Principles of Communication

Analog Modulation System (3)

LC 5-6

Lecture 9, 2008-10-10

Lecture 9 2

Contents

Linear vs. Angle ModulationPhase modulation and frequency modulation Narrowband frequency modulationSpectrum of angle modulation systemPower of angle modulation systemAngle modulator

Lecture 9 3

Linear vs. Angle Modulation MethodsAmplitude modulation (AM) methods are also called linear modulation methods, although conventional AM is not linear in the strict sense.In phase modulation systems, the phase of the carrier is changed according to the variations in the message signal.In frequency modulation system, the frequency of carrier is changed by the message signal.PM and FM are special cases of angle modulation methods. They are obviously quite nonlinear.

Lecture 9 4

Representation of PM and FM signalsThe complex envelop is

( ) cos[ ( )]c cs t A t tω θ= +

( )( ) j tcg t A e θ=

The angle modulated signal is

For PM, the phase is directly proportional to the message signal

For FM, the phase is proportional to the integral of message signal

Dp is the phase sensitivity

Df is the frequency deviation constant

( ) ( )pt D m tθ =

1 ( )( ) ( ) ( ) ( )2 2

t ff d

Dd tt D m d f t m tdtθθ σ σ

π π−∞= ⇒ = =∫

Lecture 9 5

Comparison of PM and FM

( ) ( )tf

p fp

Dm t m d

Dσ σ

−∞= ∫

( )( ) p p

ff

D dm tm t

D dt=

Lecture 9 6

Example (1)

1 2

m(t)

PM

FM

21

( )( ) dm tm tdt

=

Lecture 9 7

Example (2)Message signal ( ) cos(2 )p mm t V f tπ=

Carrier cos(2 )c cA f tπ

In PM ( ) ( ) cos(2 ) cos(2 )p p p m p mt D m t D V f t f tθ π β π= = =

In FM

( ) ( ) sin(2 ) sin(2 )2

t f pf m f m

m

D Vt D m t dt f t f t

fθ π β π

π−∞= = =∫

βp is phase modulation index

βf is frequency modulation index

Lecture 9 8

Modulation Index

1 ( ) 1 1 ( )( ) [ ( )]2 2 2i c c

d t d d tf t t t fdt dt dtψ θω θ

π π π= = + = +

[ ]( ) ( ) cos ( ) , ( ) ( )cs t R t t t t tψ ψ ω θ= = +

fFB

β Δ=

The instantaneous frequency of s(t) is

If a bandpass signal is represented by

In FM, the peak frequency deviation from the carrier frequency is1 1max[ ( )] max[ ( )]

2 2d f f pF f t D m t D Vπ π

Δ = = =

The frequency modulation index is

Vp = max[m(t)]

We can extend the definition for a general non-sinusoidal signal

B is the bandwidth of the message signal.

In PM, the peak phase deviation is max[ ( )]p p pD m t D VθΔ = =

The phase modulation index is pβ θ= Δ

Lecture 10 9

Narrowband Frequency Modulation

( ) ( 0.56

t

fD m d πτ τ−∞

⎡ ⎤ <<⎢ ⎥⎣ ⎦∫ or ）If , narrowband frequency modulation (NBFM)

cos ( ) 1

sin ( ) ( )

t

f

t t

f f

D m d

D m d D m d

τ τ

τ τ τ τ

−∞

−∞ −∞

⎡ ⎤ ≈⎢ ⎥⎣ ⎦⎡ ⎤ ≈⎢ ⎥⎣ ⎦

∫

∫ ∫

Q

( ) cos ( )d

cos cos ( )d sin sin ( )d

t

FM c f

t t

c f c f

s t A t D m

t D m t D m

ω τ τ

ω τ τ ω τ τ

−∞

−∞ −∞

⎡ ⎤= +⎢ ⎥⎣ ⎦⎡ ⎤ ⎡ ⎤= −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∫

∫ ∫

( ) cos ( ) sint

NBFM c f cs t t D m d tω τ τ ω−∞

⎡ ⎤∴ ≈ − ⎢ ⎥⎣ ⎦∫

Lecture 10 10

NBFM vs. AM

[ ]

( ) ( )1( ) sin2

( ) ( )( ) ( ) ( )2

c cc

c c

f c cNBFM c c

c c

F Fm t dt t

D F FS

ω ω ω ωωω ω ω ω

ω ω ω ωω π δ ω ω δ ω ωω ω ω ω

⎡ ⎤+ −⎡ ⎤ ⇔ −⎢ ⎥⎣ ⎦ + −⎣ ⎦⎡ ⎤− +

∴ = + + − + −⎢ ⎥− +⎣ ⎦

∫Q

[ ] [ ]1( ) ( ) ( ) ( ) ( )2AM c c c cS M Mω π δ ω ω δ ω ω ω ω ω ω= + + − + + + −

Lecture 10 11

ExampletAtm mm ωcos)( =

[ ]( ) cos cos( ) cos( )2m f

NBFM c c m c mm

A Ds t t t tω ω ω ω ω

ω= + + − −

[ ]( ) cos cos( ) cos( )2m

AM c c m c mAs t t t tω ω ω ω ω= + + + −

sAM(ω )

O－ω m ωω m

F(ω )

O－ω c－ω m －ω c －ω c＋ω m ω c－ω m ω c ω c＋ω m ω

sNBFM(ω )

O－ω c－ω m －ω c

－ω c＋ω m ω c－ω m

ω c ω c＋ω m ω

NBFM

LSB USBCarrier

AM

Carrier

USB

LSB

Lecture 10 12

Spectrum of Angle Modulated Signals

Due to the nonlinearity of angle-modulation systems the precise characterization of their spectral properties, even for simple message signals, is mathematically intractable.We will study the spectral characteristics of an angle modulation signal in the case of a sinusoidal signal

Example 5-2

Lecture 10 13

order.nth theof kindfirst theoffunction Bessel the:)(

)(21

21

1

1)(1where

)(

is envelopecomplex The

)sin()sin(

2/

2/

sin

2/

2/

sin2/

2/

sin)(

β

βθπ

θπ

ωω

π

π

θθβπ

π

θθβ

ω

ω

ωωβ

ωωβω

ωωβθ

n

ncnj

cnj

c

T

T mtjntj

cmm

T

T

tjntjc

m

T

T

tjn

mn

n

tjnn

tjc

tjc

J

JAdeAdeA

tdeeAT

dteeAT

dtetgT

c

eceAeAtg

mm

mm

mm

m

m

mmm

m

m

mm

=⎥⎦⎤

⎢⎣⎡==

=

==

===

∫∫

∫

∫∫

∑

−

−

−

−

−

−

−

−

−

−

∞

−∞=

Lecture 10 14

Angle Modulation by a Sinusoidal Signal

Bessel Function

1)( .3

2 , 0)(2/)(

1)(small very is When .2

)()1()( .1

2

1

0

=

>≈≈

≈

−=

∑∞

∞=

−

-nn

n

nn

n

J

nJJJ

JJ

β

βββ

ββ

ββ

Lecture 10 15

∑

∑∑

∞

−∞=

∗

∞

−∞=

∞

−∞=

−−=

>

−−+−=

−=−=

nmcnc

cc

nmnc

nmn

nfffJAfS

fffGffGfS

nffJAnffcfG

)()()(

0For )]()([)(

is signal modulated-angle theof spectrum The

)()()()(

isenvelopecomplex theofspectrum The

21

21

δβ

δβδ

Lecture 10 16

Carson’s Rule

Carson’s rule2( 1)TB Bβ= +

The effective bandwidth of an angle-modulated signal, which contains at least 98% of the signal power.

Lecture 10 17

Power in an Angle Modulation Signal

[ ]

[ ]

2 2 2

2 2

( ) cos ( )

1 1 cos 2 ( )2 2

c c

c c c

s t A t t

A A t t

ω θ

ω θ

= +

= + +

The normalized power of angle modulation signal:

If the carrier frequency is large so that s(t) has negligible frequency content in the region of dc, the second term is negligible

2 21( )2 cs t A=

Thus the power contained in the output of an angle modulator is independent of the message signal.

Lecture 10 18

Angle ModulatorsDirect FM – Varactor Diodes

0

00 0

00 0

12

00

12

0

0

0

Assume that ( ) and 1

1 1,2 2 ( )

1 12 ( ) 2

1 [(1 ) 1]2

[(1 ) 1]

1[(1 ) 1]2

1 ( )2

c

c

c

c

c

c c

CC K m tC

f fLC L C C

f f fL C C LC

CCLC

CfC

CfC

f K m tC

π π

π π

π−

−

ΔΔ = <<

= =+ Δ

Δ = − = −+ Δ

Δ= + −

Δ= + −

Δ≈ − −

= −

Simple LC circuitLarge frequency offsetBut, unstable

Lecture 10 19

Angle ModulatorsIndirect Method - narrowband frequency modulation (NBFM)

443442143421termsideband

cc

termcarrier

cccc

ctj

c

ttAtAttyttxtstjAeAtg

t

ωθωωωθ

θθ

sin)(cossin)(cos)()()](1[)(

valuesmallatorestrictedis)(When )(

−=−=+≈=

Lecture 10 20

Angle ModulatorsIndirect Method – Narrowband-to-Wideband Conversion

( ) cos[ ( )]( ) cos[ ( )]( ) 2cos( )( ) cos[( ) ( )] cos[( ) ( )]( ) cos[( ) ( )]

i c c

n c c

LO

m c c LO c c LO

o c c LO

s t A t ts t A n t n tf t ts t A n t n t A n t n ts t A n t n t

ω θω θ

ωω ω θ ω ω θω ω θ

= += +

== + + + − += − +

The central idea is that frequency multiplier changes both the carrier frequency and the deviation ratio by a factor of n, whereas the mixer changes the effective frequency but does not affect the deviation ratio.

FM DemodulationDifferentiator and Envelope Detector

Zero Crossing DetectorUses rate of zero crossings to estimate fi

Phase Lock Loop (PLL)Uses VCO and feedback to extract m(t)

Lecture 10 21

∫++=′t

fcfcc dmktftmkfAts0

])(22sin[)](22[)( ττππππ

Theorem

Lecture 10 22

( ) ( )

signal. modulating theof PDF theis )( where

222

)(

by edapproximat is signal WBFM theof PSD normalized them(t), ofbandwidth theis B and

12

)](max[

)(cos)(

wheresignaling, For WBFM

2

⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

>=

⎥⎦⎤

⎢⎣⎡ += ∫ ∞−

m

cf

mcf

mf

c

ff

t

fcc

f

ffD

fffD

fDAf

BtmD

dmDtAts

πππ

πβ

σσω

P

Example 5-3

Lecture 10 23

Main Points of Angle ModulationAn angle-modulation signal is a nonlinear function of the modulation, and consequently, the bandwidth of the signal increases as the modulation index increases.The discrete carrier level changes, depending on the modulating signal, and is zero for certain types of modulating waveforms.The bandwidth of a narrowband angle-modulated signal is twice the modulating signal bandwidth (the same as that for AM signaling).The real envelope of an angle-modulated signal is constant and consequently does not depend on the level of the modulating signal.

Lecture 10 24

Lecture 9 25

HomeworkLC 5-21, 5-24, 5-38, 5-40, 5-44