Communication Systems I

Part 3 :
Amplitude Modulation

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 1

Introduction
• Communication systems conveys information
through communication channel by shifting
baseband signal to a suitable frequency band.
• Modulation is a process for shifting frequency
range by a carrier signal.
• In amplitude modulation, the amplitude of the
sinusoidal carrier wave is varied according to the
baseband signal.
g (t )  a (t ) cos(2 f c t   (t ))

Amplitude Modulation Phase Modulation

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 2

 Modulation Formats
 (t )
  
t
or
t
 (t )     ( )d
0

AM
E(t )  eˆ A(t ) cos(0t  0 )

PM
E(t )  eˆ A0 cos(0t   (t ))

FM
E(t )  eˆ A0 cos( (t )t  0 )
3

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng

Amplitude Modulation
m(t)
• Sinusoidal carrier wave c(t) 0 for carrier ka
suppressed
c(t )  Ac cos(2 f ct ) ka m(t)
c(t) 1
• Amplitude-modulation wave s(t)
s(t)
s (t )  Ac [1  ka m(t )]cos(2 f ct )
– Ac denote carrier amplitude, in unit: volt
– fc denote carrier frequency
– m(t) denote the baseband signal carries the
message, in unit: volt
– ka is a constant called the amplitude sensitivity, in
(volt)-1 unit
– This absolute maximum value of kam(t) multiplied by
100 is referred to as percentage modulation.
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 4
 Amplitude Modulation – Requirement 1
• Baseband signal m(t )

• AM wave for ka m(t )  1 for all t, ka m(t )  1

– Envelope is positive.

• If ka m(t )  1 at some time t ka m(t )  1

– Over-modulation
– Phase reversed as 1 + kam(t)
crosses 0.
– Envelope distortion
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 5

Amplitude Modulation – Requirement 2

• Suppose m(t) is band-limited to the interval
W  f  W and fc >> W, where W denotes
message bandwidth.

• Fourier transform of AM wave s(t) is

s (t )  Ac [1  ka m(t )]cos(2 f c t )  Ac 1  ka m(t )  12 e j 2 fct  e  j 2 fct 
 Ac 12 e j 2 fct  e  j 2 fct   Ac ka m(t ) 12 e j 2 fct  e  j 2 fct 
thus Ac k A
S( f )   ( f  f c )   ( f  f c )  a c  M ( f  f c )  M ( f  fc )
2 2

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 6

 Amplitude Modulation
Ac k A
S( f )   ( f  f c )   ( f  f c )  a c  M ( f  fc )  M ( f  fc )
2 2

• Note:
– the message of m(t) for negative frequencies extending
from –W to 0 becomes visible for positive (measurable)
frequencies.
– for positive frequencies, signals above carrier fc is
referred as upper sideband (USB), whereas the
symmetric portion is lower sideband (LSB).
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 7

Amplitude Modulation
• Note:
– for negative frequencies, the upper sideband refers
the signals below –fc, and the lower sideband is
defined by the portion above –fc .
– for positive frequencies, the highest frequency
component is fc + W, and the lowest one is fc – W. The
difference is defined as transmission bandwidth, BT,
for the AM signal. Thus BT = 2W.

Complex
Conjugate
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 8
 Example 3.1 Single-Tone Modulation
• Considering a modulating wave m(t )  Am cos( 2f m t )
s (t )  Ac [1   cos( 2f m t )] cos( 2f c t )
•   ka Am : modulation factor, or percentage modulation.
• If Amax and Amin denote the maximum and minimum
values of the
envelope:
Amax Ac (1   )

Amin Ac (1   )

Amax  Amin

Amax  Amin

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 9

Example 3.1 Single-Tone Modulation

s (t )  Ac [1   cos(2 f mt )]cos(2 f c t )
 Ac cos(2 f c t )  12  Ac cos[2 ( f c  f m )t ]  12  Ac cos[2 ( f c  f m )t ]
The corresponding F.T.:
S ( f )  12 Ac [ ( f  f c )   ( f  f c )]
 14  Ac [ ( f  f c  f m )   ( f  f c  f m )]
 14  Ac [ ( f  f c  f m )   ( f  f c  f m )]
Assume the load resistance is 1:
Carrier power = 12 Ac
2

USB power = 18  Ac
2 2

LSB power = 18  2 Ac2

Total Sideband Power 2

Total Power 2  2
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 10
 Switching Modulator
• Assume the diode acts as an ideal switch.
• v1 (t )  c(t )  m(t )  Ac cos(2 f c t )  m(t )
– where m(t )  Ac
v1 (t ), c(t )  0
• v2 (t )  
 0, c(t )  0
v1 (t )
Transfer Function of
An ideal switch
v2 (t )

gT0 (t )

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 11

Switching Modulator
• Assume gT0 (t ) is a periodic pulse train of duty
cycle equal to one-half and period T0  1 / f c .
Thus v2 (t )  [ Ac cos(2 f ct )  m(t )]gT (t ) 0

• Representing gT0 (t ) by its Fourier series

1 2  (1) n 1
gT0 (t )    cos[2 f c t (2n  1)]
2  n 1 2n  1
Note: for period: T0, the Fourier series of g(t) is

 2n t 2n t   1 T0 2
g (t )  a0    an cos  bn sin  where  0a   T
 0
g (t ) dt
n 1  T0 T0   T 0 2

  2n t 
 2 T0 2
 na 
T  
T0 g (t )cos 
 T
 dt

 0 2 0
  
bn  2 T 2 g (t )sin  2n t  dt
T0

 
T0  0 2  T0 
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 12
 Switching Modulator
v2 (t )  [ Ac cos(2 f ct )  m(t )]gT0 (t )
 1 2  ( 1) n 1 
  Ac cos(2 f ct )  m(t )     cos[2 f ct (2n  1)]
 2  n 1 2n  1 
1 2 2 1  21  
  Ac cos(2 f ct )  m(t )    cos(2 f ct )    cos  2 (3 f c )t     cos  2 (5 f c )t    L 
2   3   5  
1 A 2  Desired AM signal
 m(t )   c  m(t )  cos(2 f ct )
2  2  
2  1 1 
 m(t )  cos  2 (3 f c )t   cos  2 (5 f c )t   L  Images
  3 5 
 
 c cos(2 f ct ) cos(2 f ct )  cos  2 (3 f c )t   cos  2 (5 f c )t   L  Delta Func.
2A 1 1
  3 5 
V2 ( f )

-W W
-5fc -4fc -3fc -2fc -fc 0 fc 2fc 3fc 4fc 5fc
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 13

Switching Modulator
m(t)
• v2 (t ) consists of two components
ka
– Desired AM signal Selected by a BPF ka m(t)
A  4  1
v2 (t )  c 1  m(t )  cos(2 f c t ) at  f c
2   Ac  s(t)
s (t )  Ac [1  ka m(t )]cos(2 fct ) c(t)
4
• amplitude sensitivity ka 
 Ac
– Unwanted components
• Delta functions at 0,  2 fc ,  4 fc , L
• Images at 0,  3 fc ,  5 fc , L
V2 ( f )

-W W
-5fc -4fc -3fc -2fc -fc 0 fc 2fc 3fc 4fc 5fc
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 14
 Envelope Detector (Demodulation)
• To demodulate an AM signal, what we need to do is to
detect the slow-varying envelope.
• Assume the diode is ideal with forward resistance rf
• We need to fast charge C and then input
slowly discharged, so
1 1 1
(rf  Rs )C  and  Rl C 
fC fC W
1 1
(rf  Rs )   Rl 
fC C WC
output

An RC
filter

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 15

Virtues, Limitations, Modifications

of Amplitude Modulation
• Virtues of amplitude modulation
– Signal is easily generated and reversed
– Modulation: switching modulator or a square law modulator
– Demodulation: envelope detector or square-law detector
• Limitations of amplitude modulation
– AM is wasteful of power and bandwidth.
• Modification of amplitude modulation
– Double sideband-suppressed carrier (DSB-SC)
modulation Save power
– Vestigial sideband (VSB) modulation
– Single sideband (SSB) modulation Save BW

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 16

 Double Sideband-Suppressed Carrier
Modulation (DSB-SC)
• DSB-SC consists of the product of the message
signal m(t) and the carrier wave c(t):
s (t )  c(t )m(t )  Ac cos(2 f ct )m(t )
m(t )  M ( f ) where W  f  W
• Fourier transform of s(t) is obtained
as 1
S( f )  Ac [ M ( f  f c )  M ( f  f c )]
2
Note: no delta
functions at fc ! Envelope detector
cannot be applied!

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 17

Ring Modulator
outer diodes on

outer diodes off

Multiplier

180 Phase change

• The outer diodes are switched on

and the inner diodes are switched off
rf : small, rb  
• The outer diodes are switched off
and the inner diodes are switched on 

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 18

 Ring Modulator
• The square-wave carrier c(t):
4  (1) n1
c(t )   cos[2 (2n  1) f ct ]
 n1 2n  1
• The ring modulator output
s (t )  c(t )m(t )
4 (1) n1

  cos[2 (2n  1) f ct ]m(t )
 n1 2n  1
BPF

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 19

Coherent Detector
• Coherent/Synchronous detection: Multiplying the
received signal s(t) by a locally generated sinusoidal wave:
v(t )  Ac cos(2 f ct   ) s (t ) where s (t )  m(t ) Ac cos(2 f c t )
 Ac Ac cos(2 f ct )cos(2 f ct   )m(t )
1 1 LPF
 Ac Ac  cos   m(t )  Ac Ac cos(4 f ct   )m(t )
2 2
1
v0 (t )  Ac Ac  cos   m(t )
2

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 20

 Coherent Detector
1
v0 (t )  Ac Ac  cos   m(t )
2
• The amplitude of this demodulated signal is
1
maximum when   v0 (t )  Ac Acm(t )
2 
and it is minimum (zero) when    :
1  2
v0 (t )  Ac Ac cos m(t )  0
2 2
• Typically, the phase of the received signal s(t) is
randomly distributed. To maintain the maximum
detected signal, we need a phase-locked loop
(PLL) circuit to lock and synchronize the phase
of local oscillator to the received signal, that is 
.
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 21

Costas Receiver
Using a negative
Feedback loop to
Form a PLL! I 12 Ac (cos  )m(t ) 1
 1 
Q 2 Ac (sin  )m(t ) tan 
Q
  tan 1
I
Quadrature
phase shift
• Best when Q/I = 0.
• If  slightly deviated
from 0, i.e.  is small,
Q
 tan   
I
•  is applied to control
the phase of VCO.
• I – channel: in-phase coherent detector.
• Q – channel: quadrature-phase coherent detector.
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 22
 Quadrature Carrier/Amplitude Multiplexing
• QAM applies two independent signals, m1(t) and m2(t), on two
carrier waves of the same frequency but differing in phase by
-90 degrees, which is expressed as:
s (t )  Ac m1 (t ) cos(2 f c t )  Ac m2 (t ) sin(2 f c t )  Re[ s (t ) Ac e j 2 fct ]
s (t )  m1 (t )  jm2 (t )
Note:
These two independent
signals, m1(t) and m2(t),
Tx: occupy the same channel
bandwidth, thus it’s a
bandwidth-conservation
scheme.  higher
spectral efficiency!

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 23

Quadrature Carrier/Amplitude Multiplexing

A
• To maintain synchronization,
we use the Costas receiver.
– Assume frequency and
phase are locked already:
At point A:
s (t )  2 cos(2 f c t )
  Ac m1 (t ) cos(2 f c t )  Ac m2 (t ) sin(2 f c t )   2 cos(2 f c t ) B
 Ac m1 (t ) cos(2 f c t )  2 cos(2 f c t )  Ac m2 (t ) sin(2 f c t )  2 cos(2 f c t )
 Ac m1 (t )  Ac m1 (t ) cos(4 f c t )  Ac m2 (t ) sin(4 f c t )
At point B:
s (t )  2sin(2 f c t )   Ac m1 (t ) cos(2 f c t )  Ac m2 (t ) sin(2 f c t )   2sin(2 f c t )
 Ac m1 (t ) cos(2 f c t )  2sin(2 f c t )  Ac m2 (t ) sin(2 f c t )  2sin(2 f c t )
 Ac m1 (t ) sin(4 f c t )  Ac m2 (t ) cos(4 f c t )  Ac m2 (t )
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 24
 Single-Sideband (SSB) Modulation
• For DSB signals, the same information is carried
in the USB and LSB if m(t) is real.  waste
bandwidth!
• SSB generation:
1. Generate a DSB signal m(t) with energy gap 2fa
centered near the origin.
2. Up-convert the DSB signal to a carrier frequency.
3. Use an ideal SSB filter to select the desired sideband.

1 2

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 25

Single-Sideband (SSB) Modulation

3 stopband passband 4 Transmitted signal

• The ideal SSB filter:

– Spectrum of SSB signal containing the upper sideband.
• In particular, the filter must only satisfy the following
requirement ：
– The desired sideband lies inside the passband of the filter.
– The unwanted sideband lies inside the stopband of the
filter.
– The transition of the filter should be  2 f a .
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 26
 Single-Sideband (SSB) Modulation
• The analysis of a SSB signal uses a technique known
1
as the Hilbert transform. H ( f )   j sgn( f )  h(t ) 
t
• The synchronization information
can be obtained by one of two
methods:
– Transmitting a low power pilot carrier
in addition to the selected sideband.
– Using highly stable oscillators in both the
transmitter and receiver for generating the
carrier frequency.

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 27

Vestigial-Sideband (VSB) Modulation

• If signals do not have an energy gap at the origin,
due to the finite roll-off of the SSB filter, the SSB
transmission leads to VSB transmission. S ( f )
Baseband M(f ) U( f ) filter VSB vestige
signal

28
-W W - fc fc - fc fc

• Amplitude response of VSB filter:

– H(f) denotes the transfer function of the filter
– M(f) is the Fourier transform of the baseband signal m(t)
– U(f) is the Fourier transform of u(t): up-converted signal
A
S ( f )  U ( f ) H ( f )  c  M ( f  fc )  M ( f  fc ) H ( f )
2
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 28
 Vestigial-Sideband (VSB) Modulation

• We need to determine the restrictions of H(f), especially

the “vestige,” so that we still can recover m(t) from s(t).

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 29

Vestigial-Sideband (VSB) DeModulation

• Coherent detection
Ac
v(t )  Ac cos(2 f c t ) s (t )  V( f )   S ( f  f c )  S ( f  fc )
2
A
 S ( f )  c  M ( f  f c )  M ( f  fc ) H ( f )
2
 V ( f )  Ac Ac M ( f )  H ( f  f c )  H ( f  f c )
4 LPF
Ac Ac
  M ( f  2 f c ) H ( f  f c )  M ( f  2 f c ) H ( f  fc )
4

 Vo ( f )  Ac Ac M ( f )  H ( f  fc )  H ( f  fc )
4
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 30
 Vestigial-Sideband (VSB) DeModulation
H( f ) H ( f  fc ) H ( f  fc )
M(f ) H ( fc )

31
- fc 0 fc - fc 0 fc 2fc -2fc -fc 0 fc
A A
Vo ( f )  c c M ( f )  H ( f  f c )  H ( f  f c ) 
4
• To obtain a distortionless m(t), Vo(f) should be a scaled
version of M(f).  [H(f – fc) + H(f + fc)] is independent of f.
 H ( f  fc )  H ( f  fc )  2 H ( fc )
where H(fc) is the value of H(f) at f = fc. H ( f  fc )  H ( f  fc )
M(f )
• Let
H ( f  f c )  H ( f  f c )  1, W  f  W
1 Ac Ac
 H ( fc ) 
2
 vo (t ) 
4
m(t ) -2fc -fc 0 fc 2fc
-W W
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 31

Vestigial-Sideband (VSB) DeModulation

• The cut-off portion of the frequency response around the
carrier frequency fc is odd symmetry. That is inside the
transition interval, f c  f v  f  f c  f v , the sum of the
values of H ( f ) at any two frequencies equally
displaced above and below fc is unity.
If f c  f1  f c  f 2 , then H ( f1 )  H ( f 2 )  1

For f > fc + W,
H(f) can be arbitrary
H ( f2 ) specified.
Required
H ( f1 )

H ( f1 )  H ( f 2 )  1
f1 f2
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 32
 Frequency Translation
• SSB modulation is a form of frequency translation,
which is also referred to as frequency changing,
mixing, or heterodyning.
Mixer

s '(t )  s1 (t )  Al cos(2 fl t )  m(t ) cos(2 f1t )  Al cos(2 fl t )

1
 Al m(t )  cos(2 ( f1  fl )t )  cos(2 ( f1  fl )t ) 
2
carrier frequency carrier frequency
translated upward translated downward
(Up-conversion) (Down-conversion)
NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 33

Frequency-Division Multiplexing (FDM)

NTHU EE364000 Communication Systems I Y.H. Huang & K.M. Feng 34