s-72.1140 transmission methods in telecommunication ... · 5 9 helsinki university of...
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Exponential Carrier Wave Modulation
S-72.1140 Transmission Methods in Telecommunication Systems (5 cr)
2 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Exponential modulation: Frequency (FM) and phase (PM) modulation
FM and PM waveformsInstantaneous frequency and phaseSpectral properties– narrow band
• arbitrary modulating waveform• tone modulation - phasor diagram
– wideband tone modulation– Transmission BW
Generating FM: de-tuned tank circuitGenerating PM: narrow band mixer modulatorGenerating FM/PM: indirect modulators (more linear operation -What linearity means here?)
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3 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
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
4 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
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:– usually transmission BW>>baseband BW– bandwidth-power trade-off (channel adaptation)– baseband and transmitted spectra does not carry a simple
relationship
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5 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
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 amplitude:
Angular phasor has the instantaneousfrequency (phasor rate)
x t A t tC C C
tC
( ) cos( ( ))( )
= +ω φθ
x t A t tPM C C
x t
tC
( ) cos( ( ) )( ),
( )
= +≤
ω φφ φ π
θ
∆ ∆
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( ) cos( ( )) Re[e ( )xp( )]C C C CCx t A t j tAθ θ= =
2 ( )f tω π=
ω 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
ω
ω
= + +
≠ + +
6 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Instantaneous frequency
Angular frequency ω (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:
( )tφ
/t s
2πAngle modulated carrier
Constant frequency carrier:0 02 rad/s, 1Hzfω π= =
1st =
( )( ) d ttdtφω = ( ) ( )
t
t dφ ω α α−∞∫=
2 1
2 1
( ) ( ) ( )( ) ds t s t s tv tdt t t
⎛ ⎞−= ≈⎜ ⎟−⎝ ⎠
Compare tolinear motion:
( )iv t
( )qv t
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7 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Frequency modulation (FM)In frequency modulation carrier instantaneous frequency is linearly proportional to modulation frequency:
Hence the FM waveform can be written as
Note that for FM
and for PM
(2 ( ) ( ) /2 )[ ]
C
C
f t d t df
tx tf
ω π θπ ∆
= == +
x t A t f x d t tC C C t
t
C t
( ) cos( ( ) ),( )
= + ≥zω π λ λθ
20 0∆
( ) ( )Cf t f f x t∆= +
integrate
( ) ( )t
t dφ ω α α−∞∫=
( ) ( )t x tφ φ∆=
8 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
AM, FM and PM waveforms
x t A t f x dFM C Ct
( ) cos( ( ) )= + zω π λ λ2∆
x t A t x tPM C C( ) cos( ( ))= +ω φ∆
Constant frequency: followsthe derivative of the modulation waveform
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9 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Narrowband FM and PM (small modulation index, arbitrary modulation waveform)
The CW presentation:The quadrature CW presentation:
The narrow band condition:
Hence the Fourier transform of XC(t) is in this case
x t A t tC C C( ) cos[ ( )]= +ω φ
x t x t t x t tC ci C cq C( ) ( )cos( ) ( )sin( )= −ω ω
x t A t A tci C C( ) cos ( ) [ ( / !) ( ) ...]= = − +φ φ1 1 2 2
x t A t A t tcq C C( ) sin ( ) [ ( ) ( / !) ( ) ...]= = − +φ φ φ1 3 3
( ) 1tφ << radx t Aci C( ) ≈ x t A tcq C( ) ( )≈ φ
X f A f f j A f f fC C C C C( ) ( ) ( ),≈ − + − >12 2
0δ Φ
cos( ) cos( )cos( )sin( ) sin( )
α β α βα β
+ =−
[ ] [ ]cos( ) ( )sin( )( ) C C CC CA tx A t tt ω φ ω≈ −F F
[ ]
[ ]0
0 0
cos(2 ) ( )1 ( )exp( ) ( )exp( )2
f t x t
X f f j jX f f j
π θ
θ θ
+
= − + + −
F[ ]
[ ]0
0 0
cos(2 )1 ( ) ( )2
f t
f f f f
π
δ δ= − + +
F
10 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Narrow band FM and PM spectra
Remember the instantaneous phase in CW presentation:
The small angle assumption produces compact spectral presentation for both FM and AM:
( ) [ ( )]( ),PM
( ) / ,FM
f tX fjf X f f
φφ∆
∆
Φ =
⎧= ⎨−⎩
F
X f A f f j A f f fC C C C C( ) ( ) ( ),≈ − + − >12 2
0δ Φ
φ φφ π λ λ
PM
FM t
t
t x tt f x d t t
( ) ( )( ) ( ) ,
=
= ≥z∆
∆20 0
x t A t tC C C( ) cos[ ( )]= +ω φ
0(0) ()) )((t
t
G Gg djωτ τω
π δ ω∫ ⇔ +
What does it mean to set thiscomponent to zero?
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11 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Assume: 1( ) sinc2 ( )2 2
fx t Wt X fW W
⎛ ⎞= ⇒ = Π⎜ ⎟⎝ ⎠
1( ) ( ) ( ), 02 2C C C C C
jX f A f f A f f fδ≈ − + Φ − >
( ) [ ( )] ( )PM PMf F t X fφ φ∆Φ = =
X fPM ( )
( ) [ ( )] ( ) /FM FMf F t jf X f fφ ∆Φ = = −
1( ) ( ) , 02 4 2
CPM C C C
j f fX f A f f A fW W
δ φ∆−⎛ ⎞≈ − + Π >⎜ ⎟
⎝ ⎠
1( ) ( ) , 02 4 2
CFM C C C
C
f f fX f A f f A ff f W W
δ ∆ −⎛ ⎞≈ − + Π >⎜ ⎟− ⎝ ⎠
X fFM ( )
Exa
mpl
e
12 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Tone modulation with PM and FM:modulation index β
Remember the FM and PM waveforms:
Assume tone modulation
Then
x t A t f x dFM C Ct
t
( ) cos[ ( ) ]( )
= + zω π λ λφ
2∆
x t A t x tPM C C
t
( ) cos[ ( )]( )
= +ω φφ
∆
x tA tA t
m m
m m
( )sin( ),cos( ),
=RST
ωω
PMFM
( ) sin( ),PM
( )2 ( ) ( / )sin( ),FM
m m
m m mt
x t A t
tf x d A f f t
β
β
φ φ ω
φπ λ λ ω
∆ ∆
∆ ∆
=⎧⎪
= ⎨ =⎪⎩
∫
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13 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
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( )= −ω ωx t A t A tci C C( ) cos ( ) [ ( / !) ( ) ...]= = − +φ φ1 1 2 2
x t A t A t tcq C C( ) sin ( ) [ ( ) ( / !) ( ) ...]= = − +φ φ φ1 3 3
x t A t tC C C( ) cos[ ( )]= +ω φ
β φ β ω<< =1, ( ) sin( ),t tm FM,PM
( ) cos( ) sin( )sin( )
cos( ) cos( )2
cos( )2
C C C C m C
CC C C m
CC m
x t A t A t tAA t t
A t
ω β ω ωβω ω ω
β ω ω
= −
= − −
+ +/
PM m
FM m m
AA f f
β φβ
∆
∆
==
sin( ) sin( ) cos( ) cos( )α β α β α β= − − +12
14 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Narrow band tone modulation: spectra and phasors
Phasors and spectra resemble AM:
x t A t A t
A t
C C CC
C m
CC m
( ) cos( ) cos( )
cos( )
= − −
+ +
ω β ω ω
β ω ω
2
2
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15 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
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( )]= +ω β ω
cos( ) cos( )cos( )sin( )sin( )
α β α βα β
+ =
−
x t A t tA t t
C C m C
C m C
( ) cos( sin( ))cos( )sin( sin( ))sin( )
=
−
β ω ωβ ω ω
cos( sin( )) ( ) ( )cos( )sin( sin( )) ( )sin( )
β ω β β ωβ ω β ω
m O mn
m mn
t J J n tt J n t
= +
=
∞
∞
∑
∑
22
neven
nodd
Jn is the first kind, order n Bessel function
/PM m
FM m m
AA f f
β φβ
∆
∆
==
16 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Wideband FM and PM spectraAfter simplifications we can write:
x t A J n tC C n C mn( ) ( )cos( )= +=−∞∞∑ β ω ω
note: ( ) ( 1) ( )nn nJ Jβ β− = −
,PM/ ,FM
m
m m
AA f f
φβ ∆
∆
⎧= ⎨⎩
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17 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Determination of transmission bandwidthThe goal is to determine the number of significant sidebandsThus consider again how Bessel functions behave as the function of β, e.g. we considerSignificant sidebands:Minimum bandwidth includes 2 sidebands (why?):Generally:
Jn ( )β ε>B fT m,min = 2
B M f MT m= ≥2 1( ) , ( )β β
M( )β β≈ + 2
A f Wm m≤ ≤1,
( ) 0.01MJ β >
( ) 0.1MJ β > ,PM/ ,FM
m
m m
AA f f
φβ ∆
∆
⎧= ⎨⎩
18 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Transmission bandwidth and deviation DTone modulation is extrapolated into arbitrary modulating signalby 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 WA f f f W Dβ
∆ ∆= == = ≡
B M D WT = 2 ( )
1,2(
2 1
2)
,m
T m D f WB D f
DW D>> =
≈ +
≈ >>
2 ( )2 , (a single pair of sideband1 s)
TB MDD W
W=≈ <<
2 1 , ,an1 d 1TB D W D D>>= − <<
( ) 2M D D≈ +
2( 2)D W≈ +
,PM/ ,FM
m
m m
AA f f
φβ ∆
∆
⎧= ⎨⎩
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19 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Example: Commercial FM bandwidthFollowing commercial FM specifications
High-quality FM radios RF bandwidth is about
Note that
under estimates the bandwidth slightly
f WD f W
B D WT
∆
∆
= ≈
⇒ = =
= + ≈
755
2 2 210
kHz, 15 kHz
kHz,(D > 2)/
( )
BT ≥ 200 kHz
B D W DT = − ≈2 1 180 kHz, >> 1
20 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
A practical FM modulator circuit realizationA tuned circuit oscillator– biased varactor diode capacitance
directly proportional to x(t)– other parts:
• input transformer• RF-choke• DC-block
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21 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Generating FM/PMCapacitance 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 22 2
kx kx k x kx− = + + <</ ...
f LC
f x t LC x tC 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 20
0
1 2
0
/ ( )
[ ( )] / { [ ( )]}[ ( )] ( )
[ ( )] ( ( ) / ) , / ( )/
π
π
π
f x t f Cx t C
f Cx tC
d tdt
CC C
C
[ ( )] ( ( ) / )
( ) ( )
/= −
≈ +LNM
OQP =
−1
1 12
12
0
1 2
0 πφ
φ π π λ λ( ) ( )t f t CC
f x dC Ct
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
ircui
t
Capacitance diodeDe-tuned resonance frequency
Resonance frequency
22 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
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 inputNarrow band mixer modulator:
Phase modulators: narrow band mixer modulator
x t A t f x dFM C Ct
( ) cos( ( ) )= + zω π λ λ2∆
x t A t x tPM C C( ) cos( ( ))= +ω φ∆
1( ) ( ) ( ), 02 2
( ) ( ) PM( ) / 2 ( ) 2 [ ( )] FM
C C C C C
C
jX f A f f A X f f f
t x td t dt f t f f x t
δ φ
φ φφ π π
∆
∆
∆
≈ − + − >
=⎧⎨ = = +⎩ Narrow band
FM/PM spectra
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23 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Indirect FM transmitterFM/PM modulator with high linearity and modulation index difficult to realizeOne 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
24 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
In[14]:= TrigReduce@Cos@ω0 t + Am Sin@ωm tDD2D
Out[14]=1
2H1 + Cos@2 Sin@t ωmD Am + 2 t ω0DL
Indirect FM transmitter:circuit realizationThe 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 tnf nφ∆
== + FM: 2 ( )
t
n f x dφ π λ λ∆ ∆
⎛ ⎞=⎜ ⎟
⎝ ⎠∫
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25 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen
Frequency detectionMethods 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 tdx t
dtA t t d t dt
C C
CC C C
( ) cos( ( ))( ) sin[ ( )]( ( ) / )
= +
= − + +
ω φ
ω φ ω φ
0
d t dt f t f f x tCφ π π( ) / ( ) [ ( )]= = +2 2∆
FM