Lecture_ch5.pptx

EE 370: Communications Engineering I Chapter 5: Angle Modulations and Demodulations
Chapter 5
Angle Modulation and Demodulation
(Figures included in this lecture notes are extracts from the text book
Modern Digital and Analog Communication Systems by B.P. Lathi and Zhi
Ding, Copyright © 2009 by Oxford University Press, Inc.)

EE 370: Communications Engineering I Chapter 5: Angle Modulations and Demodulations
1. Angle (exponential) Modulation [Non-linear]
2. Bandwidth of Angle Modulated Wave
3. Generation of FM Waves
4. Demodulation of FM Signals
6. Superheterodyne analog AM/FM Receivers
7. FM Broadcasting System
Contents

5.1 Angle Modulation
In AM signals, the amplitude of a carrier is
modulated by a signal m(t), and, hence, the
information content of m(t) is in the amplitude
variations of the carrier. Because a sinusoidal
signal is described by amplitude and angle
(angle includes frequency and phase), there
exists a possibility of carrying the same
information by varying the angle of the
carrier. This in effect is a nonlinear
modulation technique.
3

4
5.1Angle Modulation
The Concept of Instantaneous Frequency
A general sine wave signal can be expressed as
   
t
A
t 
 cos

  0


 
 t
t
   
dt
t
d
t

 
(t) is the generalized angle. For a sine wave with fixed
frequency and phase:
 can be represented as a linear function of time with a slope  : angular
speed,  = 2pf.
In general  is the derivative of the angle. That is
and    



t
d
t 




5
The Concept of Instantaneous Frequency
Figure 5.1 Concept of instantaneous frequency.

6
• Phase Modulation (PM)
   
t
m
k
t
t p
c 

 0



   
cos
PM c p
t A t k m t
 
 
 
 
     
t
m
k
dt
t
d
t p
c
i



 


The message signal is modulating the phase of the carrier signal:
without loosing generalization, we can omit the initial phase 0 and we get the
following PM signal :
i is called the instantaneous frequency of the modulated signal.

7
• Frequency Modulation (FM)
The message signal is modulating the frequency of the carrier signal:
   
t
m
k
t f
c
i 
 
    
 




t
f
c d
m
k
t 



   




t
f
c d
m
k
t
t 


     




 
  

t
f
c
FM d
m
k
t
A
t 


 cos

8
• General Concept of Angle Modulation
   
 
t
t
A
t c 

 
 cos      




t
d
t
h
m
t 



h(t) = kpd(t)  phase modulation
h(t) = kf u(t)  frequency modulation
Further, according to the equations above we can say
that the PM and FM are equivalent in certain ways 
• FM with m(t) = a PM with ∫m(t).
• PM with m(t) = a FM with m’(t).
Figure 5.3 Generalized phase modulation by means of the filter H(s) and recovery of the message
from the modulated phase through the inverse filter 1/H(s).

9
• General Concept of Angle Modulation
Figure 5.2 Phase and frequency modulation are equivalent and
interchangeable.

Figure 5.4 FM and PM waveforms.

Figure 5.5 FM and PM waveforms.

12
Power in Angle Modulated signal
Regardless the values of kp or kf, the expected
power of the modulated signal is given by: A2/2.
See example 5.1 and 5.2 of the text book

13
5.2 Bandwidth of Angle Modulated
Waves
To determine the bandwidth of an FM wave :
   
 


t
d
m
t
a 
      
c f f
c
j t k a t jk a t
j t
FM t Ae Ae e
 

 

 
 
)
   
Re
FM FM
t t
 
 
  
)
expanding the factor in power series and
substituting into the above expression we get
       
2 3
2 3
cos sin cos sin ....
2! 3!
f f
FM c f c c c
k k
t A t k a t t a t t a t t
    
 
    
 
 
 
 
f
jk a t
e

14
5.2 Bandwidth of Angle Modulated Waves
the FM signal is expressed as an unmodulated carrier plus
spectra of a(t), a2(t), … an(t), … centered at c.
Let M() be the spectrum of m(t) with bandwidth B.
The bandwidth of a(t) is also B because the integration is
equivalent to only a multiplication by 1/j .
a2(t) has a bandwidth of 2B (M()*M())
a3(t) has a bandwidth of 3B
an(t) has a bandwidth of nB
M
Conclusion: FM signal has infinite bandwidth. (theoretically)

15
Special cases:
• Narrow-Band Angle Modulation
The angle modulation is not linear in general. However, if
|kf a(t)| << 1  only the 1st two terms are important in the
above equation.
   
cos sin
FM c f c
t A t k a t t
  
 
 
 
This is a linear modulation. It is like an AM wave* with
bandwidth = 2B. This is called Narrow Band FM (NBFM).
* However the waveform is entirely different from AM

16
Narrow-Band Angle Modulation
Similarly the narrow band PM (NBPM) is given by:
The narrow band angle modulation is similar to AM (same
bandwidth, carrier plus spectrum centered on c).
The difference: in angle modulation the sideband spectrum is
p/2 phase shifted with respect to the carrier. The waveform is
completely different.
   
cos sin
PM c p c
t A t k m t t
  
 
 
 

17
• Wide-Band Angle Modulation (summary)
This is the situation where we cannot ignore the higher
order terms because (|kf a(t)| << 1) is not satisfied. (can
be due to high kf ).
In this case the bandwidth of the FM signal is found to be
given by the following approximation:
  











 B
m
k
B
f
B
p
f
FM
p
2
2
2 Carson’s rule
For truly wideband case,  f >> B  BFM  2 f

18
Wide Band Phase Modulation
All the analysis developed for the FM can be applied to
the PM by replacing mp by and kf by kp. That is
Examples 5.3 – 5.5 (pg. 217-220)
 
2 2
2
p p
PM
k m
B f B B
p
 
    
 
 
&
 
1
2 
 
B
BFM
B
f



Deviation ratio  in FM plays the role of modulation index μ in AM
Carson’s rule can be also be expressed in terms of the
deviation ratio 
where
p
m


Figure 5.6 Estimation of FM wave bandwidth.
• Wide-Band Angle Modulation

   
     





 







 



B
t
m
k
B
t
m
k
t
t
m
k
t
Bt
rect
k
f
c
k
f
c
k
f
c
4
sinc
2
1
4
sinc
2
1
cos
2





Reference to Fig. 5.6, with a staircase approximation to m(t),
  Hz
B
m
k
B
m
k
B
p
f
p
f
FM 










 2
2
2
8
2
2
1
p
p
p
Peak frequency deviation (Hz)
p
p 2
2
2
min
max p
f
f
m
k
m
m
k
f 




 Hz
B
f
BFM 2
2 


which is an overestimation due to staircase approximation
Considering NBFM Δf ≈ 0. Then above reduces to, Hz
B
BFM 4

However we previously found for NBFM, Hz
B
BNBFM 2

Therefore a better approximation is   Hz
B
f
BFM 

 2
This result is known as Carson’s rule

21
5.3 Generation of FM Wave
Narrow-Band FM and PM wave generation
Figure 5.8 (a) Narrowband PM generator.
(b) Narrowband FM signal generator.

22
Indirect Method of Armstrong
We start with the generation of a NBFM with
frequency deviation f as described previously.
Then we use a frequency multiplier ( x N ) to obtain
a WBFM. After filtering using a bandpass filter
centered at Nfc, we get an FM signal with N f.
Sometimes the frequency increase of the carrier is
not needed.
Solution: after the multiplier we insert a mixer to
down convert the carrier to the wanted one.

23
Also see Example 5.7
Figure 5.10 Block diagram of the
Armstrong indirect FM transmitter.

24
Direct Generation Using a VCO
Frequency varies linearly with control voltage. FM
wave is generated by using message signal m(t)
as the a control signal.
1) Using an OP-AMP and Hysteresis Comparator
2) Variation of L or C of a tank of a resonant circuit :
reverse biased semiconductor (i.e. diode) can be
used as a variable capacitor.
       
t
m
k
f
t
f
t
m
k
t f
c
i
f
c
i
p


2
1
or 




1) Direct Generation Using VCO
VCO circuit diagram
Varactor Diode: capacitance
is changed by the m(t)

26
2) Direct Generation using VCO

27

28
Also we can use variable inductor. It can be
achieved by winding two inductors in the same
core. Then controlling the inductance of the inner
inductor by injecting a current in the outer one.

29
5.4 Demodulation of FM Wave
In an FM signal the information resides in the instantaneous
frequency :
Method 1:
A network with a response linear to  would be able to
detect the message signal.
Example : |H()| = a + b centered around the
carrier frequency in the FM band.
 
i c f
k m t
 
 

Figure 5.12 (a) FM demodulator frequency response. (b) Output of a differentiator to the input FM wave.
(c) FM demodulation by direct differentiation. 30
|H()| = a + b
Direct
Differentiation
Method
Method 2:

31
31
    




 
  

t
f
c
FM d
m
k
t
A
t 


 cos
   
   











 

 

t
f
c
FM
FM d
m
k
t
A
dt
d
t
dt
d
t 



 cos

   
    




 


  

t
f
c
f
c
FM d
m
k
t
t
m
k
A
t 



 sin

FM signal is:
The differentiator output will be:

32
 The above expression shows that the output is
FM and AM modulated. See figure 5.12b. Since
c > kfm(t) all the time, an envelop detector can
be used to extract m(t) as shown in the previous
figure.
 Problem: A must be a constant. If not (due to
channel noise, fading, …), it must be fixed before
demodulation.
Solution: Bandpass limiter

33
Bandpass limiter

34
5.4. Demodulation of FM Wave
 Input: distorted FM signal
 Output of the hard limiter: A(t) >=0
 Output of the bandpass filter:

35
Practical Frequency Demodulators
1. Differentiation
2. Slope detection
3. Ratio detector
4. Zero-crossing detectors
5. Phase-Locked Loop (PLL):
1. Differentiation: OPAMP differentiator can be used to
convert frequency variation to amplitude variation
that can be detected using a simple envelop detector.

2. Slope detection: Any tuned circuit which has a
linear segment of positive slope in the frequency
response under or above the resonance can be used
instead of the OPAMP differentiator.
Examples: high pass RC filter
tuned RLC filter:
Limitation: narrow bandwidth.
LC
o
c
1

 


37
High pass RC filter.
Figure 5.13 (a) RC high-pass filter. (b) Segment of positive slope in amplitude response
  1
2
if
2
2
1
2



 fRC
fRC
j
fRC
j
fRC
j
f
H p
p
p
p
Thus for small RC constants such that 2πfRC<<1 the RC filter approximates a
differentiator.
H( f )=j2πfRC
 
t
FM

 
dt
t
d
k FM

0
 
f
FM
  
f
j
k FM


0
f
RC
k p
 2
and
0 

Envelope
Detector
Demodulated
Signal

38
3. Ratio detector: balanced demodulator. Not very
sensitive to the amplitude variation of the FM signal.
widely used in the past.
4. Zero-crossing detectors: frequency counter that
measures the instantaneous frequency by counting
the rate of zero-crossing.
5. Phase-Locked Loop (PLL): because of its low cost
and good performance, it is widely used in FM
receivers.

39
5.6 Superheterodyne FM/AM
Receiver
[A+m(t)]cos(ωIF t)
Or
Acos(wIF t+(t))
[A+m(t)]cos(ωct)
Or
Acos(ωct+(t))
AM superheterodyne receiver: Intermediate frequency = 455kHz and envelope
detection is used.
monophonic FM receiver: Identical to the superheterodyne AM receiver
except that the intermediate frequency is 10.7MHz and envelope detector is
replaced by a PLL or a frequency discriminator.

40
5.7 FM Broadcasting System
FCC assigned the following frequency bands for FM
broadcasting.
Frequency range: 88 to 108MHz
Separation: 200 kHz
Max. frequency deviation: 75 kHz
Old FM receivers are monophonic. (one signal m(t))
New FM receivers are stereophonic. (left and right
audio signals, i.e. two different microphones). 
more natural effect.
FCC ruled that:
 monophonic receivers must be able to receive
stereo FM signals.
 Total transmission band for the stereo FM signal =
200kHz with f = 75KHz.

41
Diagram for stereo FM transmitter
19kHz
Composite
Baseband
Signal
Left
Signal
(Base band)
Right
Signal
(Baseband)

42
Spectrum of a baseband stereo FM signal

43
FM stereo receiver

4.8 Phase-Locked Loop (PLL)
• PLL is used to track the phase and frequency of the carrier
component of an incoming signal.
• Useful for synchronous demodulation of AM signals with
suppressed carrier (no pilot).
• Can be used for demodulation of angle-modulated signals
especially under low SNR conditions.
• Also has applications in clock recovery systems in digital
receivers.

Three basic components:
1. Voltage Controlled Oscillator (VCO)
2. Multiplier-works as the phase detector(PD) or phase comparator
3. Loop filter

46
  where is a constant
VCO c o
ce t c
 
 
 
VCO c o t
  
  &
   
Therefore (5.24)
o o
t ce t
 
&
c is the free running frequency of VCO
Instantaneous frequency of the VCO :
Further, considering the output of VCO
as Bcos[ct+o(t)]:
c and B are constants of the PLL.

Although in Figure (a), incoming frequency and the VCO
Output frequency are equal (c), the analysis is also valid
when they are different as shown below.
We assume the incoming signal (input to the PLL) be
Asin[ωct+θi(t)].
If the incoming signal happens to be
Asin[ωot+ѱ(t)],
it can still be expressed as Asin[ωct+θi(t)],
where θi(t)=(ωo-ωc)t+ ѱ(t).
Thus the analysis is not restricted to equal frequencies
of the incoming signal and the free-running VCO signal.

48
These equations suggest the model in part (b) of the Figure
The multiplier output is ABsin[ωct + θi(t)]cos[ωct + θo(t)]
=(1/2)AB (sin[θi(t) - θo(t)]+sin[2ωct + θi(t) + θo(t)]).
The sum frequency term is suppressed by the loop filter.
Hence the effective input to the loop filter is (1/2)AB sin[θi(t) - θo(t)].
If h(t) is the unit impulse response of the loop filter,
eo(t)=h(t)*(1/2) AB sin[θi(t) - θo(t)]
Since , we have
where K=cB/2 and θe(t) is the phase error, defined by θe(t) = θi(t) - θo(t).
     
0
1
sin
2
t
i o
AB h t x x x dx
 
 
  
 

   
o o
t ce t
       
0
sin
t
o e
t AK h t x x dx
 
 
   


49
5.4 Demodulation of FM, PM Waves with
PLL
When the incoming FM signal is A sin[ωct + θi(t)],
Hence,
and assuming a small error θe(t),
Thus, the PLL acts as an FM demodulator. If the incoming
signal is a PM wave, and
In this case the output of the PLL should be integrated to
obtain the message signal m(t).
    .
t
i f
t k m d
  

 
     
t
o f e
t k m d t
   

 

   
i p
t k m t
     
1
.
o p
e t k m t
c
 &
       
t
m
c
k
t
e
t
m
k
t
f
o
f
o 


i.e.,


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