Chapter 5

AM, FM and Digital Modulated Systems

In this chapter, we will study the techniques of
bandpass communications.

1. Analog baseband signal (AM, FM, etc)

2. Digital baseband signals (OOK, BPSK, etc.)
3. Multiplexing (TDM, FDM, CDM)
 Basic Model for Bandpass Communication

Source Destination

m(t ) : informatio n to be sent by source (baseband)

g (t ) : processed informatio n to be modulated (baseband)
s (t ) : modulated signal to be transmitt ed (shifted frequency)
r (t ) : modulated signal corrupted in the channel (shifted frequency)
g~ (t ) : demodulate d signal approximat ing g(t)
~ (t ) : reconstruc ted signal approximat ing m(t)
m


s (t )  Re g (t )e jwct  where g  m(t ) . The goal is to fine a function
g • that satisfies the objectives of the communication.
1 1
S(f)  G  f  f c   G *   f  f c   Ps (f)  Pg  f  f c   Pg *   f  f c 
2 4
 Amplitude Modulation

g (t )  Ac 1  m(t ) 
s(t )  Ac 1  m(t )  cos wct

Definitions.
Amax  Ac
% positive modulation  100  max  m(t )  100
Ac
Ac  Amin
% negative modulation  100   min  m(t )  100
Ac
Amax  Amin max  m(t )   min  m(t ) 
% modulation  100   100
2 Ac 2
A max  max  Ac 1  m(t )  , Amin  min  Ac 1  m(t )  
Almost always, -1  min  m(t )   max  m(t )   1. Otherwise, the
transmitter and receivers become very complicated.
In general, to simplify the receiver and transmitter complexity,

 A 1  m(t )  cos wct if m(t )  1

s(t )   c
 0 if m(t )  1

This will increase the bandwidth for s (t ).

Amax  1.5 Ac
Amax  0.5 Ac
Positive modulation % = 50%
Negative modulation % = 50%
Modulation % = 50%
 1 2
Ac 1  m(t ) 
2
s (t ) 2 
2
1
 Ac 2 1  2 m(t )  m 2 (t ) 
2
1
 Ac 2 1  m 2 (t )  if m(t )  0.
2

Definition:
E  Modulation efficiency = % of total power of s (t ) containing information
m 2 (t )
E  100 (for AM signals)
1  m (t )
2

If an AM signal is 100% modulated with sinusoidal modulating signal,

1
i.e., max  m(t )   1, m 2 (t )  0.5. Thus, the modulation efficiency =
.
3
The highest possible modulation efficiency is achieved when m(t )  1 for
all t. Then, E  50%.
 Ac 2
  
2
PPEP  1  max m (t )
2
A
S ( f )  c   f  f c   M  f  f c     f  f c   M  f  f c  
2
Note. AM signals use twice the bandwidth of the original baseband signals.

Example.
Suppose an AM transmitter has the average carrier power of 5, 000W .
The AM transmitter is connected to a 50  resistor.
1 Ac 2
 5, 000  Ac  707
2 50
Suppose m(t )  cos  2 f mt  . f m <<f c . Thus the AM signal is 100% modulated.
 m(t ) 2  0.5 and max  m(t )   1.
1 Ac 2 1 Ac 2
Total transmission power =  m(t ) 2  7,500W
2 50 2 50
1 Ac 2
1  max  m(t )  20, 000W
2
PPEP 
2 50
1
E  33.33%
1  m(t ) 2
Generation of High-Power AM by Pulse Width Modulation
 Double Sideband Suppressed Carrier (DSB-SC)

The transmitter does not send the carrier signal to reduce power.
s (t )  Ac m(t ) cos wct
Ac
S( f )   M  f  f c   M  f  f c  
2
Since there is no carrier signal, % modulation =  and E  100%.
The envelop detector cannot be used.  The product detector is needed.
By not sending the carrier signal, the transmitter power can be increared for the sideband.
 Coherent Detection of DSB-SC

Coherent detection produces high quality demodulated signals.

It is difficult to do so for DSB-SC, because it has no carrier signal.
DSB-SC has a spectrum symmetric arout the center frequncy.

Two technology solutions:

- Costas Loop
- Squaring Loop
 Costas Loop

Act like an integrator.

2 2
11  11 
v3 (t )   Ao Ac  m 2 (t ) sin 2e v4 (t )   Ao Ac  m 2 (t ) sin 2 e
22  22 
Squaring Loop
 Single Sideband
Single sideband (SSB) signals have only half of the sideband of the
ordinary AM signals.
 Upper single sideband (USSB) has no spectrum for f  f c .
 Lower single sideband (LSSB) has no spectrum for f > f c .

Theorem. For SSB signals,

g (t )  Ac  m(t )  jm(t )   s (t )  Ac  m(t ) cos wct  jm(t ) sin wct  for USSB
   
g (t )  Ac  m(t )  jm(t )   s (t )  Ac  m(t ) cos wct  jm(t ) sin wct  for LSSB
   
1
where m(t )  m(t )  h(t ) and h(t )  .
t
 j f  0
h(t )  H ( f )   (H ( f ) shifts the signal phase by -90o .)
 j f 0
Spectrum for USSB Signal
Proof of Theorem (for USSB)

G ( f )  Ac M ( f )  jF  m(t ) 
  
 G ( f )  Ac  M ( f )  jH ( f ) 
2 A M ( f ) f  0 
 G( f )   c 
 0 f  0 
M ( f  fc ) f  fc   0 f   fc 
 S ( f )  Ac    Ac  
 0 f  f c M ( f  fc ) f   fc 
Generation of SSB Signals
 1 1 2 2  
2
s (t )  g (t )  Ac m (t )  m(t )
2 2

2 2  
s 2 (t )  Ac 2 m 2 (t ) because m 2 (t )  m(t ) 2

PPEP
1
 
1 2 

 max g (t )  Ac max m (t )  m(t )
2
2

2
2



2


2
For SSB-AM: R(t )  g (t )  Ac 2
m (t )   m(t ) 
2
 
 m(t )  
1  m(t )

For SSB-PM:  (t )  g (t )  tan  1
 for USSB = tan  
 m(t )   m(t ) 
vout (t )  KAc m(t )
 Vestigial Sideband (VSB)

DSB takes up too much bandwidth, and SSB is difficult to implement.

 VSB is a compromise between DSB and SSB.
 VSB partially suppresses one of the sidebands.

sVSB (t )  s (t )  hv (t )
s (t ) : DSB signal hv (t ) : impulse response for VSB filter
sVSB (t )  SVSB ( f )  S ( f ) H v ( f )

Requirement for VSB filter: H v ( f  f c )  H v ( f  f c )  C , for f  B

Ac
 SVSB ( f )   M ( f  fc ) H v ( f )  M ( f  fc ) H v ( f )
2
VSB Spectra
 vout (t )   Ao sVSB (t ) cos wct   h(t )
 1 1 
 Vout ( f )  Ao  SVSB ( f )     f  f c     f  f c    H ( f )
 2 2 
 H ( f )  1 for f  B
Ac Ao
 Vout ( f )  M ( f )  H v  f  f c   H v  f  f c   for f  B
4
 Vout ( f )  KM ( f ) for f  B due to the requirement for H v  f 
 Angle Modulation
g (t )  Ac e j (t )
where R(t )  g (t ) = Ac and  (t )  m(t ).
Phase Modulation (PM) and Frequency Modulation (FM) are special
cases of angle modulation.

s (t )  Ac cos  wct   (t ) 
For PM,  (t )  D p m(t ) D p : phase sensitivity factor 

D : frequency deviation factor 

t
For FM,  (t )  D f  m( s) ds f


Note: g (t ) is a nonlinear function of m(t ), and  (t ) is a linear function of m(t ).

1
 S ( f )  G  f  f c   G *   f  f c  
2
 G ( f )  F  g (t )  F  Ac e j (t ) 
 In general we don't know how to find G( f ).
Let m p (t ) : modulating message for PM, m f (t ) : modulating message for FM.
Dp  dm p (t )  Df t
 m f (t )  
D f  dt 
 m p (t ) 
Dp 

m f ( s) ds
Angle Modulator Circuits

RFC : Radio Frequency Choke

Definition.
Bandpass signal: s (t )  R(t ) cos (t ) where  (t )  wct   (t ).
 Instantaneous frequency:
1 1  d (t ) 
f i (t )  wi (t ) 
2 2  dt 

1  d (t ) 
For PM: fi (t )  f c 
2  dt 
1
For FM: fi (t )  f c  D f m(t )
2

Note: For angle modulated signals, the instantaneous frequency varies in

time around the center frequency f c .
FM with m(t) = Sinusoidal Signal
 1  d (t ) 
Frequency deviation: f d (t ) = fi (t )  f c 
2  dt 
 1  d (t )  
Peak frequency deviation: F = max       0, necessarily.
2
  dt 
Peak-to-peak frequency deviation:
 1  d (t )    1  d (t )  
FPP = max      min       0, necessarily.
 
2  dt  2
  dt 
1
For FM: F = D f V p where V p  max  m(t ) .
2
Note. The bandwidth requirement varies by the amplitude of m(t ) and
D f , the frequency deviation factor.
The power of FM signal is influence only by the amplitude of the
Ac 2
modulated signal, not m(t ).  power =
2
Peak phase deviation:  = max  (t ) 
For PM,   D pV p where V p  max  m(t ) .

Definition.
Phase modulation index  p  
F
Frequency modulation index  f 
B

If m(t ) is a single sinusoidal wave, then  p   f .

Example.
For PM, let m p (t )  Am sin wmt.   (t )  D p Am sin wmt   sin wmt
Thus,   D p Am   p .

For FM, m f (t )  Am cos wmt.   (t )  D p Am sin wmt   sin wmt

D f Am D f Am
Thus,     f and F  .
wm 2

For the PM and FM above, g (t )  Ac e j (t )  Ac e j  sin wmt


 G ( f )  Ac  J     f  nf 
n 
n m where J n    is the Bessel function of

the first kind of the n th order.

Bessel functions: n = 0 ,1, 2, 3, 4, 5, 6
Magnitude spectra for FM or PM

m(t )  sin wmt

BT : bandwidth within which

98% of power is contained
BT  2    1 B (Carson's rule)
 Narrowband Angle Modulation

Variation of angle is small.  (t )  0.2 rad

 g (t )  Ac e j ( t )  Ac 1  j (t ) 
 s (t )  Ac cos wct  Ac (t ) sin wct
 
 S ( f )  Ac   f  f c     f  f c    j   f  f c     f  f c  
 Dp M ( f ) for PM

where   f  f c   F  (t )    D f
 j 2 f M ( f ) for FM

 Wideband Frequency Modulation

Theorem. For WBFM, let

 t
 D f max  m(t ) 
s (t )  Ac cos wct  D f  m( s )ds and  f   1.
   2 B

 power spectrum density of s(t ),

 Ac 2   2   2 
P(f)   fm   f  fc    f m    f  fc    where f m    is p.d.f. of m(t).
2 D f   D f   Df  
Example. WBFM with m(t ): triangular function
 1
 for m  V p
f m  m    2V p where V p is the peak of the triangular function.
 0 for m  V p


 1 2   1 2 
 Ac  2
if  f  fc   Vp   if  f  f c   V p 
 P(f)    2V Df    2V p Df 
2D f  p   
 o if all other f   o if all other f 

 Ac 2   Ac 2 
 if  f c  F   f   f c  F    if   f c  F   f    f c  F  
P(f)   8F    8F 
 o   o 
 if all other f   if all other f 
D f Vp
F 
2
Spectrum of WBFM Signal
Spectrum of Wideband Binary FSK
Pre-emphasis and De-emphasis
 Frequency Division Multiplexing
Transmission of multiple messages simultaneously over wideband channel.
Any modulation type (AM, DSB, SSB, PM, FM) can be used.
FM Stereo System
 FM Broadcasting
Non commercial: 88.1 - 91.9 MHz
Commercial: 92.1 - 107.9 MHz
Today, there are analog and digital FM broadcasting with a
channel bandwidth of 200 KHz.
 Binary Modulated Bandpass Signals
 On Off Keying (OOK): Carrier sinusoid is on/off with a unipolar binary
signals. (e.g., Morse code radio trasmission)
 Binary Phase Shift Keying (BPSK): The phase of the carrier sinusoid is either
0o or 180o based on binary inputs.
 Frequency Shift Keying (FSK): Change the frequency of the carrier sinusoid
by a certain amount.
Digital Bandpass Signals
OOK: s (t )  Ac m(t ) cos wct where m(t ) is a unipolar bainary base band data.
 g (t )  Ac m(t )

Ac 2   sin  fTb  
2

 Pg ( f )   ( f )  Tb   
2 
   fTb  
1
 bit rate R  , null-to-null BW  2 B, absolute BW = .
Tb

BPSK: s(t )  Ac cos  wc t  D p m(t )  where m(t ) =  1.

s(t )  Ac cos  D p m(t )  cos wct  Ac sin  D p m(t )  sin wct
s(t )   Ac cos D p  cos wct   Ac sin D p  m(t ) sin wct
This can be made into s (t )   Ac m(t ) sin wct
 g (t )  jAc m(t )
2
 sin  fTb 
 Pg ( f )  Ac Tb 
2

  fTb 
1
 bit rate R  , null-to-null BW  2 B, absolute BW = .
Tb
PSD
Detection of OOK
Detection of BPSK
From Frequency Shift Keying section to the end of
Chapter 5, the math is quite complex.

Thus, we will focus on the concepts and

understanding how various techniques work.

Section 5.10 will be covered briefly.

Section 5.11 will not be covered.

Note that Chapter 5 is concerned with the

technology. Do not focus too much about the
mathematical details.
 Frequency Shift Keying
FSK: message m(t )  0,1 is sent as part of the frequency of bandpass signal.
(1: mark, 0:space)

s (t )  Ac cos wct  D f  m(v)dv 
t

  

s (t )  Re g (t )e jwct  where g (t )  Ac e j (t )
t
 (t )  D f  m(v)dv

 FSK Modem

FSK is very popular for telephone modems.

Example FSK Data
Spectra of
FSK Data
 PSD for
Complex
Envelope of
FSK
Detection of FSK
Quadrature Phase Shift Keying (QPSK)
(4 Level PSK)
Quadrature Amplitude Modulation (QAM)
(Multi Level AM)

s (t )  x(t ) cos wct  y (t ) sin wct

g (t )  x(t )  jy (t )  R(t )e j (t )
Generation of QAM
MSK, GMSK, QPSK, and OQPSK
Direct Sequence Spread Spectrum
Autocorrelation and PSD for an m-Sequence PN Waveform
Approximate PSD of BPSK-DS-SS Signal
Frequency-Hopped Spread Spectrum