Transcript Document
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
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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
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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
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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
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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
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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
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Waveforms
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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
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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:
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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 .
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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- 3m
c- 4m
J3()
c- 2m
c c+ m
J4()
c+ 3m
c+ 2m
c+ 4m
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.
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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= 2fm=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 + )
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