exponential carrier wave modulation

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 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


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:

– 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)


Assignment


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 CCx 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


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 ttdt ( ) ( )

t

t d

2 1

2 1

( ) ( ) ( )( ) ds t s t s tv tdt t t

Compare tolinear motion:

( )iv t

( )qv t


Assignment

( )( ) d ttdt ( ) ( )

t

t d


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


AM, FM and PM waveforms

x t A t f x dFM C Ct

( ) 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


Assignment

Briefly summarize what is the main difference between FM and PM ?

How would you generate FM by using a PM modulator?


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 radx t Aci C( ) x t A tcq C( ) ( )

X f A f f j A f f fC C C C C( ) ( ) ( ), 12 2

0

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


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 tX fjf X f f

FX f A f f j A f f fC C C C C( ) ( ) ( ),

12 2

0

PM

FM t

t

t x tt f x d t t

( ) ( )( ) ( ) ,

z

20 0


0(0) ()) )((t

t

G Gg dj

What does it mean to set thiscomponent to zero?


Assume:1( ) sinc2 ( )

2 2fx t Wt X f

W W

1( ) ( ) ( ), 02 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( ) ( ) , 02 4 2

CPM C C C

j f fX f A f f A fW W

1( ) ( ) , 02 4 2

CFM C C C

C

f f fX f A f f A ff f W W

X fFM ( )E

xam

ple


Assignment


Solution


Tone modulation with PM and FM:modulation index Remember the FM and PM waveforms:

Assume tone modulation

Then


t

( ) cos[ ( ) ]( )

z

2

x t A t x tPM C C

t

( ) cos[ ( )]( )

x tA tA t

m m

m m

( )sin( ),cos( ),

RST

PMFM

( ) sin( ),PM

( )2 ( ) ( / )sin( ),FM

m m

m m mt

x t A t

tf x d A f f t


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


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

/

PM m

FM m m

AA f f

sin( ) sin( ) cos( ) cos( ) 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( )

2

2


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 tA A A At t

A t

x t A t A t

A t

C C CC

C m

CC 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


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 tA t t

C 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

AA f f


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/ ,FM

m

m m

AA f f


Assignmentx t A J n tC C n C mn( ) ( )cos( )

,PM/ ,FM

m

m m

AA 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


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 2

B M f MT m 2 1( ) , ( )

M( ) 2

A f Wm m 1,

( ) 0.01MJ

( ) 0.1MJ ,PM/ ,FM

m

m m

AA f f


Assignment


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 WA f f f W D

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


Example: Commercial FM bandwidth Following 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 A tuned circuit oscillator

– biased varactor diode capacitance directly proportional to x(t)

– other parts:• input transformer• RF-choke• DC-block


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 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

/ ( )

[ ( )] / { [ ( )]}[ ( )] ( )

[ ( )] ( ( ) / ) , / ( )/

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

( ) ( )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


Assignment


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


( ) cos( ( ) ) z 2

x t A t x tPM C C( ) cos( ( ))

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

Narrow band

FM/PM spectra


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 tdx t

dtA t t d t dt

C C

CC C C

( ) cos( ( ))( ) sin[ ( )]( ( ) / )

0

d t dt f t f f x tC ( ) / ( ) [ ( )] 2 2

FM

35 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen25


Assignment


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


In[14]:= TrigReduce@Cos@w0 t +Am Sin@wm tDD2DOut[14]=

12H1 +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 tnf n

FM: 2 ( )

t

n f x d

exponential carrier wave modulation

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