1
Lecture 4. Analog Communications
Part II. Frequency Modulation (FM)
• Angle Modulation (FM and PM)
• Spectral Characteristics of FM signals
• FM Modulator and Demodulator
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4
2
More About Amplitude Modulation
• Simple and bandwidth efficient (compared to FM)
• High requirement on amplifiers
– Linear amplifiers are difficult to achieve in applications.
• Low fidelity performance
– Noise enhancement in quiet periods
– No tradeoff between bandwidth and fidelity performance
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4
3
Angle Modulation
An angle-modulated signal can be written as
s AnM (t )  A cos( (t ))
f (t ) 
1 d (t )

2
dt
 (t ) : Instantaneous Phase
f (t ) : Instantaneous Frequency
A typical carrier signal:  (t )  2 f c t   (t ), f (t )  f c 
d (t )
dt
• Phase Modulation (PM):
 (t )   s (t )
sPM (t )  A cos(2 f c t   s (t ))
• Frequency Modulation (FM):
t
d (t )
sFM (t )  A cos(2 ( f c t  k  s ( )d ))
 ks(t )

dt
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4
4
Phase Deviation and Frequency Deviation
Phase Modulation (PM) signal sPM (t )  A cos(2 f c t   s (t ))
Instantaneous Phase
 (t )  2 f ct   s(t )
max |  s (t ) | is called the maximum (peak) phase deviation
Peak phase deviation represents the maximum phase difference
between the transmitted signal and the carrier signal.
Frequency Modulation (FM) signal sFM (t )  A cos(2 ( f c t  k
Instantaneous Frequency
f (t ) 

t

s( )d ))
1 d (t )

 f c  ks (t ) Hz
2
dt
max | ks (t ) | is called the maximum (peak) frequency deviation
Peak frequency deviation represents the maximum departure
of the instantaneous frequency from the carrier frequency.
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4
5
Relationship between PM and FM
sPM (t )  A cos(2 f c t   s (t ))
PM
Modulator
x(t)
t
sFM (t )  A cos(2 ( f c t  k  s ( )d ))
dx(t )
dt
FM
Modulator
A cos(2 f c t  x(t ))

• Phase modulation of the carrier with a message signal is equivalent to
frequency modulation of the carrier with the derivative of the message signal.
• We will only focus on FM in the following.
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EE3008 Principles of Communications
Lecture 4
6
FM Signal
s(t)
sFM(t)
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EE3008 Principles of Communications
Lecture 4
7
Spectral Characteristics of Frequency
Modulated Signals
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EE3008 Principles of Communications
Lecture 4
8
A False Start
• Frequency Modulation (FM):
t
sFM (t )  A cos(2 ( f c t  k  s ( )d ))

Instantaneous Frequency f (t )  f c  ks (t )
Suppose that the peak amplitude of s(t) is
ms. Then the maximum and minimum values
of the instantaneous frequency of the
modulated signal would be fc+kms and fckms. Then the spectral components of the
FM signal would be within the frequency
band of [fc-kms, fc+kms] with the
bandwidth of 2kms.
Where is the
fallacy in this
reasoning?
Lin Dai (City University of Hong Kong)
The bandwidth of
the FM signal could
be arbitrarily small
by using an
arbitrarily small k!
Too good to
be true 
EE3008 Principles of Communications
Lecture 4
9
FM Sinusoidal Signal
•
Let us first assume the message signal s(t) is a sinusoidal signal:
s (t )  Am cos(2 f mt )
t
sFM (t )  A cos{2 [ f c t  k  Am cos(2 f m )d ]}

Instantaneous phase
kAm
=A cos{2 [ f c t 
sin(2 f mt )]}
2 f m
Instantaneous frequency:
=A cos[2 f c t 
Peak frequency deviation:
f
sin(2 f mt )]
fm
=A cos[2 f c t   sin(2 f mt )]
f (t )  f c  kAm cos(2 f mt )
f  max | ks (t ) | kAm
• Peak frequency deviation f  kAm is proportional to Am, the
amplitude of the message signal s(t).
•

f
is defined as the modulation index.
fm
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EE3008 Principles of Communications
Lecture 4
10
Modulation Indices of AM-DSB-C and FM
AM-DSB-C -- with a message signal s(t): m 
max[ s (t )  c]  min[ s (t )  c]
max[ s (t )  c]  min[ s (t )  c]
With a sinusoidal message signal s (t )  Am cos(2 f mt ) :
s AM  DSB C (t )  A( s (t )  c) cos(2 f c t )
m
( Am  c)  ( Am  c) Am

( Am  c)  ( Am  c)
c
=Ac(m cos(2 f mt )  1) cos(2 f c t )
FM -- With a sinusoidal message signal s (t )  Am cos(2 f mt ) :
sFM (t )  A cos[2 f c t   sin(2 f mt )]
With a general message signal s(t):
Lin Dai (City University of Hong Kong)


kAm
fm
k max | s (t ) |
Bs
EE3008 Principles of Communications
Lecture 4
11
FM Sinusoidal Signal (Cont’d)
sFM(t) can be expanded as an infinite Fourier series:
sFM (t )  A cos[2 f ct   sin(2 f mt )]

= A  J n (  ) cos[2 ( f c  nf m )t ]
n 
Jn() is called the Bessel Function of the first kind and of order n, which
is defined by
1
J n ( ) 
2

j (  sin x  nx )
e
dx


Read Reference [1] (Sec. 3.3.2) to see the derivation of sFM(t)
and more details about Bessel Function.
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4
12
A Little Bit about Bessel Function
n
For small values of  :
J n ( ) 
For large values of  :
 n 

J n ( ) 
cos    


4 2 

Symmetry property:
 J ( )
J n ( )   n
 J n (  )
2n n !
2
Lin Dai (City University of Hong Kong)
n even
n odd
J n ( )  J n ( )
EE3008 Principles of Communications
Lecture 4
13
Magnitude Spectrum of FM Sinusoidal Signals

A 
A 
j 2 ( f c  nf m ) t
  J n (  )e  j 2 ( fc  nfm )t
sFM (t )  A  J n (  ) cos[2 ( f c  nf m )t ]   J n (  )e
2 n 
2 n 
n 
S FM ( f ) 
A 
A 
J
(

)

(
f

f

nf
)

J n (  ) ( f  f c  nf m )


n
c
m
n 
2 n 
2
|SFM(f)|
A |J ()|
0
2
A
|J1()|
2
A
|J-1()| sideband
2
A
|J2()|
2…
-fc-2fm
A
|J-2()|
2 …
-fc-fm
-fc
-fc+fm -fc+2fm
A |J ()|
0
2
A
|J-1()|
2
Carrier frequency
A
|J1()|
2
A
|J2()|
2…
A
|J-2()|
2 …
0
fc-2fm
sideband
fc-fm
fc
fc+fm fc+2fm
The bandwidth of an FM sinusoidal signal is  .
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4
f
14
Power Spectrum of FM Sinusoidal Signals
S FM ( f ) 
A 
A 
J
(

)

(
f

f

nf
)

J n (  ) ( f  f c  nf m )


n
c
m
n 
n 
2
2
A2
GFM ( f ) 
4
A2
|J1()|2
4


n 
A2|J ()|2
0
4
A
|J2()|2
4
…
-fc-2fm
-fc-fm
A2
|J-1()|2
4
n 
2
-fc+fm -fc+2fm
Lin Dai (City University of Hong Kong)
A2|J ()|2
0
4
A2
|J1()|2
4
A2
|J-1()|2
4
A2 |J ()|2
2
4…
A2
|J-2()|2
4 …
…
-fc

J n (  )  ( f  f c  nf m )
GFM(f)
A2 |J ()|2
-2
4
2

A2
J n (  )  ( f  f c  nf m ) 
4
2
0
fc-2fm
fc-fm
fc
fc+fm fc+2fm
EE3008 Principles of Communications
Lecture 4
f
15
Power Spectrum of FM Sinusoidal Signals
A2
|J1()|2
4
A2|J ()|2
0
4
GFM(f)
A2
|J-1()|2
4
-fc-2fm
-fc-fm
-fc+fm -fc+2fm
A2
A2 2
Pt 
 J 0 ( )  
2
2
Power at carrier

A2 |J ()|2
2
4…
A2
|J-2()|2
4 …
…
-fc
fc-2fm
0
fc-fm
1
Power at sidebands
fc
fc+fm fc+2fm

J 02 (  )  2 n 1 J n2 (  )  1

A2
sFM (t )  A cos[2 f c t   sin(2 f mt )]  Pt 
2
Lin Dai (City University of Hong Kong)
f

2
A
 n1 J n2 ( )   n J n2 ( )  2

A2
|J1()|2
4
A2
|J-1()|2
4
A2 |J ()|2
-2
4
A2
|J2()|2
4
…
A2|J ()|2
0
4
EE3008 Principles of Communications
Lecture 4
16
Effective Bandwidth of FM Sinusoidal Signals
GFM(f)
A2 |J 
-1
4
A |J

-(1+)
4
2
≈
…
0
fc-(1+)fm
…
fc-fm
A2 |J 
0
4
A2 |J 
1
4
fc
fc+fm
…
…
A2 |J

(1+)
4
fc+(1+)fm
f
98% power
2(1+fm
Carson’s Rule: The effective bandwidth of an FM sinusoidal signal
is given by
<<1: narrowband FM
Large : wideband FM
2(1   ) f m  2( f m  f )
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
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Bandwidth Efficiency of FM Signals
• Bandwidth Efficiency of FM sinusoidal signals:
 FM
fm
1
 50%


2(1   ) f m 2(1   )
 Worse than AM systems.
• Bandwidth Efficiency of FM signals:
 FM 
Bs
1

 50%
2(1   ) Bs 2(1   )
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4
18
Edwin Howard Armstrong: Inventor of Modern FM Radio
(December 18, 1890 – January 31, 1954)
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EE3008 Principles of Communications
Lecture 4
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Bandwidth Efficiency of FM Signals
• Bandwidth Efficiency of FM sinusoidal signals:
 FM
fm
1
 50%


2(1   ) f m 2(1   )
• Bandwidth Efficiency of FM signals:
 FM 
Bs
1

 50%
2(1   ) Bs 2(1   )
 Worse than AM systems.
 Larger :
 Lower bandwidth efficiency
 Better fidelity performance
FM systems can provide much better fidelity performance than AM systems by
sacrificing the bandwidth efficiency.
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EE3008 Principles of Communications
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20
Pros and Cons of FM
• Less requirement on amplifiers (constant amplitude)
• Flexible tradeoff between channel bandwidth and fidelity
performance
• Low bandwidth efficiency
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EE3008 Principles of Communications
Lecture 4
21
FM Modulator and Demodulator
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EE3008 Principles of Communications
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Direct FM
The carrier signal used in a direct FM system can be generated by
a sinusoidal oscillator circuit where the oscillator frequency is
controllable.
For example, in the circuit shown below, the oscillator frequency
can be adjusted by tuning the capacitance of Cv.
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EE3008 Principles of Communications
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23
Indirect FM
In practice, it is very difficult to construct highly stable oscillators
that can be voltage-controlled accurately. Therefore, direct FM is
not commonly used in FM broadcast transmitters. It is only used in
applications where low equipment cost is more important than
frequency stability, e.g. radio control.
Indirect FM is more widely adopted as it is easier for practical
circuit realization. An indirect FM modulator includes two steps:
 A highly stable narrowband FM (NBFM) modulator (i.e., with a
small ) that does not require voltage-controlled oscillators; and
 A frequency multiplier to increase . This is usually done together
with frequency shifting and bandwidth expanding.
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4
24
Narrowband FM (NBFM) Modulator
NarrowBand FM (NBFM) is a special case of FM where the
modulation index, , is small (usually  << 1). Recall that an
FM signal is given by:
t
sFM (t )  A cos{2 [ f c t  k  s ( )d ]}

t
 (t )  2 k  s ( )d
 A cos(2 f c t   (t ))
 A cos(2 f c t ) cos( (t ))  A sin(2 f c t ) sin( (t ))

If | (t)| is small then we have the following approximations:
cos((t))  1 and sin((t))  (t)
As a result,
t
sFM(t)  Acos(2fct) - A(t)sin(2fct )  (t )  2 k  s( )d

t
sFM (t )  A cos(2 f c t )  A sin(2 f c t ) 2 k  s( ) d
Lin Dai (City University of Hong Kong)

EE3008 Principles of Communications

Lecture 4
25
Armstrong FM Modulator
A cos(2 f c t )
sFM (t )  A cos(2 f c t )

t
 A sin(2 f c t ) 2 k  s( ) d
t
s(t)
2 k  s ( )d

A sin(2 f c t )

Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4

26
Frequency Multiplier
The main advantage of Armstrong FM modulator is its high
frequency stability. While the Armstrong modulator is only suitable
for FM with a small . For large  , a frequency multiplier can be
used at the output of the Armstrong modulator.
In particular, let us consider a frequency doubler defined as:
eo(t) = ei2(t).
If ei(t) is an FM signal, e.g., ei(t) = cos(2fct +  sin2fmt), we have
eo(t) = cos2 (2fct +  sin2fmt) = 0.5[1+cos (22fct + 2 sin2fmt)]
Both  and carrier frequency have been doubled.
A frequency multiplier can be formed by cascading several doublers.
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EE3008 Principles of Communications
Lecture 4
27
FM Demodulator: Slope Detection
Finally, we briefly discuss the FM demodulator.
Let us take the derivative of an FM signal sFM (t )  A cos  2




t
f c t  k  s ( )d  :



t
dsFM (t )



  2 f c  2 ks (t )     A sin  2 f c t  k  s ( )d  



dt

The envelope of this signal is:
A  2 f c  2 ks (t ) 
We can then recover s(t) from this envelope signal by removing
its DC component.
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EE3008 Principles of Communications
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Summary of FM and AM
Complexity
DSB-SC
AM
FM
Bandwidth
Efficiency
high
DSB-C
low
SSB
high
VSB
high
moderate
50%
Fidelity
(evaluated by output SNR)
~ 1/Bs
~ Pt
100%
50% 
(DSB-C has a lower SNR)
Bs
 100%
Bs  
Bs
 50%
2(1   ) Bs
~ 1/Bs
~ Pt
~2
More bandwidth, better fidelity performance
Lin Dai (City University of Hong Kong)
EE3008 Principles of Communications
Lecture 4