Exponential Carrier Wave Modulation
S-72.1140 Transmission Methods in Telecommunication Systems (5 cr)
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Exponential modulation: Frequency (FM) and phase (PM) modulation
FM and PM waveforms Instantaneous frequency and phase Spectral properties
– narrow band • arbitrary modulating waveform• tone modulation - phasor diagram
– wideband tone modulation– Transmission BW
Generating FM - signals– de-tuned tank circuit– narrow band mixer modulator– indirect modulators
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Contents (cont.)
Detecting FM/PM– FM-AM conversion followed by envelope detector– Phase-shift discriminator– Zero-crossing detection (tutorials)– PLL-detector (tutorials)
Effect of additive interference on FM and PM– analytical expressions and phasor diagrams– implications for demodulator design
FM preemphases and deemphases filters
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Linear and exponential modulation
In linear CW (carrier wave) modulation:– transmitted spectra resembles modulating spectra– spectral width does not exceed twice the modulating
spectral width– destination SNR can not be better than the baseband
transmission SNR (lecture: Noise in CW systems) In exponential CW modulation:
– baseband and transmitted spectra does not carry a simple relationship
– usually transmission BW>>baseband BW– bandwidth-power trade-off (channel adaptation-suppression
of in-band noise)
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Assignment
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Phase modulation (PM) Carrier Wave (CW) signal:
In exponential modulation the modulation is “in the exponent” or “in the angle”
Note that in exponential modulation superposition does not apply:
In phase modulation (PM) carrier phase is linearly proportional to the modulation x(t):
Angular phasor has the instantaneous frequency (phasor rate)
x t A t tC C C
tC
( ) cos( ( ))( )
x t A t tPM C C
x t
tC
( ) cos( ( ) )( ),
( )
3
( ) cos( ( )) Re[e ( )xp( )]C C C CC
x t A t j tA
2 ( )C Cf t
Ct
1 2
1 2
( ) cos ( ) ( )
cos cos ( ) ( )
C C f
C f
x t A t k a t a t
A t A k a t a t
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Instantaneous frequency
Angular frequency (rate) is the derivative of the phase (the same way as the velocity v(t) is the derivative of distance s(t))
For continuously changing frequency instantaneous frequency is defined by differential changes:
( )t
/t s
2
Angle modulated carrier
Constant frequency carrier:
0 02 rad/s, 1Hzf
1st
( )( )
d tt
dt
( ) ( )t
t d
2 1
2 1
( ) ( ) ( )( )
ds t s t s tv t
dt t t
Compare tolinear motion:
( )i
v t
( )q
v t
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Assignment
( )( )
d tt
dt
( ) ( )t
t d
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Frequency modulation (FM) In frequency modulation carrier’s instantaneous frequency is
linearly proportional to modulation frequency:
Hence the FM waveform can be written as
Therefore for FM
and for PM
( ) 2 [ ] ( ) /
( ) 2 [
( )
]
( )
( )
C C C
C C
t f d t dt
t
t f x t
f xf dtt
x t A t f x d t tC C C t
t
C t
( ) cos( ( ) ),( )
z
20 0
( ) ( )Cf t f f x t
integrate
( ) ( )t
t d
( ) ( )t x t
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AM, FM and PM waveforms
x t A t f x dFM C C
t
( ) cos( ( ) ) z 2
x t A t x tPM C C
( ) cos( ( ))
constant frequency followsderivative of the modulation waveform
Instantaneous frequency directlyproportional to modulation waveform
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Assignment
Briefly summarize what is the main difference between FM and PM ?
How would you generate FM by using a PM modulator?
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Narrowband FM and PM (small modulation index, arbitrary modulation waveform) The CW presentation: The quadrature CW presentation:
Narrow band (small angle) condition:
Hence the Fourier transform of XC(t) is
x t A t tC C C( ) cos[ ( )]
x t x t t x t tC ci C cq C( ) ( )cos( ) ( )sin( )
x t A t A tci C C( ) cos ( ) [ ( / !) ( ) ...] 1 1 2 2
x t A t A t tcq C C( ) sin ( ) [ ( ) ( / !) ( ) ...] 1 3 3
( ) 1t rad
x t Aci C( ) x t A t
cq C( ) ( )
X f A f fj
A f f fC C C C C( ) ( ) ( ),
1
2 20
cos( ) cos( )cos( )
sin( ) sin( )
cos( ) ( )sin( )( ) C C CC CA tx A t tt F F
0
0 0
cos(2 ) ( )
1( )exp( ) ( )exp( )
2
f t x t
X f f j jX f f j
F
0
0 0
cos(2 )
1( ) ( )
2
f t
f f f f
F
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Narrow band FM and PM spectra
Remember the instantaneous phase in CW presentation:
The small angle assumption produces compact spectral presentation both for FM and AM:
( ) [ ( )]
( ),PM
( ) / ,FM
f t
X f
jf X f f
F
X f A f fj
A f f fC C C C C( ) ( ) ( ),
1
2 20
PM
FM t
t
t x t
t f x d t t
( ) ( )
( ) ( ) ,
z
20 0
x t A t tC C C( ) cos[ ( )]
0(0) (
)) )
((t
t
GGg d
j
What does it mean to set thiscomponent to zero?
14 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Assume:1
( ) sinc2 ( )2 2
fx t Wt X f
W W
1( ) ( ) ( ), 0
2 2C C C C C
jX f A f f A f f f
( ) [ ( )] ( )PM PMf F t X f
X fPM
( )
( ) [ ( )] ( ) /FM FMf F t jf X f f
1( ) ( ) , 0
2 4 2C
PM C C C
j f fX f A f f A f
W W
1( ) ( ) , 0
2 4 2C
FM C C CC
f f fX f A f f A f
f f W W
X fFM
( )E
xam
ple
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Assignment
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Solution
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Tone modulation with PM and FM:modulation index Remember the FM and PM waveforms:
Assume tone modulation
Then
x t A t f x dFM C C
t
t
( ) cos[ ( ) ]
( )
z
2
x t A t x tPM C C
t
( ) cos[ ( )]( )
x tA t
A tm m
m m
( )sin( ),
cos( ),RST
PM
FM
( ) sin( ),PM
( )2 ( ) ( / )sin( ),FM
m m
m m mt
x t A t
tf x d A f f t
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Tone modulation in frequency domain: Phasors and spectra for narrowband case Remember the quadrature presentation:
For narrowband assume
x t x t t x t tC ci C cq C( ) ( )cos( ) ( )sin( )
x t A t A tci C C( ) cos ( ) [ ( / !) ( ) ...] 1 1 2 2
x t A t A t tcq C C( ) sin ( ) [ ( ) ( / !) ( ) ...] 1 3 3
x t A t tC C C( ) cos[ ( )]
1, ( ) sin( ),t tm
FM,PM
( ) cos( ) sin( )sin( )
cos( ) cos( )2
cos( )2
C C C C m C
C
C C C m
C
C m
x t A t A t t
AA t t
At
/
PM m
FM m m
A
A f f
sin( ) sin( ) cos( ) cos( ) 1
2
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Narrow band tone modulation: spectra and phasors
Phasors and spectra resemble AM:
x t A tA
t
At
C C C
C
C m
C
C m
( ) cos( ) cos( )
cos( )
2
2
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Assignment
( ) cos( )cos( ) cos( )
cos( ) cos( )2 2
cos( )
C C m m C C C
C m C mC m C m
C C
x t A A t t A t
A A A At t
A t
x t A tA
t
At
C C C
C
C m
C
C m
( ) cos( ) cos( )
cos( )
2
2
(i) Discuss the phasor diagrams and explain phasor positions based onanalytical expressions (ii) Comment amplitude and phase modulation in both cases
AM
NB-FM
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FM and PM with tone modulation and arbitrary modulation index
Time domain expression for FM and PM:
Remember:
Therefore:
x t A t tC C C m( ) cos[ sin( )]
cos( ) cos( )cos( )
sin( )sin( )
x t A t t
A t tC C m C
C m C
( ) cos( sin( ))cos( )
sin( sin( ))sin( )
cos( sin( )) ( ) ( )cos( )
sin( sin( )) ( )sin( )
m O mn
m mn
t J J n t
t J n t
2
2neven
nodd
Jn is the first kind, order n Bessel function
/PM m
FM m m
A
A f f
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Wideband FM and PM spectra After simplifications we can write:
x t A J n tC C n C mn( ) ( )cos( )
note: ( ) ( 1) ( )nn nJ J
,PM
/ ,FMm
m m
A
A f f
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Assignment
x t A J n tC C n C mn( ) ( )cos( )
,PM
/ ,FMm
m m
A
A f f
Assuming other parameters fixed, what happens to spectralwidth and line spacing when(i) Frequency deviation increases in FM(ii) Modulation frequency increases in FM(iii) Phase deviation decreases in PM
24 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Determination of transmission bandwidth The goal is to determine the number of significant sidebands Thus consider again how Bessel functions behave as the
function of , e.g. we consider Significant sidebands: Minimum bandwidth includes 2 sidebands (why?): Generally:
Jn( )
B fT m,min
2B M f M
T m 2 1( ) , ( )
M( ) 2
A f Wm m 1,
( ) 0.01MJ
( ) 0.1MJ ,PM
/ ,FMm
m m
A
A f f
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Assignment
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Transmission bandwidth and deviation D
Tone modulation is extrapolated into arbitrary modulating signal by defining deviation by
Therefore transmission BW is also a function of deviation
For very large D and small D with
that can be combined into
1,/ /
m mm m A f W
A f f f W D
B M D WT2 ( )
1,2(
2 1
2)
,m
T m D f WB D f
DW D
2 ( )
2 , (a single pair of sideband1 s)TB M
D
D W
W
2 1 , ,an1 d 1TB D W D D
( ) 2M D D
2( 2)D W
,PM
/ ,FMm
m m
A
A f f
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Example: Commercial FM bandwidth Following commercial FM specifications
High-quality FM radios RF bandwidth is about
Note that
under estimates the bandwidth slightly
f W
D f W
B D WT
75
5
2 2 210
kHz, 15 kHz
kHz,(D > 2)
/
( )
BT200 kHz
B D W DT 2 1 180 kHz, >> 1
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A practical FM modulator circuit
A tuned circuit oscillator– biased varactor diode capacitance
directly proportional to x(t)– other parts:
• input transformer• RF-choke• DC-block
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Generating FM/PM Capacitance of a resonant circuit can be made to be a function
of modulation voltage.
That can be simplified by the series expansion
( - )1 12
38
11 2
2 2
kxkx k x
kx / ...
f LC
f x t LC x t
C x t C Cx t
f x t f Cx t C f LC
CC
CC
CC C C
1 2
1 2
1 1 2
0
0
1 2
0
/ ( )
[ ( )] / { [ ( )]}
[ ( )] ( )
[ ( )] ( ( ) / ) , / ( )/
f x t f Cx t C
fCx t
Cd t
dt
CC C
C
[ ( )] ( ( ) / )
( ) ( )
/
LNM
OQP
1
112
12
0
1 2
0
( ) ( )t f tCC
f x dC C
t
f
z2 22
0
Remember that the instantaneous frequency is the derivative of the phase
Note that this applies for arelatively small modulationindex
De-
tune
d ta
nk c
ircu
it
Capacitance diode
De-tuned resonance frequency
Resonance frequency
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Assignment
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Integrating the input signal to a phase modulator produces frequency modulation! Thus PM modulator applicable for FM
Also, PM can be produced by an FM modulator by differentiating its input
Narrow band mixer modulator:
Phase modulators: narrow band mixer modulator
x t A t f x dFM C C
t
( ) cos( ( ) ) z 2
x t A t x tPM C C
( ) cos( ( ))
1( ) ( ) ( ), 0
2 2( ) ( ) PM
( ) / 2 ( ) 2 [ ( )] FM
C C C C C
C
jX f A f f A X f f f
t x t
d t dt f t f f x t
Narrow band
FM/PM spectra
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Frequency detection
Methods of frequency detection– FM-AM conversion followed by envelope detector– Phase-shift discriminator– Zero-crossing detection (tutorials)– PLL-detector (tutorials)
FM-AM conversion is produced by a transfer function having magnitude distortion, as the time derivative (other possibilities?):
As for example
x t A t t
dx t
dtA t t d t dt
C C
C
C C C
( ) cos( ( ))
( )sin[ ( )]( ( ) / )
0
d t dt f t f f x tC
( ) / ( ) [ ( )] 2 2 FM
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35 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen25
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Assignment
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Indirect FM transmitter FM/PM modulator with high linearity and modulation index difficult to realize One can first generate a small modulation index signal that is then applied
into a nonlinear circuit
Therefore applying FM/PM wave into non-linearity increases modulation index should be filtered away, how?
Mat
hem
atic
a®-e
xpre
ssio
ns
41 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
In[14]:= TrigReduce@Cos@w0 t +Am Sin@wm tDD2DOut[14]=
1
2H1 +Cos@2 Sin@t wmDAm+2 t w0DL
Indirect FM transmitter:circuit realization The frequency multiplier produces n-fold multiplication of
instantaneous frequency
Frequency multiplication of tone modulation increases modulation index but the line spacing remains the same
2 1
1
( ) ( )
c
f t nf t
nf n
FM: 2 ( )
t
n f x d