2 AMPLITUDE MODULATION
A system of modulation in which the envelope of the transmitted wave contains a component similar to
the waveform of the signal to be transmitted.
The envelope of the modulated carrier has the same shape as the message waveform, achieved by adding
the translated message that is appropriately proportional to the unmodulated carrier.
2.1 DOUBLE-SIDEBAND FULL-CARRIER MODULATION
2.1.1 Single-Tone Modulation
2.1.1.1 Time Domain Representation
A single-tone modulating signal may be represented as x(t) = Am cos m t. It has a waveform shown in
Figure 2.1.
1.2
0.8
0.4
0
0
200
400
600
800
1000
-0.4
-0.8
-1.2
Figure 2.1 A 1-kHz single-tone modulating signal
A carrier wave is a sinusoidal voltage or current generated in a transmitter and is subsequently modulated by a
baseband or modulating signal. The carrier signal can have the equation c (t) = Ac cos c t. Its waveform is
given in Figure 2.2.
1
0.8
0.6
0.4
0.2
0
-0.2 0
20
40
60
80
100
120
140
160
180
200
-0.4
-0.6
-0.8
-1
Figure 2.2 A 10-kHz carrier wave
The waveform for the standard AM or double-sideband full-carrier AM is shown in Figure 2.3. It is described by
the equation, xc (t) = Ac [ 1 + m cos m t ] cos c t, where m is the modulation index.
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1.5
0
0
1000
-1.5
Figure 2.3 A DSBFC AM signal
1.5
Ac
Amax
0
0
1000
Amin
-1.5
Figure 2.4 An AM signal with modulating and carrier signals shown
From Figure 2.4, the amplitude of the waveform may be represented by the equation
A = Ac + Am cos m t
A = Ac [ 1 + m cos m t ]
where m =
Am 1
.
Ac
Hence, the AM equation for single-tone modulation is
xc (t) = Ac [ 1 + m cos m t ] cos c t
1
m
Am  Amax  Amin 

Ac
Amax  Amin
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2.1.1.2 Frequency Domain Representation
The equation of xc (t) can be rewritten as
xc ( t )  Ac cos  c t 

mAc
cos ( c   m ) t  cos ( c   m ) t
2

since 2 cos A cos B = cos (A + B) + cos (A - B). Hence, the frequency domain representation of xc (t) is shown in
Figure 2.5, where fc - fm is the lower side frequency and fc + fm is the upper side frequency.
Ac
2
Ac
2
mAc
4
-fc - fm
mAc
4
-fc
mAc
4
-fc + fm
fc - fm
mAc
4
fc
fc + fm
Figure 2.5 AM spectrum of single-tone modulation
2.1.1.3 Power Calculations
Average Carrier Power, Pc
Pc 
1 VC2 1 2
 IC R
2 R
2
Average Side Frequency Power, PS B
2
PS
B
1  m VC  1
1
1
m2
 

 mVC 2   mIC  2  PC

2  2  R 8R
8
4
Total AM Power, PT
PT = Pc + 2 PS B
 m2 
PT  PC 1  
2

2.1.1.4 Current Calculations
The total power and carrier power can be represented by the following equations:
PT  I T2 R
PC  IC2 R
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IC is the unmodulated carrier current and IT is the total, or modulated, current of an AM transmitter. These
currents are usually applied or measured at the antenna. Hence, R is the antenna resistance.
PT IT2  m2 

 1  
PC IC2 
2
IT  IC 1 
m2
2
2.1.2 Multi-Tone Modulation
xc (t) = Ac [ 1 + m1 cos m1 t + m2 cos m2 t + m3 cos m3 t + ... ] cos c t
mA
xC ( t )  AC cos C t  1 C  cos( C   m1 ) t  cos( C   m1 ) t 
2

m2 AC
mA
cos( C   m2 ) t  cos( C   m2 ) t   3 C  cos( C   m3 ) t  cos( C   m3 ) t   ...

2
2
The total transmitted power is
m12
m22
m32
PT  PC 
PC 
PC 
PC  ...
2
2
2
 mt2 
PT  PC 1 

2 

where mt2  m12  m22  m32  ...
To prevent overmodulation,
mt  1
m1 , m2 , m3 , ...  1
In general,
xc (t) = Ac [ 1 + m x(t) ] cos c t
where
m = modulation index
x(t) = modulating or baseband signal
xc (t) = modulated or transmitted signal
We observe that the envelope of xc (t) has the same shape as the baseband signal x(t) provided two
requirements are satisfied.
1) The amplitude m x(t) is always less than unity or
 m x(t)  < 1
for all values of t. This condition ensures that the function 1 + m x(t) is always positive.
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When not satisfied, the carrier wave becomes overmodulated, resulting in carrier phase reversals and
therefore, envelope distortion. It is, therefore, apparent that by avoiding overmodulation, a one-to-one
relationship is maintained between the envelope of the AM wave and the modulating wave for all values of t.
2) The carrier frequency fc is much greater than the highest frequency component of x(t) or
fc >> W
where W is the message bandwidth. If this condition is not satisfied, an envelope cannot be visualized
satisfactorily.
The envelope has the shape of x(t) providing that a) the carrier frequency is much greater than the rate of
variation of x(t), otherwise, an envelope cannot be visualized, and b) there are no phase reversals in the
modulated wave, that is, the amplitude Ac[1 + m x(t)]does not go negative.
Preserving the desired relationship between envelope and message thus requires fc >> W and m  1.
With m = 1, known as 100 percent modulation, the modulated amplitude varies between 0 and 2 Ac.
Overmodulation, m > 1, results in carrier phase reversals and envelope distortion.
2.1.3 AM Spectrum
xc (t) = Ac [ 1 + m x(t) ] cos c t
If the spectrum of the baseband signal is as shown in Figure 2.6, then, similar to Sec. 2.1.1.2 on Frequency
Domain Representation, the spectrum of the AM wave is shown in Figure 2.7.
X(f)
0
-W
W
Figure 2.6 Baseband spectrum
XC(f)
mAC
2
-fC -W -fC
-fC +W
0
fC -W
fC
fC +W
Figure 2.7 AM spectrum
PROPERTIES OF THE AM SPECTRUM
1) There is symmetry about the carrier frequency, with the amplitude being even. For positive frequencies, the
portion of the spectrum lying above fc is referred to as the upper sideband, whereas the symmetrical portion
below fc is referred to as the lower sideband. For negative frequencies, the upper sideband is represented by the
portion above -fc and the lower sideband by the portion above -fc. Hence the designation double sideband
amplitude modulation. The condition fc >> W ensures that the sideband do not overlap.
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2) The transmission bandwidth, BT, for an AM wave is exactly twice the message bandwidth, W, that is,
BT = 2W
The result points out that AM is not attractive when one must conserve bandwidth. After all, the message could
be sent at baseband with one-half of the AM bandwidth.
2.2 DOUBLE-SIDEBAND SUPPRESSED-CARRIER MODULATION (DSBSC)
From Figure 2.7, it is clear that the carrier wave is completely independent of the information-carrying
signal or baseband signal x(t). This means that the transmission of the carrier wave represents a waste of power,
which points to a shortcoming of AM. That is, only a fraction of the total transmitted power is affected by x(t).
To overcome this shortcoming, we can suppress the carrier component from the modulated wave. This results in
a double-sideband suppressed carrier modulation, or DSB for short.
Dropping the carrier term and the now meaningless modulation index, we have the equation defining a
DSB signal as
xc (t) = Ac x(t) cos c t
Unlike AM, the modulated wave is zero in an absence of modulation. That is, if x(t) is 0, then xc (t) is 0. Since
the carrier has been suppressed, the average transmitted power is
PT  2 PSB 
m2
PC
2
The DSB spectrum is simply the translated message spectrum, as shown in Figure 2.8. The frequency
spectra of AM and DSB are similar, but the time-domain picture is another story. Figure 2.9 shows the DSB
waveform using single-tone modulation.
The modulated wave undergoes a phase reversal whenever the baseband signal x(t) crosses zero. The
DSB envelope does not have the same shape as the message since negative values of x(t) are reflected in the
carrier phase. Full recovery of the message, therefore, entails an awareness of these phase reversals and an
envelope detector would not be sufficient. Stated in another way, DSB involves more than just “amplitude”
demodulation.
XC(f)
mAC
2
-fC -W -fC
-fC +W
0
fC -W
fC
fC +W
Figure 2.8 DSB spectrum
DSB conserves power but requires complex detection circuitry. Conversely, AM demodulation is simply
envelope detection but at the cost of greater transmitted power.
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1
0
0
1000
-1
Figure 2.9 DSB waveform
2.3 SINGLE-SIDEBAND SUPPRESSED-CARRIER MODULATION (SSBSC)
AM and DSB are wasteful of bandwidth. They require a transmission bandwidth equal to twice the
message bandwidth. The upper and lower sidebands are related to each other by virtue of their symmetry about
the carrier frequency; that is, given the amplitude and phase spectra of either sidebands, we can uniquely
determine one from the other. This means that insofar as the transmission of information is concerned, only one
sideband is necessary, and if the carrier and the other sideband are suppressed at the transmitter, no information
is lost.
The average transmitted power of SSB is
PT  PSB
m2

PC
4
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2.4 FREQUENCY-DOMAIN REPRESENTATION
2.5 TIME-DOMAIN REPRESENTATION OF SINGLE-TONE MODULATION
1
0
0
1000
-1
21
1
0
0
1000
-1
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EXERCISES
1) A broadcast AM transmitter radiates 50 kW of carrier power. What will be the radiated power at 85 percent
modulation?
2) A certain AM transmitter has an unmodulated RF carrier power of 1 kW. Determine the total power, the
power in each sideband, and the peak power for each of the following modulation percentages using single-tone
modulation: a) 25%, b) 50%, and c) 100%.
3) When the modulation percentage is 75, an AM transmitter produces 10 kW. How much of this is the carrier
power? How much is the power per sideband? What would be the percentage power saving if the carrier and
one of the sidebands were suppressed before transmission took place?
4) At what depth of modulation can an AM transmitter save 90% of power if the carrier and one sideband are
suppressed?
5) The antenna current of an AM transmitter is 8 A when only the carrier is sent, but it increases to 8.93 A when
the carrier is sinusoidally modulated. Determine the antenna current when the depth of modulation is 0.8.
6) When a broadcast AM transmitter is 50% modulated, its antenna current is 12 A. What will the current be
when the modulation depth is increased to 0.9?
7) The input impedance at the base of a certain 10 kW commercial broadcasting station antenna is 50 resistive.
For single-tone modulation, the rms ammeter at the base of the antenna reads 16 A. Determine the percentage of
modulation.
8) A certain transmitter radiates 9 kW with the carrier unmodulated, and 10.125 kW when the carrier is
sinusoidally modulated. Calculate the modulation index. If another sine wave, corresponding to 40 percent
modulation, is transmitted simultaneously, determine the total radiated power.
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9) The antenna current of an AM broadcast transmitter modulated to a depth of 40% by an audio sine wave, is
11 A. It increases to 12 A as a result of simultaneous modulation by another audio sine wave. What is the
modulation index due to this second wave?
10) A 360-W carrier is simultaneously modulated by two audio waves with modulation percentages of 55 and
65, respectively. What is the total sideband power radiated?
11) The output current of a 60% modulated AM generator is 1.5 A. To what value will this current be if the
generator is modulated additionally by another audio wave whose modulation index is 0.7? What will be the
percentage power saving if the carrier and one of the sidebands are now suppressed?
12) A 60 dBm AM transmitter is to be modulated simultaneously by two audio waves of indices 0.5 and 0.7.
What is the total current if the antenna resistance is 75?
13) An SSB transmission contains 80 kW. It is required that the system be replaced by a standard amplitudemodulated signal with the same power content. Determine the power content of each of the sidebands in the new
system if the percentage modulation is 80%.
14) The antenna current of an AM broadcast transmitter modulated to a depth of 38.5% by an audio wave is
10.5 amperes. It increased by 4% as a result of simultaneous modulation caused by another wave. Determine the
percent modulation caused by the second wave.
15) At a certain depth of modulation, the power used is 22% of the total power when the carrier is suppressed. If
another simultaneous wave is added and the carrier and one of the sidebands are suppressed, the power saved is
85%. What are the modulation indices?
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