Chapter4

4 Amplitude Modulation (AM)

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4.2 Double-Sideband Suppressed Carrier AM (DSB-SC) . . . . . . . . . . . 6
4.2.1 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2.2 Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Pilot Carrier . . . . . . . . . . . . . . . . . . . . . . . . . 11
Phase Locked Loop . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Double-Sideband Large Carrier AM . . . . . . . . . . . . . . . . . . . 15
4.3.1 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.2 Carrier and Sideband Power in AM . . . . . . . . . . . . . . . . 18
4.3.3 Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 Quadrature AM (IQ) . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.5 Single-Sideband AM (SSB) . . . . . . . . . . . . . . . . . . . . . . . 22
4.5.1 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5.2 Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.6 Vestigial-Sideband AM (VSB) . . . . . . . . . . . . . . . . . . . . . . 29
4.6.1 Video Transmission in Commercial TV Systems . . . . . . . . . . 31

Dr. Tanja Karp 1

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Dr. Tanja Karp 2

4.1 Introduction
Modulation: Process by which a property or a parameter of a signal is varied in proportion
to a second signal.
Amplitude Modulation: The amplitude of a sinusoidal signal with ﬁxed frequency and
phase is varied in proportion to a given signal.

Purpose:
• Adaptation of the information signal to the transmission channel
• Shift of the information signal to an assigned frequency band

|F(ω)|

ω
−ωc −W W ωc

|Φ(ω)|

ω
−ωc ωc

• Eﬃcient antenna design: size is at least 1/4th of signal wavelength
⇒ antennas for lowpass signals would be too large (f = 3 kHz, λ = 100, 000 m).
Dr. Tanja Karp 3

• Simultaneous transmission of several information signals (e.g. radio broadcasting)

|F(ω)|

ω

|Φ(ω)|

ω
−ωc3 −ωc2 -ωc1 ωc1 ωc2 ωc3

Dr. Tanja Karp 4

2

1.8

1.6

1.4

1.2 POTS

1

0.8

0.6
DSL upstream DSL downstream
0.4

0.2

0
0 50 100 150 200 250 300 350 400 450 ...1Mhz
frequency in kHz

Dr. Tanja Karp 5

4.2 Double-Sideband Suppressed Carrier AM (DSB-SC)
4.2.1 Modulation
Generation of DSB-SC modulated signal:
f (t) φ(t) φ(t) = f (t) cos(ωct)

φ(t): modulated transmit signal
cos(ωct) f (t): modulating signal, real valued
cos(ωct): carrier signal, ωc: carrier frequency in rad/sec
Spectrum of DSB-SC modulated signal:
1 1
φ(t) = f (t) cos(ωct) ◦—• Φ(ω) = F (ω − ωc) + F (ω + ωc)
2 2

Dr. Tanja Karp 6

|F(ω)|

ω
−ωc −W W ωc

|Φ(ω)|
lower sideband upper sideband
0.5|F(0)|
ω
−ωc ωc
2W 2W
• Carrier frequency has to be larger than twice the bandwidth ω ≥ 2W .
• Bandwidth of the modulated signal φ(t) is twice as large as the bandwidth of the
modulating signal f (t).
• No separate carrier is present in φ(t).
• Upper sideband: spectral content for positive frequencies above ωc.
Lower sideband: spectral content for positive frequencies below ωc.
• Information in upper and lower sideband are redundant since Φ(ωc +ω) = Φ∗(ωc −ω),
or equivalently: |Φ(ωc + ω)| = |Φ(ωc − ω)| and ∠Φ(ωc + ω) = −∠Φ(ωc − ω)

Dr. Tanja Karp 7

4.2.2 Demodulation

φ(t) = f (t) cos(ωct) LP filter fˆ(t)

ωp > W
ωs < 2ωc −W
2 cos(ωct)

Before lowpass filtering:
2
φ(t) 2 cos(ωct) = 2f (t) cos (ωct) = f (t) (1 + cos(2ωct))
1 1
F{φ(t) 2 cos(ωct)} = F (ω) + F (ω − 2ωc) + F (ω + 2ωc)
2 2
After lowpass filtering:
ˆ
F (ω) = F (ω)

|F {2 f (t) cos2 (ωct)}|
LP filter

ω
−2ωc −W W 2ωc
2W 2W

The oscillators at the transmitter and receiver have to be synchronized, i.e. the carrier
frequency ωc as well as the phase must be identical (coherent demodulation).
Dr. Tanja Karp 8

Influence of Frequency and Phase Offset:
The oscillator at the receiver has a constant phase offset of θ0 as well as a slightly different
carrier frequency of ωc + ∆ω when compared to the one at the transmitter.

φ(t) = f (t) cos(ωct) LP filter fˆ(t)
ωp > W
ωs < 2ωc −W
2 cos((ωc + ∆ω)t + θ0 )

φ(t) 2 cos((ωc + ∆ω)t + θ0) = 2f (t) cos(ωct) cos((ωc + ∆ω)t + θ0)

= f (t) cos((2ωc + ∆ω)t + θ0) + f (t) cos(∆ωt + θ0)

ˆ
f (t) = f (t) cos(∆ωt + θ0)
1 1
= f (t) exp(j∆ωt) exp(jθ) + f (t) exp(−j∆ωt) exp(−jθ)
2 2

ˆ 1 1
F (ω) = exp(jθ)F (ω − ∆ω) + exp(−jθ)F (ω + ∆ω)
2 2

Dr. Tanja Karp 9

Phase error only (i.e. ∆ω = 0):
ˆ
f (t) = f (t) cos(θ0) ◦—• ˆ
F (ω) = F (ω) cos(θ0)

ˆ
⇒ The recovered signal is scaled by a constant. For θ0 = ±90◦ we have f (t) = 0.

Frequency error only (i.e. θ0 = 0):
ˆ ˆ 1 1
f (t) = f (t) cos(∆ωt) ◦—• F (ω) = F (ω − ∆ω) + F (ω + ∆ω)
2 2
⇒ The recovered signal is still modulated by a cosine signal of low frequency ∆ω .

Dr. Tanja Karp 10

Pilot Carrier

• send a sinusoidal tone whose frequency and phase is proportional to ωc

• sent outside the passband of the modulate signal
• Receiver detects the tone, translates to correct frequency(doubling) and demodulates

Dr. Tanja Karp 11

Example - Commercial Stereo FM Stations

Transmitter

• need to transmit left(L) and right(R) as well as (L+R) for monophonic
• (L+R) occupies 0 − 15kHz
• so does (L-R), so shift up using DSB-SC with ωc = 38kHz
• place pilot tone at 19kHz

Dr. Tanja Karp 12

Receiver

• narrow bandpass ﬁlter at 19kHz and then double to 38kHz
• after demodulation using pilot tone, we have
Left channel = (L + R) + (L − R) = 2L
Right channel = (L + R) − (L − R) = 2R

Dr. Tanja Karp 13

Phase Locked Loop(PLL)

• Pilot Tone Problem -BP filters drift in tuning, bad at rejecting noise
• Solution: Phase Locked Loop(PLL)

• Operation when Voltage Controlled Oscillator(VCO) frequency(ωV CO ) is close to ωc
– low-frequency component of output is proportional to magnitude and sign of phase
difference
– this voltage adjusts ωV CO to keep phase difference a minimum
• Bandwidth of PLL determined by LPF
– Small BW ⇒ good noise rejection but receiver may never lock
– Large BW ⇒ good lock but bad noise rejection

Dr. Tanja Karp 14

4.3 Double-Sideband Large Carrier AM
4.3.1 Modulation

• Reduces complexity of receiver
• Since this type of AM is used in commercial broadcast stations, usually termed AM
• Similar to DSB-SC, except that we incorporate the carrier
– carrier must be larger than the rest of the signal
– ruins low-frequency response of the system, so must not require frequency response
down to 0.

φAM = f (t) cos(ωct) + A cos(ωct)
1 1
ΦAM (ω) = F (ω + ωc) + F (ω − ωc) + πAδ(ω + ωc) + πAδ(ω − ωc)
2 2

Dr. Tanja Karp 15

• if A is large enough signal recovery is done with envelope detection
[A + f (t)] ≥ 0 for all t

Dr. Tanja Karp 16

• Let f (t) = cos(ωmt), we deﬁne m to control the amount of modulation
peak DSB-SC amplitude
m=
peak carrier amplitude

φ(t) = A cos(ωct) + mA cos(ωmt) cos(ωct)
= A[1 + m cos(ωmt)] cos(ωct)

• percentage of modulation for DSB-LC signal with sinusoidal modulation
A(1 + m) − A(1 − m)
%mod = × 100% = m × 100%
A(1 + m) + A(1 − m)

• we call m the modulation index
• in order to detect the signal with no distortion we require m ≤ 1
Dr. Tanja Karp 17

4.3.2 Carrier and Sideband Power in AM

• carrier provides no information so it is just wasted power
• for an AM signal φAM (t) = A cos(ωct) + f (t) cos(ωct) the power is
2
φ2 (t) = A cos2(ωct) + f 2(t) cos2(ωct) + 2Af (t) cos2(ωct)
AM

2
= A cos2(ωct) + f 2(t) cos2(ωct)
2
= A /2 + f 2(t)/2

• so we can express the total power as,
1 2 1 2
Pt = Pc + Ps = A + f (t)
2 2
so that the fraction of the total power contained in the sidebands is
Ps f 2(t)
µ= =
Pt A2 + f 2(t)

Dr. Tanja Karp 18

• so when f (t) = cos(ωmt) we get
1 2 1 1 2 2
φ2 (t)
AM = A + ( )( )m A
2 2 2
m2
µ=
2 + m2

• so for best case, i.e., m = 1, 67% of the total power is wasted with the carrier

4.3.3 Demodulation

• the price we pay for wasted power is a tradeoﬀ for simple receiver design
• receiver is simply an envelope detector

Dr. Tanja Karp 19

4.4 Quadrature AM (IQ)
|F(ω)|

ω
−ωc −W W ωc

|Φ(ω)|
0.5|F(0)|
ω
−ωc ωc
2W 2W

• for real signal f (t), F (ω) = F ∗(−ω)
• using this symmetry we can transmit two signals that form a complex signal with same
bandwidth
• we use two sinusoidal carriers, each exactly 90◦ out of phase
remember, ejωt = cos(ωt) + j sin(ωt)
• transmitted over the same frequency band,

Dr. Tanja Karp 20

φ(t) = f (t) cos(ωct) + g(t) sin(ωct)
2
φ(t) · cos(ωct) = f (t) cos (ωct) + g(t) sin(ωct) cos(ωct)
1 1 1
= f (t) + f (t) cos(2ωct) + f (t) sin(2ωct)
2 2 2
2
φ(t) · sin(ωct) = f (t) cos(ωct) sin(ωct) + g(t) sin (ωct)
1 1 1
= f (t) sin(2ωct) + g(t) − cos(2ωct)
2 2 2

Dr. Tanja Karp 21

4.5 Single-Sideband AM (SSB)

• remember for real f (t), F (−ω) = F ∗(ω)
• a single sideband contains entire information of the signal
• let’s just transmit the upper/lower sideband.

Dr. Tanja Karp 22

4.5.1 Modulation

• one way is to generate DSB signal, and then suppress one sideband with filtering
• hard to do in practice, can’t get ideal filters
• assume no low-frequency information ⇒ no components around ωc
• use heterodyning(frequency shifting), only need to design on sideband filter
• another way is the use of phasing
• assume a complex, single-frequency signal, f (t) = ejωmt with carrier signal f (t) =
ejωct
• multiplying we get φ(t) = f (t)ejωct = ejωmtejωct
• using the frequency-translation property of the Fourier Transform, our spectrum be-
comes

Φ(ω) = 2πδ(ω − (ωc + ωm))

Dr. Tanja Karp 23

• to make the signal φ(t) realizable, we take the R{φ(t)}
jωm t jωc t jωm t jωc t
R{φ(t)} = R{e }R{e } − I{e }I{e }
= cos(ωmt) cos(ωct) − sin(ωmt) sin(ωct)

• So the upper side band is
φSSB+ (t) = cos(ωmt) cos(ωct) − sin(ωmt) sin(ωct)

Dr. Tanja Karp 24

• likewise the lower sideband is
φSSB− (t) = cos(ωmt) cos(ωct) + sin(ωmt) sin(ωct)

• in general we write,
ˆ
φSSB (t) = f (t) cos(ωct) ± f (t) sin(ωct)

ˆ
where f (t) is f (t) shifted by 90◦

Dr. Tanja Karp 25

4.5.2 Demodulation

Synchronous detection, analogous to DSB-SC
Influence of Frequency and Phase Offset:
The oscillator at the receiver has a constant phase offset of θ as well as a slightly different
carrier frequency offset of ∆ω giving
φd(t) = cos[(ωc + ∆ω)t + θ]

Dr. Tanja Karp 26

ˆ
φSSB (t)φd(t) = [f (t) cos(ωct) ± f (t) sin(ωct)] cos[(ωc + ∆ω)t + θ]
1
= f (t){cos[(∆ω)t + θ] + cos[(2ωc + ∆ω)t + θ]}
2
1ˆ
= ± f (t){sin[(∆ω)t + θ] − sin[(2ωc + ∆ω)t + θ]}
2
1 1ˆ
eo(t) = f (t) cos[(∆ω)t + θ] f (t) sin[(∆ω)t + θ]
2 2
Phase error only (i.e. ∆ω = 0):
1 ˆ
eo(t) = [f (t) cos θ f (t) sin θ]
2
To understand this better we re-write the above equation as
1 ˆ jθ
eo(t) = R{[f (t) ± j f (t)]e ]}
2
⇒ So phase error in the receiver oscillator results in phase distortion.

Dr. Tanja Karp 27

Frequency error only (i.e. θ = 0):
1 ˆ
e0(t) = [f (t) cos(∆ω)t f (t) sin(∆ω)t]
2
or
1 ˆ j∆ωt
eo(t) = R{[f (t) ± j f (t)]e }
2
⇒ Demodulated signal contains spectral shifts and phase distortions.

Dr. Tanja Karp 28

4.6 Vestigial-Sideband AM (VSB)
• compromise between DSB and SSB.
• partial suppression of one sideband
1 1
ΦV SB (ω) = [ F (ω − ωc) + F (ω + ωc)]HV (ω)
2 2

• after synchronous detection we have
1 1
Eo(ω) = F (ω)HV (ω + ωc) + F (ω)HV (ω − ωc)
4 4
1
= F (ω)[HV (ω + ωc) + HV (ω − ωc)]
4

Dr. Tanja Karp 29

thus for reproduction of f (t) we require

[HV (ω − ωc) + HV (ω + ωc)]LP = constant

• magnitude can be satisﬁed, but phase requirements are hard to satisfy
• use when phase is not important

Dr. Tanja Karp 30

4.6.1 Video Transmission in Commercial TV Systems

• video requires 4M Hz bandwidth to transmit
• so DSB would require 8M Hz per channel
• use VSB to decrease the needed bandwidth to 5M Hz

Dr. Tanja Karp 31

4.7 Summary
|F(ω)|

ω
−ωc −W W ωc

|Φ(ω)|
0.5|F(0)|
ω
−ωc ωc
2W 2W

Double Sideband-Suppressed Carrier(DSB-SC)

• spectrum at ωc is a copy of baseband spectrum with scaling factor of 1/2
• information is sidebands is redundant
• for coherent detection, we must have same frequency and phase of carrier signal
• detection can be done with pilot tone, PLL

Dr. Tanja Karp 32

Double Sideband-Large Carrier(DSB-LC)

• same as DSB-SC, with an addition of a carrier term
• detection is a simple envelope detector
• Wastes, at best case, 67% of the power in the carrier term
• frequency response at low-frequencies are ruined

Quadrature Amplitude Modulation(QAM)

• eﬃcient utilization of bandwidth

Dr. Tanja Karp 33

• forms a complex signal with two sinusoidal carriers of same frequency, 90◦ out of phase

Single Sideband Modulation(SSB)

• suppress either upper or lower sideband for more eﬃcient bandwidth utilization
• generated by ﬁltering DSB-SC
Dr. Tanja Karp 34

• can also use phasing to cancel the “negative” frequencies
• can use either suppressed carrier, pilot tone, or large carrier AM also

Vestigial Sideband(VSB)

• compromises DSB and SSB
• transmitter and receiver ﬁlters must be complementary, i.e., they must add to a constant
at baseband
• phase must not be important

Dr. Tanja Karp 35

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