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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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


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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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

∆ ∆

3

( ) 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

= + +

≠ + +


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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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∆=


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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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


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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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


Tone modulation with PM and FM:modulation index β

Remember the FM and PM waveforms:

Assume tone modulation

Then


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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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


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

8


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

∆

∆

==


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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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 ∆

∆

⎧= ⎨⎩


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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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


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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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


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


( ) 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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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


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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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

S-72.1140 Transmission Methods in Telecommunication ... · 5 9 Helsinki University of...

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