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Analog Communications
Frequency Modulation (FM)

• Angle Modulation (FM and PM)

• Spectral Characteristics of FM signals
• FM Modulator and Demodulator
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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
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Angle Modulation

An angle-modulated signal can be written as

sAnM (t )  A cos( (t ))  (t ) : Instantaneous Phase

1 d (t ) f (t ) : Instantaneous Frequency
f (t )  
2 dt

1 d (t )
A typical carrier signal:  (t )  2 fct   (t ), f (t )  f c  
2 dt
• Phase Modulation (PM):
 (t )   s(t ) sPM (t )  A cos(2 fct   s(t ))

• Frequency Modulation (FM):

1 d (t ) t
  ks (t ) sFM (t )  A cos(2 ( f ct  k  s( )d ))
2 dt 
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Phase Deviation and Frequency Deviation

Phase Modulation (PM) signal sPM (t )  A cos(2 f ct   s(t ))
Instantaneous Phase:  (t )  2 fct   s(t )
Phase Deviation: |  s (t ) |
Maximum (Peak) Phase Deviation: max |  s (t ) |
t
Peak phase deviation represents the maximum phase difference
between the transmitted signal and the carrier signal.
t
Frequency Modulation (FM) signal sFM (t )  A cos(2 ( f ct  k 

s( )d ))
Instantaneous Frequency: f (t )  1  d (t )  f  ks(t ) Hz
2
c
dt
Frequency Deviation: | ks (t ) |
Maximum (Peak) Frequency Deviation: max | ks (t ) |
t
Peak frequency deviation represents the maximum departure
of the instantaneous frequency from the carrier frequency.
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Relationship between PM and FM

sPM (t )  A cos(2 fct   s(t ))
x(t) PM
Modulator

t A cos(2 fct  x(t ))

sFM (t )  A cos(2 ( f ct  k  s( )d ))

dx(t ) FM
dt Modulator

• 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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FM Signal

s(t)

sFM(t)
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Spectral Characteristics of Frequency

Modulated Signals
 8

A False Start

• Frequency Modulation (FM):

t
sFM (t )  A cos(2 ( f ct  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 The bandwidth of
of the instantaneous frequency of the the FM signal could
modulated signal would be fc+kms and fc- be arbitrarily small
kms. Then the spectral components of the by using an
FM signal would be within the frequency arbitrarily small k!
band of [fc-kms, fc+kms] with the
bandwidth of 2kms.
Where is the Too good to
fallacy in this be true 
reasoning?
 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 ct  k  Am cos(2 f m )d ]} Instantaneous phase


kAm Instantaneous frequency:

=A cos{2 [ f ct  sin(2 f mt )]}
2 f m f (t )  fc  kAmcos(2 f mt )
f
=A cos[2 f ct  sin(2 f mt )] Peak frequency deviation:
fm
f  max | ks(t ) | kAm
=A cos[2 fct   sin(2 f mt )]

• 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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Modulation Indices of AM-DSB-C and FM

max[ s(t )  c]  min[ s(t )  c]

AM-DSB-C -- with a message signal s(t): m 
max[ s(t )  c]  min[ s(t )  c]

With a sinusoidal message signal s(t )  Am cos(2 f mt ) :

( Am  c)  ( Am  c) Am
sAM  DSBC (t )  A(s(t )  c) cos(2 fct ) m 
( Am  c)  ( Am  c) c
=Ac(m cos(2 f mt )  1) cos(2 fct )

FM -- With a sinusoidal message signal s(t )  Am cos(2 f mt ) :

kAm
sFM (t )  A cos[2 fct   sin(2 f mt )] 
fm
k max | s(t ) |
With a general message signal s(t): 
Bs
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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 (  sin x  nx )
J n ( )  e dx
2 

Read Reference to see the derivation of sFM(t)

and more details about Bessel Function.
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A Little Bit about Bessel Function

n
For small values of  : J n ( ) 
2n n !

2   n 
For large values of  : J n ( )  cos     
  4 2 

 J ( ) n even
Symmetry property: J n ( )   n J  n ( )  J n ( )
 J n (  ) n odd
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Magnitude Spectrum of FM Sinusoidal Signals


A  A 
sFM (t )  A  J n (  ) cos[2 ( f c  nf m )t ]   J n (  )e   J n (  )e j 2 ( fc  nfm )t
j 2 ( fc  nf m ) t

n 
2 n  2 n
A  A 
S FM ( f )  
2 n
J n (  ) ( f  f c  nf m )   J n ( ) ( f  fc  nfm )
2 n

A |J ()| |SFM(f)| A |J ()| Carrier frequency

0 0
2 2
A A A A
|J1()| |J-1()| sideband |J-1()| |J1()|
2 2 2 2 sideband
A A A A
|J2()| |J-2()| |J-2()| |J2()|
2… 2 … 2 … 2…
-fc-2fm -fc-fm -fc -fc+fm -fc+2fm 0 fc-2fm fc-fm fc fc+fm fc+2fm f

The bandwidth of an FM sinusoidal signal is  .

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Power Spectrum of FM Sinusoidal Signals

A  A 
S FM ( f ) 
2
 n 
J n (  ) ( f  f c  nf m ) 
2
 n 
J n (  ) ( f  f c  nf m )
 
A2 A2
GFM ( f )   J n (  )  ( f  f c  nf m )   J n (  )  ( f  f c  nf m )
2 2

4 n  4 n 

A2|J ()|2 GFM(f) A2|J ()|2

0 0
4 4
A2 A2 A2
|J1()|2 |J-1()|2 A2
|J-1()|2 |J1()|2
4 4 4 4
A2 A2 |J ()|2 A2 A2 |J ()|2
|J2()|2 -2 |J-2()|2 2
…
4 4
… 4 … 4…
-fc-2fm -fc-fm -fc -fc+fm -fc+2fm 0 fc-2fm fc-fm fc fc+fm fc+2fm f
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Power Spectrum of FM Sinusoidal Signals

A2|J ()|2 GFM(f) A2|J ()|2
0 0
4 4
A2 A2 A2
|J1()|2 |J-1()|2 A2
|J-1()|2 |J1()|2
4 4 4 4
A2 A2 |J ()|2 A2 A2 |J ()|2
|J2()|2 -2 |J-2()|2 2
…
4 4
… 4 … 4…
-fc-2fm -fc-fm -fc -fc+fm -fc+2fm 0 fc-2fm fc-fm fc fc+fm fc+2fm f

 
2
A2 2 A2 A
 n1 J n2 ( )   n J n2 ( )  2
 1
Pt   J 0 ( )  
2 2
J 02 (  )  2 n1 J n2 (  )  1

Power at carrier Power at sidebands 

A2
sFM (t )  A cos[2 fct   sin(2 f mt )]  Pt 
2
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Effective Bandwidth of FM Sinusoidal Signals

A2 |J |2
GFM(f) 0
4 A2 |J |2
2
A2 |J |2
A |J 1 A2 |J
-(1+)| (1+)|
2 -1
4 2
4
4 … … 4
≈

0 fc-(1+)fm … fc-fm fc fc+fm … fc+(1+)fm f

98% power

2(1+fm

Carson’s Rule: The effective bandwidth of an FM sinusoidal signal

is given by
2(1   ) f m  2( f m  f )
<<1: narrowband FM
Large : wideband FM
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Bandwidth Efficiency of FM Signals

• Bandwidth Efficiency of FM sinusoidal signals:

fm 1
 FM    50%
2(1   ) f m 2(1   )  Worse than AM systems.

• Bandwidth Efficiency of FM signals:

Bs 1
 FM    50%
2(1   ) Bs 2(1   )
 18

Edwin Howard Armstrong: Inventor of Modern FM Radio

(December 18, 1890 – January 31, 1954)

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Bandwidth Efficiency of FM Signals

• Bandwidth Efficiency of FM sinusoidal signals:

fm 1
 FM    50%
2(1   ) f m 2(1   )  Worse than AM systems.

 Larger :
• Bandwidth Efficiency of FM signals:
 Lower bandwidth efficiency
Bs 1
 FM    50%  Better fidelity performance
2(1   ) Bs 2(1   )

FM systems can provide much better fidelity performance than AM systems by

sacrificing the bandwidth efficiency.
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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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FM Modulator and Demodulator

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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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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.
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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 [ fct  k  s( )d ]}

t
 A cos(2 fct   (t ))  (t )  2 k  s( )d

 A cos(2 fct ) cos( (t ))  A sin(2 f ct )sin( (t ))
If | (t)| is small then we have the following approximations:
cos((t))  1 and sin((t))  (t)
As a result,
sFM(t)  Acos(2fct) - A(t)sin(2fct )  (t )  2 k  s( )d
t

 t
sFM (t )  A cos(2 f ct )  A sin(2 f ct ) 2 k  s( )d
 
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Armstrong FM Modulator

A cos(2 fct )

sFM (t )  A cos(2 f ct )

 t
 A sin(2 f ct ) 2 k  s ( )d
 

t
A sin(2 fct )
2 k  s( )d

s(t)
 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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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 f ct  k  s( )d  :
t

dsFM (t )
dt

   t

 

  2 f c  2 ks(t )     A sin  2 f ct  k  s( )d  


The envelope of this signal is:
A  2 fc  2 ks(t ) 
We can then recover s(t) from this envelope signal by removing
its DC component.
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Summary of FM and AM

Complexity Bandwidth Fidelity

Efficiency (evaluated by output SNR)

DSB-SC high
50% ~ Pt
AM DSB-C low
SSB high 100% (DSB-C has a lower SNR)
Bs
VSB high 50%   100%
Bs  
Bs
FM moderate  50% ~ Pt ~ 2
2(1   ) Bs

More bandwidth, better fidelity performance