lecture 8 angle modulation - university of ottawa
TRANSCRIPT
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9-Jan-18
Angle Modulation
Angle modulation: frequency modulation (FM) or phase
modulation (PM).
Basic idea: vary frequency (FM) or phase (PM) according
to the message signal.
While AM is (almost) linear, FM or PM is highly nonlinear.
FM/PM provide many advantages (main – noise
immunity) over AM, at a cost of larger bandwidth.
Demodulation may be complex, but modern ICs allow
cost-effective implementation.
Example: FM radio (high quality, not expensive
receivers).
Lecture 8
Lecture 8, ELG3170 : Introduction to Communication Systems © S. Loyka 1(28)
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Angle Modulation: Basic Definitions
• Angle-modulated signal (PM or FM) can be expressed as:
• Phase modulation:
• Frequency modulation:
( ) ( )( )cosc
x t A t= ψ
( ) ( ), ( ) ( )c
t t t t m tψ = ω +ϕ ϕ = ∆ϕ ⋅
( ) ( )0
, ( ) ( )
t
ct t d t m tψ = ω + Ω τ τ Ω = ∆Ω⋅∫
• Max phase deviation:
• Max frequency deviation:
• Normalized message signal:
Max ( ) Max ( )c
t t t∆ϕ = ϕ = ψ −ω
Max ( ) Max ( )c
t t∆Ω = Ω = ω −ω
( ) 1m t ≤
Note: deviation is w.r.t. unmodulated value.
[ ] 0( ) (0)
ct tψ ≈ ω +Ω +ϕfor a short period of time (small t):
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Angle Modulation: Parameters
• Instantaneous frequency:
• Instantaneous phase:
• Effect of mod. signal amplitude:
( )( )
( ) ( )
( ) ( )
,
,
c c
c c
d t dm td t PM
t dt dtdt
t m t FM
ϕψ ω + = ω + ∆ϕ
ω = = ω +Ω = ω + ∆Ω⋅
( ) ( )
( )
( ) ( )0
0 0
( ),
,
c ct
t t
c c
t t t m t PM
t dt d t m d FM
ω +ϕ = ω + ∆ϕ⋅
ψ = ω τ τ = ω + Ω τ τ = ω + ∆Ω τ τ
∫∫ ∫
( ) ( ), max ( ) 1M t A m t m t= ⋅ =
,
,
p
f
k A PM
k A FM
∆ϕ =∆Ω =
, - modulation constants,
Hz/V & rad./V
f pk k
measured in lab 3.
Lecture 8, ELG3170 : Introduction to Communication Systems © S. Loyka 3(28)
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Angle Modulation: Examplesrectangular pulse sawtooth pulse
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Example: Sinusoidal Modulating Signal
• Assume that
• Instantaneous phase:
• Modulated signal:
• Modulation indices:
( )( ) cosm
m t t= ω
( )( )
( )
cos ,
sin ,
c m
c m
m
t t PM
tt t FM
ω + ∆ϕ⋅ ωψ = ∆Ωω + ω ω
( )
( )
( )
cos cos ,
cos sin ,
c c m
c c m
m
A t t PM
x t
A t t FM
ω + ∆ϕ⋅ ω = ∆Ω
ω + ω ω
,
,
p
fm
PM
FM
β = ∆ϕ
∆Ωβ = ω
Valid in general case
as well, with
-> max.
modulating frequencym
ω
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Spectrum of Angle-Modulated Signal
• Consider sinusoidal modulating signal:
• Complex envelope is expanded in Fourier series:
• Expansion coefficients are
• Finally,
( ) ( ) ( )sincos sin Re m c
j t j tc c m cx t A t t A e e
β⋅ ω ω = ω +β⋅ ω =
( ) ( )sinm m
j t jn t
c c n
n
C t A e A c e
∞β⋅ ω ω
=−∞
= = ∑
( )
0
2 sinsin
0
1 1( )
2
mT m
m m
u tj u nuj t jn t
n n
m
c e e dt e du JT
=ω π β −β ω − ω= = = β
π∫ ∫
( ) ( ) ( )cosc n c m
n
x t A J n t
∞
=−∞
= β ω + ω ∑
- Bessel function of 1st kind & n-th order,( )n
J β ( ) ( )( 1)nn n
J J−
β = − β
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Spectrum of Angle Modulation:
Increasing n moves the peak to the right!
( )n
J β
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n 0.1β = 0.2β = 0.5β = 1β = 2β = 5β = 8β = 10β = n
0 0.998 0.990 0.938 0.765 0.224 -0.178 0.172 -0.246 0
1 0.050 0.100 0.242 0.440 0.577 -0.328 0.235 0.043 1
2 0.001 0.005 0.031 0.115 0.353 0.047 -0.113 0.255 2
3 0.020 0.129 0.365 -0.291 0.058 3
4 0.002 0.034 0.391 -0.105 -0.220 4
5 0.007 0.261 0.186 -0.234 5
6 0.001 0.131 0.338 -0.014 6
7 0.053 0.321 0.217 7
8 0.018 0.223 0.318 8
9 0.006 0.126 0.292 9
10 0.001 0.061 0.207 10
11 0.026 0.123 11
12 0.010 0.063 12
13 0.003 0.029 13
14 0.001 0.012 14
15 0.004 15
16 0.001 16
the last significant
spectral component:
[ ]1n = β+
( ) ( ) ( )cosc n c m
n
x t A J n t
∞
=−∞
= β ω + ω ∑
Spectrum of Angle Modulation: ( )n
J β
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Spectrum: Examples
100 150 200 250 3000
0.5
1
.
0.3β =
100 150 200 250 3000
0.5
1
.
1β =
5β =
0 100 200 300 4000
0.1
0.2
0.3
0.4
.
0 100 200 300 4000
0.2
0.4
.
10β =
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Bandwidth of Angle-Modulated Signal
• Power bandwidth (98% of the power) of angle-modulated
signal (Carson’s rule):
• Power bandwidth of PM and FM signals:
• These expressions hold for a general modulating signal
as well, - the max. modulating frequency.
• Angle modulation with large index expands spectrum!
( )2 1m
∆ω ≈ β+ ω
( )( )
( )
2 1 ,2 1
2 ,
m
m
m
PM
FM
∆ϕ+ ω∆ω≈ β+ ω = ∆Ω+ω
mω
Lecture 8, ELG3170 : Introduction to Communication Systems © S. Loyka 10(28)
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Narrowband Angle Modulation
• Modulation index is low,
• Modulated signal can be expressed as:
• Similar to AM signal, the bandwidth is (both, PM & FM)
( ) ( )
( ) ( )
cos sin
cos cos cos2 2
c c m
c c
c c c m c m
x t A t t
A AA t t t
= ω +β⋅ ω = β β
= ω + ω +ω − ω −ω
1β<<
2 ( )x
S f
f
cf( )c
f F− ( )cf F+
0,180
2
cA β 0, 0
2
cA β
cA
2m
∆ω≈ ω
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Wideband Angle Modulation
• Modulation index is high,
• The signal bandwidth is:
• Different for PM and FM!
• Wideband FM: the bandwidth is twice the frequency
deviation. Does not depend on the modulating
frequency.
• Wideband PM: the bandwidth depends on modulating
frequency.
• Modulation index bandwidth expansion factor.
1β >>
2 ,2
2 ,
m
m
PM
FM
∆ϕ⋅ω∆ω≈ βω =
∆Ω
β =
Lecture 8, ELG3170 : Introduction to Communication Systems © S. Loyka 12(28)
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9-Jan-18
PM Modulator
variable phase
shifter
( )m t
cos( )c
A tω
~
( ) ( )cos ( )c c
x t A t m t= ω +β⋅
RF oscillator
General
principle:
Practical
implementation:
L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.
0
0
,
modulation constant
resonant frequency
ck f f f f
k
f
∆ϕ ≈ ∆ ∆ = −
=
=
Q.: sketch ( )f∆ϕ ∆
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9-Jan-18
FM Modulator
RF oscillator
( )m t
( ) ( ) 0
0
cos
t
c cx t A t d
= ω + Ω τ τ + ϕ
∫variable
resonant
circuit
General
principle:
Practical
implementation:
Difficulty: frequency
stability.
Suitable for
narrowband FM only.
L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.
Lecture 8, ELG3170 : Introduction to Communication Systems © S. Loyka 14(28)
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9-Jan-18
Narrowband Angle Modulator
( ) [ ]cos ( )
cos sin ( )
c c
c c c c
x t A t m t
A t A t m t
= ω +β⋅ ≈
≈ ω − ω ⋅β⋅
( )pm tβ
( )fm tβ
Small modulation index: 1β≪
sin ( )c c
A t m tω ⋅β⋅
Message signal
Out
J.Proakis, M.Salehi, Communications Systems Engineering, Prentice Hall, 2002
Q.: can an AM modulator
be used instead?
Lecture 8, ELG3170 : Introduction to Communication Systems © S. Loyka 15(28)
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9-Jan-18
Indirect Wideband Angle Modulator
Frequency multiplier:
( )2 1cos ( ) 1 cos 2 ( )
2t tψ = + ψ
( )1cos 2 ( )
2tψ
BPF
J.P
roakis
, M
.Sale
hi, C
om
munic
ations S
yste
ms E
ngin
eering, P
rentice H
all, 2002
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Indirect Wideband FM Transmitter
(Amstrong)
B.P. Lathi, Modern Digital and Analog Communications Systems, Oxford University Press
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Direct Wideband Angle Modulator
L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.
Explain how it operates
Hint: consider it without feedback first
Explain why feedback is required
Explain why frequency divider is required
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FM Demodulators
• FM-to-AM conversion:
( ) ( )0for
2
c
c c
BH f H f f f f= + α − − <
( ) ( ) 0
0
cos
t
c cx t A t d
= ω + Ω τ τ +ϕ
∫ ( ) ( ) [ ]0
( ) cos *c
y t A H t= +αΩ
( )tΩ∼
• Possible candidate: (differentiator) ( ) 2H f f= π
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FM Slope Detector• Circuit diagram:
• Magnitude frequency response:
fre
qu
en
cy b
an
d fo
r lin
ea
r F
M-
AM
co
nve
rsio
n
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Balanced Discriminator: Block Diagram
L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.
Lecture 8, ELG3170 : Introduction to Communication Systems © S. Loyka 21(28)
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Balanced Discriminator: Circuit Diagram
L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.
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Phased Locked Loop (PLL) Detector
( ) [ ]sin ( )in in c inv t A t t= ω +ϕ
( ) [ ]0 0 0cos ( )
cv t A t t= ω +ϕ
( ) ( )0 2( ) ( )
VCO c c
dt t t v t
dtω = ω +ϕ = ω +α
Informally,
0( ) ( )
int tϕ ≈ ϕ
2
1( ) ( )
in
dv t t
dt≈ ϕα
( ) [ ]1 2
1 0sin ( ) ( ) (2 )term
2in c
A Av t t t= ϕ −ϕ + ω
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PLL Detector: Linear Model
( )in
tϕ
0( )tϕ
( )t∆ϕ ( )1v t
0 2( ) ( )
dt v t
dtϕ = α
( ) [ ]
( )
1 2
2 0
1 2
0
sin ( ) ( )2
( ) ( ) ( )2
in
in d
A Av t t t
A At t K t
= ϕ −ϕ ≈
≈ ϕ −ϕ = ∆ϕ
[ ]2
1 1( ) ( ) ( ) ( )
in in
d dv t t t t
dt dt= ϕ − ∆ϕ ≈ ϕα α
( ) [ ]1 2
1 0sin ( ) ( )
2
(2 )term
in
c
A Av t t t= ϕ −ϕ +
+ ω
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FM Demodulator: Lab 3
explain its operation !
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Comparison of AM and FM/PM
• AM is simple (envelope detector) but no
noise/interference immunity (low quality).
• AM bandwidth is twice or the same as the modulating
signal (no bandwidth expansion).
• Power efficiency is low for conventional AM.
• DSB-SC & SSB – good power efficiency, but complex
circuitry.
• FM/PM – spectrum expansion & noise immunity. Good
quality.
• More complex circuitry. However, ICs allow for cost-
effective implementation.
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Important Properties of Angle-Modulated
Signals: Summary
• FM/PM signal is a nonlinear function of the message.
• The signal’s bandwidth increases with the modulation
index.
• The carrier spectral level varies with the modulation
index, being 0 in some cases.
• Narrowband FM/PM: the signal’s bandwidth is twice that
of the message (the same as for AM).
• The amplitude of the FM/PM signal is constant (hence,
the power does not depend on the message).
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Summary
Angle modulation: PM & FM
Spectra of angle-modulated signals. Modulation index.
Narrowband (low-index) & wideband (large-index) modulation.
Signal bandwidth.
Relation between PM and FM.
Generation of angle-modulated signals. Narrowband & wideband
modulators.
Demodulation of PM and FM signals. Slope detector & balanced
discriminator. PLL detector.
Comparison of AM and FM/PM.
Homework: Reading: Couch, 5.6, 4.13, 4.14. Study carefully all
the examples and make sure you understand them.
Lecture 8
Lecture 8, ELG3170 : Introduction to Communication Systems © S. Loyka 28(28)