Chapter 2 Continuous-Wave Modulation2 1 Introd ction2.1 Introduction

1

2.2 Consider a carrier (2.1) )2cos()( tfAtc cc π=

1+k m(t) S(t)X

1+kam(t) S(t)

The output of the modulator Accos(2πfct)

Where m(t) is the baseband signal , ka is the amplitude sensitivity. [ ] (2.2) )2cos()(1)( tftmkAts cac π+=

( ) g a p y

(2 4)2(2.3) t allfor ,1 )( .1

ftmka <

)( offreqency hightest theis where(2.4) .2

tmWWfc >>

2

Recall(2.2) )2cos()()2cos()( tftmkAtfAts caccc ππ +=

1 [ ]

[ ])()(1)2cos()(

)()(21)2cos(

ffMffMtftm

fffftf ccc

++⇔

++−⇔

π

δδπ

[ ]

[ ] [ ] (2.5) )()(2

)()(2

)(

)()(2

)2cos()(

ffMffMAkffffAfs

ffMffMtftm

ccca

ccc

ccc

++−+++−=

++−⇔

δδ

π

[ ] [ ])( of Transform Fourier theis )( where

22tmfM

1 Negative frequency component of m(t) becomes visible1.Negative frequency component of m(t) becomes visible.2.fc-W < M(f) < fc lower sideband

f < M(f) < f +W upper sideband3

fc < M(f) < fc+W upper sideband3.Transmission bandwidth BT=2W

Virtues and Limitations of Amplitude ModulationTransmitter

Receiver

Major limitations

4

Major limitations 1.AM is wasteful of power.2.AM is wasteful of bandwidth.

2 3 Linear Modulation Schemes2.3 Linear Modulation Schemes

Linear modulation is defined by

(2.7) )2sin()()2cos()()( −= tftstftsts cQcI ππ

componentQuadrature)(component phase-In)(

==

tsts

Q

I

Three types of linear modulation:

componentQuadrature)(tsQ

1.Double sideband-suppressed carrier (DSB-SC) modulation2 Single sideband (SSB) modulation2.Single sideband (SSB) modulation3.Vestigial sideband (VSB) modulation

5

Notes:1 sI(t) is solely dependent on m(t)1.sI(t) is solely dependent on m(t)2.sQ(t)is a filtered version of m(t).

Th l difi i f ( ) i l l d ( )6

The spectral modification of s(t) is solely due to sQ(t).

Double Sideband-Suppressed Carrier (DSB-SC) Modulation(2 8))2cos()()( tftmAts π= (2.8) )2cos()()( tftmAts cc π=

The Fourier transform of S(t) is 7

( )(2.9) )()(

21)( ⎥⎦

⎤⎢⎣⎡ ++−= ccc ffMffMAfs

Coherent Detection (Synchronous Detection)

The product modulator output is )()2cos(' )( tstfAtv cc ϕπ +=

(2 10))()('1)()4('1)()2cos()2cos('

tAAttfAA

tmtftfAA cccc

φφ

φππ

++

+=

Let V(f) be the Fourier transform of v(t)

(2.10) )()cos('2

)()4cos('2

tmAAtmtfAA ccccc φφπ ++=

1

filtered out

(2.11) )( cos'21)(0 tmAAtv cc φ= (Low pass filtered)

8

Costas ReceiverI-channel and Q-channel are coupled together toI channel and Q channel are coupled together to form a negative feedback system to maintain synchronization

0φ ≈

2 2 2 21 1cos sin ( ) ( )sin(2 )4 8

c cA m t A m t

φ

φ φ φ=

The phase control signal ceases with modulation

1 4

2 2( ) (sin2 2 )cA m t φ φ φ≈ ≈The phase control signal ceases with modulation.

(multiplier +(multiplier + very narrow band LF)

9

Quadrature-Carrier Multiplexing (or QAM)T DSB SC i l h h lTwo DSB-SC signals occupy the same channel bandwidth, where pilot signal (tone ) may be

d dneeded.

)2sin()()2cos()()( 21 tftmAtftmAts cccc ππ += )2sin()()2cos()()( 21 tftmAtftmAts cccc ππ +

10

Single-Sideband Modulation (SSB)The lower sideband and upper sideband of AM signal contain same information .The frequency-discrimination method consists of a

d d l (DSB SC) d b d filproduct modulator (DSB-SC) and a band-pass filter.The filter must meet the following requirements:a The desired sideband lies inside the passbanda.The desired sideband lies inside the passband.b.The unwanted sideband lies inside the stopband. c.The transition band is twice the lowest frequency of

the message.

11To recover the signal at the receiver, a pilot carrier or a stable oscillator is needed (Donald Duck effect ).

Vestigial Sideband Modulation (VSB)When the message contains near DC component

The transition must satisfy

liihb h 1)()(.a ffHffH cc =++−

(2.13) for 1)()( :linearisresponsephase b.The

W fWffHffH cc ≤≤−=++−

12(2.14) fWB νT +=

Consider the negative frequency response:g q y p( )H f

c vf f− cf c vf f+ cf W+c vf f− +cf−c vf f− −cf W− −

Here, the shift response │H(f-fc)│ is( )cH f f−

f0 2 c vf f− 2 cf 2 c vf f+ 2 cf W+vf0vf−W

13

and │H(f+fc)│ is( )H f f+( )cH f f+

vf− 0 vf W2 c vf f− +2 cf−2 c vf f− −2 cf W− −

14

So, we get │H(f-fc)│ +│ H(f+fc)│ is( )H f f( )cH f f−

2 c vf f− 2 cf 2 c vf f+vf0vf−W−

( )H f f+( )cH f f+

vf− 0 vf W2 c vf f− +2 cf−2 c vf f− −

15

Consider –W<f<W we get:

vfvf−W− 0 W

Which is equal to

W− W

So, │H(f-fc)│ + │H(f+fc)│ =1 for -W<f<W16

(2.15) )2sin()('21)2cos()(

21)( tftmAtftmAts cccc ππ ±=

± corresponds to upper or lower sideband22

ff

HQ(f)m(t) m’(t)

[ ] (2.16) for )()( )( W f WffHffHjfH ccQ ≤≤−+−−=

17

Television Signals (NTSC)

18

2.4 Frequency Translation

cos(2 )A f tπl l

Up conversionf2=f1+fl , fl=f2-f1

Down conversionf f f f f f

19

f2=f1-fl , fl=f1-f2

2.5 Frequency-Division Multiplexing (FDM)

20

2.6 Angle ModulationBasic Definitions:Better discrimination against noise and interference g(expense of bandwidth).

[ ] (2.19))(cos)( tAts ic θ=The instantaneous frequency is

[ ] (2.19) )(cos)( tAts ic θ

)(lim)( Δti tftf =

2)()(lim

)()(

0Δ

Δ0Δ

ii

t

tti

tttt

ff

πθθ

⎥⎦⎤

⎢⎣⎡

Δ−Δ+

=→

→

(2.21) )(21

20Δ

i

t

dttd

tθ

π

π

=

⎥⎦⎢⎣ Δ→

(2 22)2)( is )( carrier, dunmodulate anFor

2i

tftt

dt

φπθθ

π

+=21constant is where

(2.22) 2)(

c

cci tftφ

φπθ +=

1. Phase modulation (PM))(2)( tmktft pci += πθ

[ ] (2 23))(2cosmodulator theofy sensitivit phase :

)()(

tmktfAs(t)k

fp

pci

+= π

πθ

2. Frequency Modulation (FM)[ ] (2.23) )(2cos tmktfAs(t) pcc += π

(2 24)( ) ( )f t f k m t= + (2.24)

π π0

( ) ( )

( ) 2 2 ( )

i c f

t

i c f

f t f k m t

t f t k m dθ τ τ

= +

= + ∫ (2.25)

π π (2.26)

:frequency sensitivity of the modulator0

cos 2 2 ( )t

c c fs(t) A f t k m d

k

τ τ⎡ ⎤= +⎢ ⎥⎣ ⎦∫:frequency sensitivity of the modulator

compare (2.23) and (2.26)

f

p

k

k m'⇒ π0

2 ( )t

f(t) k m dτ τ= ∫

22generating FM signal generating PM signal

2.7 Frequency ModulationFM is a nonlinear modulation process , we can not apply Fourier transform to have spectral analysis directly.Fourier transform to have spectral analysis directly.

1.Consider a single-tone modulation which produces a b d FM (k i ll)narrowband FM (kf is small)

2.Next consider a single-tone and wideband FM (kf is large)

(2 27))2()(l t tfAt (d t i i ti )

)2cos( )((2.27) )2cos()(let

mmfci

mm

tfAkftftfAtm

+==

ππ (deterministic)

d i tifΔ(2.28) )2cos( mc

f

Akftfff Δ+= π

23 deviationfrequency :Δ mf Akf =

)(2)( (2.25), Recall0

ττπθ dftt

ii = ∫(2.30) )2sin(2 π tf

fftπf mc

Δ+=

(2.31) index Modulation β ffm

Δ=

(2.32) )2sin(2)(

( )

πβθ

β

tftπftf

mci

m

+=[ ]

dill hiFMN b d(2.33) )2sin(2cos)( ( ))()(

βπβπ

βtftfAts

ff

mcc

mi

+=(2.19) =>

radian. onenlarger thais, FM Widebandradian.one ansmaller this, FM Narrowband

ββ

24

gβ

Narrowband FM[ ])2i (2)( ffA β[ ]

[ ] [ ] )34.2( )2sin(sin)2sin()2sin(cos)2cos( )2sin(2cos)(

tftfAtftfAtftfAts

mccmcc

mcc

πβππβππβπ

−=+=

[ ] 1)2sin(cossmall, isBecausetfmπβ

β≈

[ ](2.35) )2sin()2sin()2cos()(

)2sin()2sin(sintftfAtfAts

tftf

mcccc

mm

ππβππβπβ

−≈≈

( ))()()()( fff mcccc β

25

Th f Fi 2 21 iThe output of Fig 2.21 is )2sin()()2cos()(' tfdmkAtfAts cfccc πττπ ∫−=

s(t) differs from ideal condition in two respects:∫

1.The envelope contains a residual AM.(FM has constant envelope)(FM has constant envelope)

2.θi(t) contains odd order harmonic distortions

)!7!5!3

(sin753

L+−+−=xxxxx

For narrowband FM, ≤ 0.3 radians. !7!5!3

β

26

(2.35) )2)sin(2(sin)2(cos)((2.35)Recall

tftfAtfAts mcccc −≈ ππβπ

[ ] [ ]{ }(2.36) )(2cos)(2cos21)2(cos

( ))) (()()(

tfftffAtfA

fff

mcmcccc

mcccc

−−++≈ ππβπ

β

[ ])()()(

(2.2) )2(cos)(1 )()2cos()( ,wavemodulatingsinusoidal with AMFor

AM

ffkftftmkAts

tftm

cac

m

+==

ππ

[ ] [ ]{ } (2.37) )(2cos)(2cos21)2(cos

)2cos()2(cos)2(cos

tfftffAtfA

tftfAktfA

mcmcccc

mccacc

−+++=

+=

ππμπ

πππ

2

Narrow band FM

27AM

Wideband FM (large β)

[ ] (2.33) )2sin(2cos)( += mcc tftfAts πβπ

[ ]))2sin(2exp(Re)(sincosexp

+=+=

tfjtfjAtsxjx(jx)

πβπ[ ][ ] (2.38) ))(2exp()(~Re

))2sin(2exp(Re)(=

+

c

mcc

tfjtstfjtfjAts

ππβπ

[ ]bydefinedenvelopecomplextheis)(~ andpart real thedenotes Re where

ts)]2sin(exp[)(~

bydefined envelopecomplex theis)(= mc tfjAts

tsπβ (2.39)

(2.40) )2exp()(~ ∑∞

= mn t nfjcts π

( )

28

−∞=n

Complex Fourier Transform

1212

( )exp( 2 )m

m

f

n m mfc f s t j nf t dtπ

−= −∫ %

[ ] (2.41)1212

exp sin(2 ) 2 )m

m

f

m c m mff A j f t j nf t dtβ π π

−= −∫

[ ]

Let (2.42)2

( i )

m

c

x f tA j d

π

β

=

(2 43)π

∫ [ ] exp ( sin )2

cnc j x nx dxβ

π= − (2.43)

Define the th order Bessel function of the first kind asnπ−∫

2A3, x2

2 22 ( ) 0)d y dyx x n y

dx dx+ + − =（

[ ] (2.44)1( ) exp ( sin )2n

dx dx

J j x nx dxπ

πβ β

π −= −∫2

( )n c nc A Jππ

β=

∫

∞

29( ) ( )c nn

s t A J β=−

=% (2.45)exp( 2 )mj nf tπ∞

∞∑

[ ] (2.47) )(2exp)(Re)( mcnc tnffjJAts ⎥⎦

⎤⎢⎣

⎡ += ∑∞

∞−

πβ

[ ] (2.48) )(2cos)( mcnc tnffJA += ∑∞

∞−

πβ

[ ] (2 49))()()()(

is )( of ransform Fourier tThe

ffffffJAfS

ts

∑∞

δδβ [ ] (2.49) )()()(2

)( mcmcnc nfffnfffJAfS +++−−= ∑

∞−

δδβ

30Figure 2.23 Plots of Bessel functions of the first kind for varying order.

Properties of ( )

1. ( ) ( 1) ( ), for all (2.50)n

n

J

J J n

β

β β= −1. ( ) ( 1) ( ), for all (2.50)2.If is small ( ) 10

n nJ J n

J

β ββ

β

−

≈( )

( )2

0

1J

βββ ≈

( ) 0 2 (2.51)

3 ( ) 12

nJ n

J

β

β∞

≈ >

=∑-

3. ( ) 1

Observation o

nJ β∞

=∑f FM

1.An FM signal contains components.2.For small , the FM signal is effectively composed of a carrier and

2 3 ,c m m mf , f , f , fβ

L

a single pair of side freqencies at narrowband FM3.The am

c mf f± ⇒

plitude of carrier depends on 2

β

31A (2.54)

22 21 ( )

2 2c

c nP A J β∞

−∞

= = ∑

Example 2.2

32

Transmission Bandwidth of FM signalsWith a specified amount of distortion , the FM signal is p , geffectively limited to a finite number of significant side frequencies.

A.Carson’s rule , = , (2.55)12 2 2 (1 )T m m

fB f f f f ff

β ββ

Δ≈ Δ + = Δ + Δ =

mfβ

33

B. , ( ) 0.01 , maxmax max2 2T m n T

fB n f J B nββΔ

= >> =

Universal curve for evaluating the 1 percent bandwidth of an FM wave

34

Example 2.3In north America, the maximum value of frequency deviation is fi d t 75kH f i l FM b d ti b di If t k

fΔfixed at 75kHz for commercial FM broadcasting by radio. If we take the modulation frequency W=15kHz, which is typically the “maximum” audio frequency of interest in FM transmission, we find that corresponding value of the deviation ratio is

51575D ==

Using Carson’s rule of Equation (2.55) , replacing by D , and replacing fm by W , the approximate value of the transmission

15β

bandwidth of the FM signal is obtained asBT=2(75+15)=180kHz

On the other hand use of the curve of Figure 2 26 gives theOn the other hand , use of the curve of Figure 2.26 gives the transmission bandwidth of the FM signal to be

BT=3.2 =3.2x75=240kHzfΔIn practice , a bandwidth of 200kHz is allocated to each FM transmission . On this basis , Carson’s rule underestimates the transmission bandwidth by 10 percent , whereas the universal curve

35

transmission bandwidth by 10 percent , whereas the universal curve of Figure 2.26 overestimates it by 20 percent.

Generation of FM signals

(2.56)21 2( ) ( ) ( ) ( )n

nv t a s t a s t a s t= + + +L

Frequency Multiplier

( )

0cos 2 2 ( )

tc c fs t A f t k m dπ π τ τ⎡ ⎤= +⎢ ⎥⎣ ⎦∫

The frequency multiplier output

(2 58)'( ) 'cos 2 2 ( )t

c c fs t A nf t nk m dπ π τ τ⎡ ⎤= +⎢ ⎥⎣ ⎦∫36

(2.58)

0

( ) cos 2 2 ( )

'( ) ( )

c c f

i c f

s t A nf t nk m d

f t nf nk m t

π π τ τ+⎢ ⎥⎣ ⎦= +

∫ (2.59)

Varactor diode VCO FM modulator

32-1

Crosby Direct FM Transmitter

32-2

Demodulation of FM signalsThe frequency discrimination consists of a slope circuitThe frequency discrimination consists of a slope circuit followed by an envelope detector

Consider Fig 2 29a the frequency response of a slopeConsider Fig 2.29a , the frequency response of a slope circuit is

22 ),

2(2⎪

⎧ +≤≤−+− Tc

Tc

Tc

BffBfBffaj π

22

),2

(222

),2

(

)(1

⎪

⎪⎪⎪

⎨ +−≤≤−−−+= Tc

Tc

Tc

ccc

BffBf Bffaj

fffffj

fH π (2.60)elsewhere ,0

222

⎪⎪⎪

⎩ 33

1 1( ) 2 ( ) , 0cH f f H f f− = >%

2 2( ) 2 ( ) , 0cH f f H f f− = >%

34

Appendix 2 3 Hilbert TransformAppendix 2.3 Hilbert TransformFourier Transform-frequency-selectiveHilb T f h l iHilbert Transform-phase-selective

(±900shift)Let g(t)⇔G(f)Denote the Hilbert transform of g(t) as g( )

(A2.31) )(1)(ˆ

ττ

τπ

dtgtg ∫

∞

∞−=

)(ˆ1ansformHilbert tr inverse Theτπ t∫ ∞ −

(A2.32) )(ˆ1)(

τ

ττ

πd

tgtg ∫

∞

∞− −−=

35

(A2.33)1 sgn( )j ftπ

⇔ −

(A2.34)

1 0sgn( ) 0 0

ff f

⟩⎧⎪= =⎨ ( )

f f

g ( )1 0

( )

f ff

⎨⎪− ⟩⎩

The Fourier transform of is

(A2.35)

( )ˆ( ) sgn( ) ( )

g t

G f j f G f= −

H(f)g(t) )(ˆ tg

( )

36

Properties of the Hilbert Transform( i d i i )(time domain operation)

If g(t) is real

ˆ)(is)(ˆoftransform2.Hilbert

spectrummagnitudesamethehave)(and )(ˆ.1tgtg

tgtg− (take H.F of and ( )g t

)(ˆ)g( 0)(g)g(3.

)(is )(oftransform2.Hilbert

tgtdttt

tgtg

⊥⇒=∫∞

compare with A2.32)( )g t

)()g()(g)g(-

g∫ ∞

37

For a band-pass system , we consider x(t) ⇔ X( f )x(t) ⇔ X( f )

X( f ) is limited within ± W HzW fW << fc

(A2.48)( ) ( )cos(2 ) ( )sin(2 )I c Q cx t x t f t x t f tπ π= −

The complex evelope of x(t) is (A2.49)( ) ( ) ( )I Qx t x t j x t= +% ( )( ) ( ) ( )I Qj

band pass system f

x(t) y(t)system, fc

H(f) 2B

fc

f

(A2.50)( ) ( )cos(2 ) ( )sin(2 )I c Q ch t h t f t h t f tπ π= −38

(A2.51) )()()(~ response implusecomplex theDefine

QI tj hthth +=

[ ] (A2 52))2exp()(~Re)(

)( oftion representacomplex The

( ))()()( QI

tfjthth

th

j

= π[ ]

)*2(h(A2 52)F

functions pass-low are )(h~ and )(,)(

(A2.52) )2exp()( Re)(

QI

c

j

tthth

tfjthth = π

(A2.53) )2exp()(*~)2exp()(~)(2

)*2,(havewe(A2.52) From

cc t fjtht fjthth

zzvjuvz

−+=

+=+=

ππ

(A2.54) )(*~)(~)(2

(A2.53) toansformFourier trApply

cc ffHffHfH −−+−=

)()(*real is )( Since

fHfHth

−=

(A2 55)0)(2)(~ with tolimited is )(H~ and c

ffHffH

fBBff

>=−⇒

<≤

)'(2)'(~ , )( from )(~obtain can We

(A2.55) 0,)(2)(

c

c

ffHfHfHfH

ffHffH

+=

>=⇒

39

band-pass h(t)x(t) y(t)system

[ ] (A2 57)( ) Re ( )exp( 2 )t t j f t%[ ] (A2.57)

(A2 58)

( ) Re ( )exp( 2 )

( ) ( )

cy t y t j f t

h x t d

π

τ τ τ∞

=

= −∫ (A2.58)

Define the pre-envelope of as

( ) ( )

( )

h x t d

h t

τ τ τ−∞∫

( ) ( ) ( ),h t h t j h t∧

+ = + :)(th∧

Hilbert T. of )(th

( ) ( ) sgn( ) ( )

2 ( )H f H f f H f

H f f+ = +

> 0⎧⎪

( )( ) (0)

0

f fH f H + = A0 ( 2.37)

0f

f

⎧⎪ =⎨⎪⎩ 0 0f⎪ <⎩

40[ ] [ ]A A2.59)2.58 ( ) Re ( ) Re ( ) (y t h x t dτ τ τ∞

+ +−∞⇒ = −∫

[ ]+

+

=+=

thththjthh)(Re)(

)(ˆ)( Recall

[ ]+= txtx )(Re)(

To prove (A2.60)

∫ +

∞

∞− +

⎤⎡

⎥⎦⎤

⎢⎣⎡ − τττ dtxh )()(Re

p ( )

∫∫

∫∞∞

∞

∞− ⎥⎦⎤

⎢⎣⎡ −+−+=

ττττττ

τττττ

dtxhdtxh

dtxjtxhjh

)(ˆ)(ˆ)()(

)](ˆ)()][(ˆ)([Re

∫∫∫∫∫

∞

∞− −

∞

∞−

∞

∞

∞−∞−

−−=

−−−=

τττττ

ττττττ

τπ dudtxuh

dtxhdtxh

u ,)(ˆ)()d-)x(th(

)()()()(

11-

ντdd

t =− , ντ −= t

∫∫∫∫∫

∫∫∫

∞∞

∞

∞− −−−

∞

∞−

∞

∞−

∞∞∞

+−= νντττ νπ duuhdxdtxh ut )()(ˆ)()( 11ντ dd −=

∫∫∫

∞

∞

∞−

∞

∞−

−=

−+−=

τττ

τττ

dtxh

duutxuhdtxh

)()(2

)()()()(

∫∫

∞

∞− ++

∞−

−= τττ

τττ

dtxh

dtxh

)](Re[)](Re[2

)()(2

41

b(A2 58)

[ ] [ ]−= ∫ +

∞

+ (A2.59) )(Re)(Re)(

becomes(A2.58)

τττ dtxhty [ ] [ ]

⎥⎦⎤

⎢⎣⎡ −= ∫

∫

+

∞

∞ +

+∞− +

)()(Re21

( ))()()(

τττ dtxh

y

⎥⎦⎤

⎢⎣⎡ −−=

⎥⎦⎢⎣

∫

∫∞

∞

∞−

))(2exp()(~)2exp()(~Re21

2

ττπττπτ dtfjtxfjh cc

⎥⎦⎤

⎢⎣⎡ −=

⎥⎦⎢⎣

∫

∫∞

∞

∞−

)(~)(~)2exp(Re21

2

τττπ dtxhtfj c ⎥⎦⎢⎣ ∫ ∞−2

42

Comparing (A2.57) and (A2.61) we have

(A2.62)

(A2 63)

2 ( ) ( ) ( )

2 ( ) ( ) ( )

y t h t x t dτ τ∞

−∞= −∫ %% %

%

t

or (A2.63)We can represent bandpass signals and systems

2 ( ) ( ) ( )y t h t x t= ∗% %

by the equivalent lowpass functions

and (without the factor )

( ), ( )( ) exp( 2 )

x t y t

h t j f t

% %

%and (without the factor ) ( ) exp( 2 )ch t j f tπ

43

[ ] [ ] (A2 64))()()()()(~2 tjxtxtjhthty QIQI +∗+= [ ] [ ][ ]

[ ])()()()(

(A2.64) )()()()()(2txthtxth

tjxtxtjhthtyQQII

QIQI

∗−∗=+∗+=

[ ](A2.66) )(~)(~)(~let (A2.65) )()()()(

tyjtytytxthtxthj

QI

QIIQ

+=∗+∗+

(A2 68))()()()((t)2y(A2.67) )()()()((t)2y

Q

I

txthtxthtxthtxth

QIIQ

QQII

∗+∗=∗−∗=

(A2.68) )()()()((t)2yQ txthtxth QIIQ ∗+∗=

44

45

Procedure for evaluating the response of a band pass systemof a band-pass system

)(~by)(Replace1 txtx[ ] )2exp()(~ Re)(

)(by )(Replace 1.tfjtxtx

txtxcπ=

[ ]~

)2exp()(~ Re)( .2 tfjthth cπ=

[ ])2exp()(~Re)(4)(~*)(~)(~2Obtain .3tfjtyty

txthtycπ=

=[ ])2exp()(Re)( .4 tfjtyty cπ=

46

To simplify the analysis 1 shift to the right by to align to the band pass frequency( )H f f%1. shift to the right by to align to the band-pass frequency

2. set , for (2.61)

1

1 1

( )

( ) 2 ( ) 0c

cH f f

H f f H f f− = >%

Recall

2 ( )TBj πa f f− + T TB Bf f f⎧ − ≤ ≤ +⎪

1

2 ( )2

( )

cj πa f f

H f j

+

= (2.60)

2 2

2 ( )2 2 2

c c

T T Tc c c

f f f

B B Bπa f f f f f

≤ ≤ +⎪⎪⎪ + − − − ≤ ≤ − +⎨⎪

elsewhere2 2 2

0

c c c⎨⎪⎪⎪⎩

From (2.60) and (2.61), we getB B B

⎩

⎧

(2.62)elsewhere1

4 ( )( ) 2 2 2

0

T T TB B Bj a f fH f

π⎧ + − ≤ ≤⎪= ⎨⎪⎩

%

elsewhere0⎪⎩

47

t

s t

s t A f t k m dπ π τ τ⎡ ⎤= +⎢ ⎥∫

Recall FM signal ( )

( ) cos 2 2 ( )c c fs t A f t k m dπ π τ τ⎡ ⎤= +⎢ ⎥⎣ ⎦

⎡ ⎤

∫The complex envelope is

0( ) cos 2 2 ( )

t

c fs t A j k m d

s t

π τ τ⎡ ⎤= ⎢ ⎥⎣ ⎦∫%

%1

(2.63)

Let denote the complex envelop

0( ) exp 2 ( )

( ) e of the slope ckt. response output.1 p p( )y t h t x t

S f H f S f

= ∗

=

%% %

% %%

p p pRecall (A2.63) 2 , we have

upper arm of Fig 2 30 in text)1

( ) ( ) ( )1( ) ( ) ( ) (

T T T

S f H f S f

B B Bj a f S f fπ

=

⎧ + − ≤ ≤⎪⎨

%

upper arm of Fig 2.30 in text)11( ) ( ) ( ) (2

2 ( ) ( )2 2 2

(2.64)= ⎨

⎪⎩

elsewhere

2 2 20

d s t⎡ ⎤%

( )T

d s ts t a j B s tdt

π⎡ ⎤⇒ = +⎢ ⎥⎣ ⎦% % (2.65)

From (2.63) and (2.65) , we have

1( )( ) ( )

48

tfT c f

T

ks t j B aA m t j k m d

Bπ π τ τ

⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦∫%

(2.66)1

0

2( ) 1 ( ) exp 2 ( )

[ ]

(2.67))(22cos)(2

1

)2exp()(~Re)(

11

⎥⎤

⎢⎡ ++⎥

⎤⎢⎡

+=

=

∫t

ff

T

c

dmktftmk

aAB

t fjtsts

πττπππ

π

(2.67) 2

)(22cos)(10 ⎥⎦⎢⎣

++⎥⎦

⎢⎣

+ ∫fcT

cT dm ktftmB

aAB ττπππ

0sin 2 2 ( )

tc ff t k m dπ π τ τ⎡ ⎤− +⎢ ⎥⎣ ⎦∫⎣ ⎦

is a hybrid-modulated signal (amplitude , frequency)

However, provided that we choose 1, for all

1( )2

( )f

s tk

m t tB

<

using an envelope detector, we have2

TB

k⎡ ⎤ 1

2( ) 1 ( )f

T cT

ks t B aA m t

Bπ

⎡ ⎤= +⎢ ⎥

⎣ ⎦% (2.68)

The bias term can be removed by a second frequencyB aAπThe bias term can be removed by a second frequency

discriminator with ( ) , where ( )2 2 1( ).T cB aA

H f H f H f

π

= −% %

49

Balanced Frequency Discriminator

~~

Let the transfer function of the second branch of Fig 2.30 be (complementary slope circuit)

(2 70))(2

1)(~

(2.69) )()(

2

12

tmk

aABts

fHfH

fπ ⎥⎤

⎢⎡

=

−=

)(~)(~)(

(2.70) )(1)(

210

2

tststs

tmB

aABtsT

cTπ

−=

⎥⎦

⎢⎣

−=

(2.71) )(4 )()()(0

tmaA k cfπ=50

FM Stereo MultiplexingTwo factors which influence FM stereo standards1.Operation within the allocated FM channels.

2 Compatible with monophonic radio receiver2.Compatible with monophonic radio receiver.

[ ] [ ] (2.72) )2cos()4cos()()()()()( t fKt ftmtmtmtmtm ccrlrl ππ +−++=51

Stereo FM

Figure 9-40. FM stereo generation block diagram.

51-1

Stereo FM

In Figure 9-40, audio signals from both left and right mircrophones are combined in an linear matrixing network to produce an L+R signal and an L-R signal.

Both L+R and L-R are signals in the audio band and must be separated before modulating the carrier for transmission. Thi i li h d b t l ti th L R di i lThis is accomplished by translating the L-R audio signal up in the spectrum.

As seen in Figure 9-40, the frequency translation is achieved by amplitude-modulating a 38-kHz subsidiary carrier in a balanced modulator to produce DSB SCcarrier in a balanced modulator to produce DSB-SC.

51-2

Stereo FM Transmitter

Stereo FM transmitter using frequency-division multiplexing.51-3

Stereo FM Transmitter

Stereo FM transmitter: (a) block diagram; (b) resulting spectrum.SAC: Subsidiary Communication Authorization 51-4

Stereo FM

The stereo receiver will need a frequency-coherent 38-kHz q yreference signal to demodulate the DSB-SC.

To simplify the receiver, a frequency- and phase-coherent signal is derived from the subcarrier oscillator by frequency di i i (÷2) t d il tdivision (÷2) to produce a pilot.

The 19 kHz pilot fits nicely between the L+R and DSB SC LThe 19-kHz pilot fits nicely between the L+R and DSB-SC L-R signals in the baseband frequency spectrum.

51-5

Stereo FMAs indicated by its relative amplitude in the baseband composite signal, the pilot is made small enough so that i FM d i i f h i i l b 10% f hits FM deviation of the carrier is only about 10% of the total 75-kHz maximum deviation.

After the FM stereo signal is received and demodulated to baseband, the 19-kHz pilot is used to phase-lock an p poscillator, which provides the 38-kHz subcarrier for demodulation of the L-R signal.

A simple example using equal frequency but unequal amplitude audio toned in the L and R microphones is usedamplitude audio toned in the L and R microphones is used to illustrate the formation of the composite stereo (without pilot) in Figure 9-41.

51-6

Stereo FM

Figure 9-41. Development of composite stereo signal. The 38 kHz alternatelyFigure 9 41. Development of composite stereo signal. The 38 kHz alternately multiplies L-R signal by +1 and –1 to produce the DSB-SC in the balanced AM modulator (part d). The adder output (shown in e without piot) will be filtered to reduce higher harmonics before FM modulation. 51-7

Stereo FM

Spectrum of stereo FM signal.

SCA: Subsidiary communication authorizationy(commercial-free program)

51-8

51-9

Reference : G. M. Miller “Modern Electronic Communication” 5th Edition, Prentice Hall

2 8 Nonlinear Effects in FM Systems2.8 Nonlinear Effects in FM Systems1.Strong nonlinearity, e.g., square-law modulators ,

h d li i f l i lihard limiter, frequency multipliers.2.Weak nonlinearity, e.g., imperfections

Nonlinear input-output relation

(2.73) )()()()( 32

3210 tvatvatvatv iii ++= ( ))()()()( 3210 iii

v (t) v (t)Nonlinear

Channel (device)

vi(t) v0(t)

52

signal FMFor [ ])(2cos)( tt fAtvt

cci φπ +=

∫[ ] [ ])(2cos)(2cos)(

)(2)(22

0

ttfAattfAatv

dm kt f

φπφπ

ττπφ

+++=

= ∫[ ] [ ]

[ ] (2.74) )(2cos

)(2cos)(2cos)(33

3

210

tt fAa

ttfAattfAatv

cc

cccc

φπ

φπφπ

++

+++=

[ ])(2cos)43(

21 3

312

2 tt fAaAaAa cccc φπ +++=

[ ])(24cos21 2

2 tt fAa cc φπ ++

[ ] (2.75) )(36cos41

23

3 tt fAa cc φπ ++4

53

WfffB mT 2222 , rule sCarson' +Δ=+Δ=W W W WfΔ4fΔ2W W W WfΔ4fΔ2

f 2ffc 2fc

In order to seperate the desired FM signal from the second harmonic , we have2 (2 )f f W f f W− Δ + > + Δ +2

(2.76)Th t t f th b d fil

(2 )3 2

c c

c

f f W f f Wf f W

Δ + > + Δ +

> Δ +

t iThe output of the band-pass fil

[ ]

ter is

no effect to33'( ) ( )cos 2 ( ) ( ( ))v t a A a A f t t m tπ φ= + +[ ] no effect to

An FM system is extremely sensitive to phase nonlinearities.

0 1 3( ) ( )cos 2 ( ) ( ( ))4c c cv t a A a A f t t m tπ φ= + +

Common type of source : AM-to -PM conversion.54

2.9 Super Heterodyne Receiver(Carrier frequency tuning filtering amplification and demodulation)(Carrier-frequency tuning , filtering , amplification , and demodulation)

AM radio receiver

f =f -f (2 78)

AM radio receiver

fIF=fLO-fRF (2.78)

A FM t li it t lit d i tiA FM system may use a limiter to remove amplitude variations.

55

Commercial FM Broadcast、Allocations and Sidebands

56

2.10 Noise in CW modulation System1.Channel model: additive white Gaussian noise (AWGN)2.Receiver model: a band-pass filer followed by an ideal demodulator

The PSD of w(t) is denoted by 20NThe PSD of w(t) is denoted by .2

57

(2.79) )2sin()()2cos()()(:tionrepresenta noise narrowbandin noise filtered The

−= t ftntftntn cQcI ππ

(2.80))()()(ison demodulatifor signal filtered The

( ))()()()()(

+= tntstx

ff cQcI

)(ofpower average)SNR(

ratio noise-to-signal channel The(2.80) )()()( +

ts

tntstx

ratio noise-to-signaloutput The)( ofpower average)(pg)SNR( C =

tn

output at the noise ofpower averagesignal ddemodulate theofpower average)SNR( O =

(2.81) (SNR)(SNR)merit of Figure

C

O=

58

2.11 Noise in Linear Receiver Using Coherent DetectionThe DSD-SC systemy

( ) cos(2 ) ( ) ( ) ( )c c Ms t CA f t m t m t S fπ= ⇔

(2.83)( )

c c M

W

MWP S f df

−= ∫

C,DSB(SNR)2 2

2cPC A

WN=

(baseband) (2.84)

0

2 2c

WNC A P

= (baseband) (2.84)

C:system dependent scaling factor02WN

59

(2 85))2sin()()2cos()()()2cos()()()(

tftntftntmtfCAtntstx

++=

πππ

)2()cos()(

(2.85))2sin()()2cos()()()2cos(

t ftxtv

tftnt ftntmtfCA

c

cQcIcc

=

−+=

π

πππ

)(21)(

21 tntmCA Ic +=

[ ] )4sin()(21)4cos()()(

21 t ftnt ftntmCA cQcIc −++ ππ

11componentsfrequency high

:indicates(2 86)

(2.86) )(21)(

21)(filter pass-Low tntmCAty Ic +=⇒

d t th tb thj t dl t li)(2output.receiver at the additive are )( and )( 1.

:indicates(2.86)

ttntm I

detector.coherent by therejectedcompletely is )( 2. tnQ

60

4power ))(21( signaloutput average The 22= PACtmCA cc

2Let 42

= WBT

212)

21(power ))(

21( noise average The 00

2 == WNWNtnI

(2.87) 2

4)SNR(22

0

22

SCDSBO, ==− WNPAC

NW

PACcc

(SNR)

22O

00SCSO, WNNW

(2.88) 1(SNR)(SNR)

SC-DSBC

O=

bandwidth.andeperformancbetweenoff-tradeNo2.SC-DSB ofmerit of figure same thehas SSBCoherent 1.

problem! Serious bandwidth.andeperformancbetween offtradeNo 2.

61

2.12 Noise in AM Receivers Using Envelope Detection

[ ] (2 89))2cos()(1)( tftmkAts π+= [ ])2cos()()2cos(

(2.89) )2cos()(1)(t ftmkAt fA

tftmkAts

caccc

cac

πππ

+=+=

(2 90))1()SNR(22

AMCPkA ac +

=

:filter theofoutput At the

(2.90) 2

)SNR(0

AMC,WN

=

[ ] (2.91) )2sin()()2cos()()( )()()(

tftntftntmkAAtntstx

cQcIacc ππ −++=+=

[ ] ( ))()()()()( ff cQcIacc

62

[ ]{ } (2.92) )()()(

)( of envelope)(2

1 22 c tntntmkAA

txty

QIac +++=

=

[ ]{ } ( ))()()(c QIac

)( )()( Assume tntntmkAA QIacc >>++

1

)()()(2

WNA

tntmkAAty

c

Iacc

>>

++=

(carrier power > noise power)

1 2.2

.1 0

k

WN

a

c

<

>> (carrier power > noise power)

(2.94) 2

(SNR)0

22

AMO,WN

PkA ac≈

(2.95) 1(SNR)

(SNR)2

2

C

O

PkPk

a

a

AM +≈

63

64

Threshold Effect

noise power > carrier power

65

noise power > carrier power

2.13 Noise in FM Receivers

The discriminator consists of a slope network and an envelope detector.

)2sin()()2cos()()(Let t ftnt ftntn cQcI ππ −=

[ ] ( ))()()()( f[ ] (2.130) )()2(cos)()( ttftrtn c ψπ +=

[ ] (2.131) )()()( is envelope The 2122 tntntr QI +=

)( ⎤⎡(2.132)

)()(

tan)( is phase The 1⎥⎦

⎤⎢⎣

⎡= −

tntn

tI

Qψ

(1.115) 0 ),2

exp()(

.2over ddistributeuniformis)(andd,distributeRayleigh is)(where

2

2

2R

π

≥−= rrrrf

tΨtr

(1.114) 2 0 ,21)(

2 22

πψπ

ψ

σσ

<<=Ψf 66

The incoming FM signal s(t) is defined by

(2 133))(22cos)( ⎤⎡ + ∫t

dktfAt (2.133) )(22cos)(0 ⎥⎦

⎤⎢⎣⎡ += ∫fcc dmktfAts ττππ

[ ] (2.135) )(2cos tt fA cc φπ +=

(2.134) )(2)( where

0 ττπφ dm kt

t

f ∫=

outputfilterbandpassAt the

[ ] [ ] (2 136))(2)()(2)()()(

outputfilter bandpass At the

ffAtntstx

φ+=

[ ] [ ] (2.136) )(2cos)()(2cos tt ftrttfA ccc ψπφπ +++=

r

φψ −

ψ

r

φθ −

A

[ ])( where

⎫⎧

>> trAc

θ φ cA

[ ][ ] )137.2(

)()(cos)()()(sin)( tan)()( 1

⎭⎬⎫

⎩⎨⎧

−+−

+= −

tttrAtttrtt

c φψφψφθ

67

Note that the envelope of x(t) is of no interest to us (limiter)

[ ] )1382()()(i)()()(

)( Because

tttrtt

trAc

φφθ +

>>

[ ] )138.2( )()(sin)()()( ttA

ttc

φψφθ −+≈

[ ](2.139))()(sin)()(2 tttrdm kt

f φψττπ −+= ∫ [ ]( ))()()(0 Ac

f φψ∫

)(12.40) (Fig isoutput tor discrimina The

dθ

(2 140))()(

)(21)(

ttkdt

tdtv

+

=θ

π

hnoise additivemessage

(2.140) )()( tntmk df +=

[ ]{ } (2.141) )()(sin)(2

1)(

where

tttrdd

Atn

dφψ −= [ ]{ } ( ))()()(

2)(

dt Ac

dφψ

π68

),2(0,over ddistributeuniformly is )()( Assume tt πφψ −

)(i lifWsignal. message oft independen is )(then

),( ,y)()(tnd

φψ

[ ]{ } (2 142))(sin)(1)( ttrdtn ψ≈

as )(simplify may We tnd

[ ]{ } (2.142) )(sin)(2

)( ttrdt A

tnc

d ψπ

≈

h)(d)(fd fi itiF tt[ ] (2.143) )(sin)()(

havewe,)(and)(ofdefinitionFromttrtn

ttr

Q ψψ

=Q

(2.144) )(

21)(

dtdn

Atn Q

d ≈ ( )2

)(dtAc

d π

The quadrature component appearsThe quadrature component appears69

From (2.140)The average output signal power = kf

2PRecall d TFRecall

fjdtd TF

π2.

⇔

nQ(t) nd(t)d1dtAcπ2)( fS

QN )( fSdN

(2 145))()(2

fSffSnoise is enhanced at high frequency

(2.145) )()( 2 fSA

fSQd N

cN =

70

Assume that nQ(t) has ideal low-pass characteristic with bandwidth BTwith bandwidth BT

(2 146))(2

0 TBffNfS ≤ (2.146) 2

, )( 20 T

cN f

AffS

d≤=

2 If WBT >

output receiver At the

2If W>

(2.147) , )( 2

20

0Wf

AfNfSc

N ≤=

71

)( ofpower Average

22

00

W

Wc

dffANtn = ∫−

(2.148) 3

2 2

30

AWN

=

effect quieting noise 1

3

2

c

A

A

∝cA

(2.149) 23

)SNR( 30

22

FM, WNPkA fc

O =2A

when increasing carrier power

0The average power of is ,

the a erage noise po er in message band idth is

( )2cAs t

WN

FM

the average noise power in message bandwidth is

SNR (2 150)

02

( ) cC

WN

A⇒ ,FM SNR (2.150)

0

( )2

cC

WN⇒ =

(2 151)3)SNR( 2 Pk fO

⇒2

,(2.29) ( ) ( )f m o FMf k A SNR fΔ = ∝ Δ

(2.151) )SNR( 2

FMW

f

C=⇒

72

⎥⎤

⎢⎡ Δ

+= )2sin(2cos)( tfftfAts ππ

Example 2.5 Single-Tone Modulation⎥⎦

⎢⎣

+= )2sin(2cos)( tff

tfAts mm

cc ππ

)2sin()(2 , may write We0

t fffdm k mm

t

f πττπ Δ=∫

)2cos()( sideboth t fk

ftmdtd

mf

πΔ=⇒

2)(il d)1()(fTh fP ΔΩ 22

)(is load)1(across)(ofpower averageThefk

fPtm =Ω

,FM3 3From (2.149), SNR

2 2 2 2

3

( )( ) ,4 4

c cO

A f A fN W N W W

ββ

Δ Δ= = =

03

04 4N W N W W

(2.152) 23)(

23

)SNR()SNR( 22 β=

Δ=⇒

WfO

2.4) Example (from 31

)SNR()NR( , AM tocompare

C

O=

S

22)SNR( FM WC

3)SNR( C AM

2

e.performancbetter has FM , 31

23When 2 >β

FM. widebandand FM narrowbandbetween n transitio theas 5.0 Define

471.032

=

=>⇒

β

β

73

FM Threshold Effect (When CNR is low)

When there is no signal, i.e., carrier is unmodulated. The composite signal at the frequency discriminator input

[ ] (2.153)

( ) t 1

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

( )

cc I c Q

Q

x t A n t f t n t f t

n t

π π

θ −

= + −

( ) tan

Occasionally,

1 ( )( )

Q

c I

tA n t

θ =+

may sweep around the origin , ( >1 ( ) )cP r t Ay,

'

y p g , ((t) increases or decreases 2

Th di i i t t t i l t

( ) )

( )

c

tθ π

θThe discriminator output is equal to ( )2

tθπ

( )r(t)

P1

nQ(t) x(t)

Ac0 P2

θnI(t) 74

Figure 2.44 Illustrating impulselike components in θ′ (t) =dθ (t)/dt produced by changes of 2π in θ (t); (a) and (b) are graphs of θ(t) and θ′ (t), respectively.

75

d tr t A t t d t ψψ π ψ ψ> < ≤ + >

A positive-going click occurs , when

0( )( ) ( ) ( ) ( )cr t A t t d tdt

ψ π ψ ψ> < ≤ + > , , 0

A negative-going click occurs when

( ) ( ) ( ) ( )

cd tr t A t t d t

dtψψ π ψ ψ> > − > + < , , 0( )( ) ( ) ( ) ( )

The carrier-to -noise ratio is definA

ed by 2c

T

AB N

ρ = (2.154)

The output signal to noise ratio is calculated as02

B

The output signal-to-noise ratio is calculated as1. The average output signal power is calculated assuming

TBfΔ = a sinusoidal modulation which produces . (noise free)

22

Th t t i i l l t d h2. The average output noise power is calculated when no signal is present (The carrier is unmodulated). 76

Figure 2.45 Dependence of g poutput signal-to-noise ratio on input carrier-to-noise ratio for FM i I I hFM receiver. In curve I, the average output noise power is calculated assuming ancalculated assuming an unmodulated carrier. In curve II, the average output noise g ppower is calculated assuming a sinusoidally modulated

i B th I d IIcarrier. Both curves I and II are calculated from theory.

(2.155),20or 20When 0

22

NBAAT

cc ≥≥=ρ

avoided bemay effects threshold

( ),22 0

0NB TT

ρ

77

The procedure to calculate minimum ( 20)cA ρ ≥p 1. Given and W, determine

( )c

TBρ

β (using Figure 2.26 or Carson's rule)

2

2. Given , we have 202

0 02c

TAN B N≥

Capture Effect:2

The receiver locks onto the stronger signald th k and suppresses the weaker one.

78

FM Threshold Reduction (tracking filter)

FM d d l t ith ti f db k (FMFB)• FM demodulator with negative feedback (FMFB)

• Phase locked loopp

Figure 2.46 FM h h ld iFM threshold extension.

Figure 2.47 gFM demodulator with negative feedback.

79

Pre-emphasis and De-emphasis on FM

Figure 2 48 (a) Power spectral density of noise at FM receiver oFigure 2.48 (a) Power spectral density of noise at FM receiver o(b) Power spectral density of a typical message signal.

Figure 2.49 Use of pre-emphasis and de-emphasis in an FM system.80

(2.156),1)(d ≤≤−= WfWfH

isoutput tor discriminaat the PSDThe

(2.156) , )(

)(pe

de ≤≤ WfWfH

fH

(2.146) 2

, )(

p

2

20 ≤=

BfA

fNfS TNd

(2 157))()()(

2

22

02 ≤=BffHfNfSfH

A

T

cd

(2 158))( noiseoutput Average

(2.157) 2

, )()()(

220

de2de

∫⎟⎟⎞

⎜⎜⎛

≤=

dffHfN

ffHA

fSfH

W

cNd

isfactortimprovemenThe

(2.158) )(emphsis-de power with de

220 ∫=⎟⎟

⎠⎜⎜⎝ −

I

dffHfA W

c

(2.162) 2isfactor t improvemenThe

3

=WI

I

( ) )(3

w

w-

22∫ dffHf de

81

Example 2.6 Figure 2.50 (a) Pre-emphasis filter.(b) De-emphasis filter.

isresponsefilteremphsis-presimpleA

1)(

isresponsefilter emphsispresimpleA

0pe +=

fj ffH

1 is responsefilter emphsis-deA

0f

1

1)(0

de+

=

fj ffH

(2.161) )(t)(3

)(

3

21

3

0

2

3

⎤⎡==

−

∫ WWf

W

dffWI

W )(tan)(3)(1

3 0

1

02

0

⎥⎦⎤

⎢⎣⎡ −

+−∫ fW

fW

ffdff

W 82

Preemphasis for FMThe main difference between FM and PM is in the relationship between frequency and phase.

f = (1/2π).dθ/dt. A PM detector has a flat noise power (and voltage) output

f ( t l d it ) Thi iversus frequency (power spectral density). This is illustrated in Figure 9-38a. However an FM detector has a parabolic noise powerHowever, an FM detector has a parabolic noise power spectrum, as shown in Figure 9-38b. The output noise voltage increases linearly with frequency.g y q yIf no compensation is used for FM, the higher audio signals would suffer a greater S/N degradation than the lower frequencies. For this reason compensation, called emphasis, is used for broadcast FM.

83

Preemphasis for FM

Figure 9-38. Detector noise output spectrafor (a). PM and (b). FM.

84

Preemphasis for FMA preemphasis network at the modulator input provides a constant increase of modulation index mfprovides a constant increase of modulation index mffor high-frequency audio signals. Such a network and its frequency response are q y pillustrated in Figure 9-39.

Fig. 9-39. (a)Premphasis network, and (b) Frequency response.

85

Preemphasis for FMWith the RC network chosen to give τ = R1C = 75μs in North America (150μs in Europe), a constant input audio signal will

lt i l t t i i th VCO i t lt fresult in a nearly constant rise in the VCO input voltage for frequencies above 2.12 kHz. The larger-than-normal carrier deviations and mf will preemphasize high-audio frequencies. f p p g q

At the receiver demodulator output, a low-pass RC network with τ = RC = 75μs will not only decrease noise at higherwith τ = RC = 75μs will not only decrease noise at higher audio frequencies but also deemphasize the high-frequency information signals and return them to normal amplitudes relative to the low frequencies.

The overall result will be nearly constant S/N across the 15-The overall result will be nearly constant S/N across the 15kHz audio baseband and a noise performance improvement of about 12dB over no preemphasis. Phase modulation systems d t i h ido not require emphasis.

86

Pre-emphasis and De-emphasis on FM

P h i d d h i ( ) h ti di (b) tt tiPreemphasis and deemphasis: (a) schematic diagrams; (b) attenuation curves

87

Pre-emphasis and De-emphasis on FM

Example of S/N without preemphasis and deemphasis. 88

Pre-emphasis and De-emphasis on FM

Example of S/N with preemphasis and deemphasis. 89

D lb d i h iDolby dynamic preemphasis90

Figure 2.55 Comparison of the noise performance of various CW modulation systems. Curve I: Full AM, μ = 1. Curve II: DSB-SC, SSB. Curve III: FM, β = 2. Curve IV: FM, β = 5. (Curves III and IV include 13-dB pre-emphasis, de-emphasis improvement.)

91

In making the comparison, it is informative to keep inmind the transmission bandwidth requirement of themodulation systems in question. Therefore, we definenormalized transmission bandwidth as

BB T

WB T

n =

T bl 2 4 V l f B f i CW d l ti hTable 2.4 Values of Bn for various CW modulation schemes

FM

AM, DSB-SC SSB β = 2 β = 5

Bn 2 1 8 16

92