Angle Modulation
Professor Z Ghassemlooy
Electronics & IT Division
Scholl of Engineering
Sheffield Hallam University
U.K.
www.shu.ac.uk/ocr
Z. Ghassemlooy
Contents
 Properties of Angle (exponential) Modulation
 Types
– Phase Modulation
– Frequency Modulation
 Line Spectrum & Phase Diagram
 Implementation
 Power
Z. Ghassemlooy
Properties
 Linear CW Modulation (AM):
– Modulated spectrum is translated message spectrum
– Bandwidth  message bandwidth
– SNRo at the output can be improved only by increasing
the transmitted power
 Angle Modulation: A non-linear process:– Modulated spectrum is not simply related to the
message spectrum
– Bandwidth >>message bandwidth. This results in
improved SNRo without increasing the transmitted
power
Z. Ghassemlooy
Basic Concept
 First introduced in 1931
A sinusoidal carrier signal is defined as: c ( t )  E c cos [  c t   c ( t )]
For un-modulated carrier signal the total instantaneous angle
is:
 c (t )   c t   c (t )
j ( t )
Thus one can express c(t)
c ( t )  E c cos  c ( t )  E c Re [ e c ]
as:
Thus:
• Varying the frequency fc 
Frequency modulation
• Varying the phase c

Phase modulation
Z. Ghassemlooy
Basic Concept - Cont’d.
 In angle modulation: Amplitude is constant, but angle
varies (increases linearly) with time
c(t)
(red)
Unmodulated
carrier
47/2
35/2
23/2
11/2
-/2
Amplitude
Ec
Slope = c/t
Initial
phase c
Frequency-modulated
angle
Phase-modulated
angle
0
t=0
Unmodulated
carrier
1
2
3
t
(ms)
4
t
2
0
-1
Z. Ghassemlooy
m(t)
Phase Modulation (PM)
 c (t )  m (t )  K p m (t )
PM is defined If
K p  180
0
c ( t ) PM  E c cos [  c t  K p m ( t )]
Thus
Where Kp is known as the phase modulation index
Instantaneous phase
i(t)
Ec
c(t)
 i (t )  K p m (t )
Instantaneous frequency
c(t)
c(t)
 i (t ) 
Rotating Phasor diagram
Z. Ghassemlooy
d  c (t )
dt

  c   c (t )
Frequency Modulation (FM)
The instantaneous frequency is;
 i (t )   c  K f m (t )
Where Kf is known as the frequency deviation (or frequency modulation index).
Note: Kf < fc to make sure that f(t) >0.

Note that
Instantaneous phase
 c (t )  K f m (t )

Integrating
 i (t )   c   c (t )
t
 c (t )   c t  K
f
 m ( t ) dt  0
0
t
Substituting c(t) in c(t) results in: c ( t ) FM  E c cos [  c t  K f  m ( t ) dt ]
0
Z. Ghassemlooy
Waveforms
Z. Ghassemlooy
Important Terms
 Frequency swing
f p  p  K f E mp  p
 Carrier Frequency Deviation (peak)
fc  fd  K f Em
 Rated System Deviation (i.e. maximum deviation allowed)
FD =
75 kHz, FM Radio, (88-108 MHz band)
25 kHz, TV sound broadcast
5 kHz, 2-way mobile radio
2.5 kHz, 2-way mobile radio
m 
 Percent Modulation
 Modulation Index

fd
FD
fd
fm
Z. Ghassemlooy
 100 %
FM Spectral Analysis
Let modulating signal m(t) = Em cos mt
Substituting it in c(t)FM expression and integrating it results in:
t
c ( t ) FM  E c cos [  c t  K
f
 m ( t ) dt ]  E c cos [  c t 
0
Since

fd
fm
K
f
m
E m sin  m t ]
and  f c  f d  K f E m
c ( t ) FM  E c cos [  c t   sin  m t ]  E c cos  c t cos ( sin  m t )  E c sin  c t sin ( sin  m t )
the terms cos ( sin mt) and sin ( sin mt) are defined in
trigonometric series, which gives Bessel Function Coefficient as:
Z. Ghassemlooy
Bessel Function Coefficients
cos ( sin x) = J0 () + 2 [J2() cos 2x + J4() cos 4x + ....]
And
sin ( sin x) = 2 [J1() sin x + J3() sin 3x + ....]
where Jn() are the coefficient of Bessel function of the 1st kind, of the order n
and argument of .
Z. Ghassemlooy
FM Spectral Analysis - Cont’d.
Substituting the Bessel coefficient results in:
c ( t ) FM / E c  cos  c t [ J 0 (  )  2 J 2 ( ) cos 2  m t  2 J 4 ( ) cos 4  m t  ........]
 sin  c t [ 2 J 1 (  ) sin  m t  2 J 3 (  ) sin 3  m t  .........]
Expanding it results in:
c ( t ) FM  E c J 0 (  ) cos  c t
Carrier signal
 E c { J 1 (  )[cos (  c   m ) t  cos (  c   m ) t )]
 E c { J 2 (  )[cos (  c  2  m ) t  cos (  c  2  m ) t )]
 E c { J 3 (  )[cos (  c  3  m ) t  cos (  c  3  m ) t )]
Side-bands signal
(infinite sets)
 .......... .......... .........}

Since
J  n (  )  )  1) J n (  )
n
Then
c ( t ) FM  E c
 J n ( ) cos (  c 
n  
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n  m )t
FM Spectrum
J0()
Side bands
J1()
J4()
J2()
J2()
c- 3m
c- 4m
J3()
c- 2m
c c+ m
J4()
c+ 3m
c+ 2m
c+ 4m
Side bands
Bandwidth (?)
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J3()
FM Spectrum - cont’d.
• The number of side bands with significant amplitude depend on 
see below
 = 0.5
 = 1.0
c
 = 2.5
c
=4
c
c
Bandwidth
Most practical FM systems have 2 <  < 10
Generation and transmission of pure FM requires infinite bandwidth,
whether or not the modulating signal is bandlimited. However practical
FM systems do have a finite bandwidth with quite well pwerformance.
Z. Ghassemlooy
FM Bandwidth BFM
 The commonly rule used to determine the bandwidth is:
– Sideband amplitudes < 1% of the un-modulated carrier can be
ignored. Thus Jn()> 0.01
For large values of ,
BFM = 2nfm= 2fm=2 (fc/ fm).fm = 2 fc
For small values of  ,
BFM = 2fm
General case: use Carson equation
For limited
cases
BFM  2(fc + fm)
BFM  2 fm (1 + )
Z. Ghassemlooy