed as a percentage: percent modulation = m x 100%
We can also determine the modulation index m from the maximum and minimum values of the envelope of v AM
Vmax  Vmin
2
V  Vmin
Vc  max
2
V
V  Vmin
m  m  max
Vc Vmax  Vmin
Vm 
When Vm is not less than Vc (or m > 1), overmodulation
occurs. Overmodulation is distortion of the signal and so it is
hard to recover at the receiver, as is depicted to the right.
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Example
If a carrier signal vc=9sin(25000t) V is modulated by a sine wave vm=7.5sin(2440t) V, what is the percentage
modulation of the resulting signal?
Solution:
Now that we have a basic understanding of how Amplitude Modulation works in the time domain, let’s look at
AM in the frequency domain.
4. AM in Frequency Domain
Recall the equation for the amplitude modulated waveform is given by: vAM  Vc  Vm sin  2 f mt  sin  2 f ct  .
We already know the frequency domain representations of the modulating signal (vm) and the carrier signal (vc),
but what does the amplitude modulated signal looks like in the frequency domain?
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Applying the trig identity for product of two sine waves…
Wait – hold it - WHAT identity? This one!: sin A sin B 
cos( A  B) cos( A  B)

2
2
Are you back with me? Let’s try again: Applying the trig identity for product of two sine waves to
vAM  Vc  Vm sin  2 f mt  sin  2 f ct  results in:
v AM  Vc sin 2 f ct 
fc (= 5 kHz)
fc - fm (= 4.560 kHz)
fc + fm (= 5.440 kHz)
0.5
0.4
Voltage (V)
Now there are 3 frequencies:
Vm
V
cos 2 ( f c  f m )t  m cos 2 ( f c  f m )t
2
2
And the amplitudes at fc - fm and fc + fm are halved.
0.3
0.2
0.1
This means the resulting frequency domain looks like this:
0
0
1000
2000
3000
Frequency (Hz)
4000
5000
6000
What happened? The process of modulating a signal creates sidebands. For an AM signal, sidebands have half
the amplitude of the original message and are reflected about the carrier.
What does this mean for the AM signals bandwidth? It increased! Why is this a concern? Bandwidth is the #2
limiting factor in communications systems… and we’re going to want to push as much information as we’re
able though the minimum amount of bandwith possible.
Let’s look at a slightly more complex example – the single frequency component of tuning fork just isn’t
driving this bandwidth issue home.
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How does Amplitude Modulation impact bandwidth? It doubles the bandwidth of the original signal.
BW = fUSB – fLSB
Where:
and
fUSB is the frequency of the Upper Sideband (fUSB = fc + fm)
fLSB is the frequency of the Lower Sideband (fLSB = fc - fm)
Example
A standard AM broadcast station (980 AM) transmits the sound of a two chime doorbell (A=440 Hz, F= 349
Hz). Show the frequency domain representation transmitted signal & determine the bandwidth. Which of the
two chimes determines the bandwidth?
Solution:
5. AM Power
The AM signal consists of the carrier and the two sidebands, and each of these produces power. The
total transmitted power PT is the sum of the carrier power Pc and power in the two sidebands PLSB and PUSB.
PT  Pc  PLSB  PUSB
Recall that the average power dissipated by resistor R with a sinusoidal source of amplitude Vpk is given
P
2
Vrms
R
V

pk
/ 2
R
9

2

Vpk2
2R
Since the vAM is composed of three sinusoids
vAM  Vc sin 2 fct 
Vm
V
cos 2 ( fc  f m )t  m cos 2 ( fc  f m )t
2
2
the total average power dissipated by the antenna with resistance R is given as:
PT  Pc  PLSB  PUSB


Vc / 2
 
2
Vm / 2 2
R
Vm2 Vm2



2 R 8R 8R
R
 
2
Vm / 2 2

2
R
Vc2
Remembering that the modulation index is m = Vm /Vc we can write
PT
2
2
2
Vc   mVc   mVc  Vc2  m2 m2 





1

2R
8R

2R 
8R
4

4 
The common term is just the carrier power, thus the total power can also be written
 m2 
PT  PC 1 

2 

In reality it is difficult to determine AM power by measuring the output voltage. However, antenna current is
easy to measure and output power can be expressed
PT 
IT2 R
where IT  I c
AM Power Efficiency
Sideband power is maximized by setting m = 1.
 m2 
PT  Pc 1 

2 

Pc
m2
Pc
4
10
m2
Pc
4
m2
1
2
For m = 1, what percentage of the total power is dedicated to the sidebands?
At maximum modulation, the sideband power is at most 33% of the total transmitted power.
Percentage of total power (% PT)
100
80
60
Power in carrier ( Pc)
40
20
Power in sidebands ( PSB)
0
100
90
80
70
60
50
40
30
20
10
0
Percentage modulation (% m)
Two-thirds of the power is wasted in the carrier. Further, 100% modulation only occurs at peaks in the
modulating signal, thus the average sideband power is considerably worse than the ideal. Notwithstanding
these problems, AM is still widely used because it is simple and effective.





AM broadcast radio
CB radio
TV broadcasting
Air traffic control radios
Garage door opens, keyless remotes
Example
An AM signal is 100% modulated (modulation index = 1). The carrier power is 100 W.
(a) What is the total power in the sidebands?
(b) What is the total power of the AM signal?
(c) What is the percentage of power in the sidebands?
What’s the point? AM is by no means the only form of modulation (though it’s probably the easiest to work
through and visualize). The intent of this course is not to make you all communications engineers, but you do
need to have enough background in modulation to understand the implications (esp. to bandwidth) moving
forward. You’ll see this again in a few lessons, with digital applications!
LCDR Jennie Wood, CDR Hewitt Hymas
Help us improve these notes! Send comments, corrections and clarifications to jwood@usna.edu
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