1

Chapter 2 Continuous-Wave Modulation

2.1 Introduction

2

2.2 Amplitude Modulation

The output of the modulator

Where m(t) is the baseband signal , ka is the amplitude sensitivity.

frequencycarrier :

amplitudecarrier :

(2.1) )2cos()(

c

c

cc

f

A

tfAtc

(2.2) )2cos()(1)( tftmkAts cac

)( offreqency hightest theis where

(2.4) .2

(2.3) t allfor ,1 )( .1

tmW

Wf

tmk

c

a

X 1+kam(t) S(t)

Accos(2fct)

3

Recall

1.Negative frequency component of m(t) becomes visible.

2.fc-W M(f) fc lower sideband

fc M(f) fc+W upper sideband

3.Transmission bandwidth BT=2W

(2.2) )2cos()()2cos()( tftmkAtfAts caccc

)( of Transform Fourier theis )( where

(2.5) )()(2

)()(2

)(

)()(2

1)2cos()(

)()(2

1)2cos(

tmfM

ffMffMAk

ffffA

fs

ffMffMtftm

fffftf

ccca

ccc

ccc

ccc

4

Virtues and Limitations of Amplitude Modulation

Transmitter

Receiver

Major limitations

1.AM is wasteful of power.

2.AM is wasteful of bandwidth.

5

2.3 Linear Modulation Schemes

Linear modulation is defined by

Three types of linear modulation:

1.Double sideband-suppressed carrier (DSB-SC) modulation

2.Single sideband (SSB) modulation

3.Vestigial sideband (VSB) modulation

component Quadrature)(

component phase-In)(

(2.7) )2sin()()2cos()()(

ts

ts

tftstftsts

Q

I

cQcI

6

Notes:

1.sI(t) is solely dependent on m(t)

2.sQ(t)is a filtered version of m(t).

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

7

Double Sideband-Suppressed Carrier (DSB-SC) Modulation

The Fourier transform of S(t) is

(2.8) )2cos()()( tftmAts cc

(2.9) )()(2

1)(

ccc ffMffMAfs

8

Coherent Detection (Synchronous Detection)

The product modulator output is

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

(2.10) )()cos('2

1)()4cos('

2

1

)()2cos()2cos('

)()2cos(' )(

tmAAtmtfAA

tmtftfAA

tstfAtv

ccccc

cccc

cc

(2.11) )( cos'2

1)(0 tmAAtv cc

filtered out

(Low pass filtered)

9

Costas Receiver

I-channel and Q-channel are coupled together to

form a negative feedback system to maintain synchronization

The phase control signal ceases with modulation.

1

4

2 2 2 2

2 2

0

1 1cos sin ( ) ( )sin(2 )

4 8

( ) (sin2 2 )

c c

c

A m t A m t

A m t

(multiplier +

very narrow band LF)

10

Quadrature-Carrier Multiplexing (or QAM)

Two DSB-SC signals occupy the same channel

bandwidth, where pilot signal (tone ) may be

needed.

)2sin()()2cos()()( 21 tftmAtftmAts cccc

11

Single-Sideband Modulation (SSB)

The lower sideband and upper sideband of AM signal

contain same information .

The frequency-discrimination method consists of a

product modulator (DSB-SC) and a band-pass filter.

The filter must meet the following requirements:

a.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.

To recover the signal at the receiver, a pilot carrier or a stable oscillator

is needed (Donald Duck effect ).

12

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

The transition must satisfy

(2.14)

(2.13) for 1)()(

:linear is response phase b.The

1)()(.a

fW B

W fWffHffH

ffHffH

νT

cc

cc

Consider the negative frequency response: H f

c vf f cf c vf fcf Wc vf f cf

c vf f cf W

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

2 c vf f 2 cf 2 c vf f 2 cf Wvf0vfW

13

and │H(f+fc)│ is cH f f

vf 0 vf W2 c vf f 2 cf2 c vf f 2 cf W

14

So, we get │H(f-fc)│ +│ H(f+fc)│ is

cH f f

2 c vf f 2 cf 2 c vf f 2 cf Wvf0vfW

cH f f

vf 0 vf W2 c vf f 2 cf2 c vf f 2 cf W

15

Consider –W<f<W we get:

vfvfW 0 W

Which is equal to

W W

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

16

17

± corresponds to upper or lower sideband

(2.15) )2sin()('2

1)2cos()(

2

1)( tftmAtftmAts cccc

HQ(f) m(t) m’(t)

(2.16) for )()( )( W f WffHffHjfH ccQ

18

Television Signals (NTSC)

19

2.4 Frequency Translation

Up conversion

f2=f1+fl , fl=f2-f1

Down conversion

f2=f1-fl , fl=f1-f2

cos(2 )A f t

20

2.5 Frequency-Division Multiplexing (FDM)

21

2.6 Angle Modulation

Basic Definitions:

Better discrimination against noise and interference

(expense of bandwidth).

The instantaneous frequency is

(2.19) )(cos)( tAts ic

constant is where

(2.22) 2)(

is )( carrier, dunmodulate anFor

(2.21) )(

2

1

2

)()(lim

)(lim)(

0Δ

Δ0Δ

c

cci

i

i

ii

t

tt

i

tft

t

dt

td

t

ttt

tftf

22

1. Phase modulation (PM)

2. Frequency Modulation (FM)

(2.23) )(2cos

modulator theofy sensitivit phase :

)(2)(

tmktfAs(t)

k

tmktft

pcc

p

pci

(2.24)

π π

π π (2.26)

:frequency sensitivity of the modulator

compare (2.23) and (2.26)

0

0

( ) ( )

( ) 2 2 ( )

cos 2 2 ( )

i c f

t

i c f

t

c c f

f

p

f t f k m t

t f t k m d

s(t) A f t k m d

k

k m'

π0

2 ( )t

f(t) k m d

(2.25)

generating FM signal generating PM signal

23

2.7 Frequency Modulation

FM is a nonlinear modulation process , we can not apply

Fourier transform to have spectral analysis directly.

1.Consider a single-tone modulation which produces a

narrowband FM (kf is small)

2.Next consider a single-tone and wideband FM

(kf is large)

deviationfrequency :Δ

(2.28) )2cos(

)2cos( )(

(2.27) )2cos()(let

mf

mc

mmfci

mm

Akf

tfff

tfAkftf

tfAtm

(deterministic)

24

radian. one nlarger tha is , FM Wideband

radian. one ansmaller th is , FM Narrowband

(2.33) )2sin(2cos)(

(2.32) )2sin(2 )(

(2.31) index Modulation

(2.30) )2sin(2

)(2)( (2.25), Recall0

tftfAts

tftπft

f

f

tff

ftπf

dft

mcc

mci

m

m

m

c

t

ii

(2.19) =>

25

Narrowband FM

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

)2sin()2sin(sin

1)2sin(cos

small, is Because

)34.2( )2sin(sin)2sin()2sin(cos)2cos(

)2sin(2cos)(

tftfAtfAts

tftf

tf

tftfAtftfA

tftfAts

mcccc

mm

m

mccmcc

mcc

26

The output of Fig 2.21 is

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

1.The envelope contains a residual AM.

(FM has constant envelope)

2.i(t) contains odd order harmonic distortions

For narrowband FM, ≤ 0.3 radians.

)2sin()()2cos()(' tfdmkAtfAts cfccc

)!7!5!3

(sin753

xxx

xx

β

27

(2.37) )(2cos)(2cos2

1)2(cos

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

(2.2) )2(cos)(1 )(

)2cos()( , wavemodulating sinusoidal with AMFor

(2.36) )(2cos)(2cos2

1)2(cos

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

(2.35) Recall

AM

tfftffAtfA

tftfAktfA

tftmkAts

tftm

tfftffAtfA

tftfAtfAts

mcmcccc

mccacc

cac

m

mcmcccc

mcccc

Narrow band FM

AM

28

Wideband FM (large β)

(2.40) )2exp()(~

)]2sin(exp[)(~by defined envelopecomplex theis )(~ andpart real thedenotes Re where

(2.38) ))(2exp()(~Re

))2sin(2exp(Re)(

sincosexp

(2.33) )2sin(2cos)(

n

mn

mc

c

mcc

mcc

t nfjcts

tfjAts

ts

tfjts

tfjtfjAts

xjx(jx)

tftfAts

(2.39)

Complex Fourier Transform

29

(2.41)

Let (2.42)

12

12

12

12

( )exp( 2 )

exp sin(2 ) 2 )

2

exp ( sin )2

m

m

m

m

f

n m mf

f

m c m mf

m

cn

c f s t j nf t dt

f A j f t j nf t dt

x f t

Ac j x nx dx

2

(2.43)

Define the th order Bessel function of the first kind as

A3, x

(2.44)

22 2

2( ) 0)

1( ) exp ( sin )

2

( )

( ) ( )

n

n c n

c n

n

n

d y dyx x n y

dx dx

J j x nx dx

c A J

s t A J

(2.45)exp( 2 )m

j nf t

（

30

(2.49) )()()(2

)(

is )( of ransform Fourier tThe

(2.48) )(2cos)(

(2.47) )(2exp)(Re)(

mcmcnc

mcnc

mcnc

nfffnfffJA

fS

ts

tnffJA

tnffjJAts

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

31

-

Properties of ( )

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

2.If is small

( ) 1

( )2

( ) 0 2 (2.51)

3. ( ) 1

Observation o

0

1

2

n

n

n n

n

n

J

J J n

J

J

J n

J

f FM

1.An FM signal contains components.

2.For small , the FM signal is effectively composed of a carrier and

a single pair of side freqencies at narrowband FM

3.The am

2 3 ,c m m m

c m

f , f , f , f

f f

plitude of carrier depends on

A (2.54)

22 21

( )2 2

cc n

P A J

32

Example 2.2

33

Transmission Bandwidth of FM signals

With a specified amount of distortion , the FM signal is

effectively limited to a finite number of significant side

frequencies.

A.Carson’s rule

, = , (2.55)1

2 2 2 (1 )T m m

m

fB f f f f f

f

34

B.

, ( ) 0.01 , maxmax max2 2

T m n T

fB n f J B n

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

35

Example 2.3

In north America, the maximum value of frequency deviation is fixed 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

Using Carson’s rule of Equation (2.55) , replacing by D , and replacing fm by W , the approximate value of the transmission bandwidth of the FM signal is obtained as

BT=2(75+15)=180kHz

On 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=240kHz

In 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 of Figure 2.26 overestimates it by 20 percent.

515

75D

f

f

36

Generation of FM signals

(2.56)

( )

The frequency multiplier output

(2.58)

21 2

0

0

( ) ( ) ( ) ( )

cos 2 2 ( )

'( ) 'cos 2 2 ( )

'( ) ( )

nn

t

c c f

t

c c f

i c f

v t a s t a s t a s t

s t A f t k m d

s t A nf t nk m d

f t nf nk m t

(2.59)

Frequency Multiplier

Varactor diode VCO FM modulator

32-1

Crosby Direct FM Transmitter

32-2

Demodulation of FM signals

The frequency discrimination consists of a slope circuit

followed by an envelope detector

Consider Fig 2.29a , the frequency response of a slope

circuit is

elsewhere ,022

),2

(2

22 ),

2(2

)(1

Tc

Tc

Tc

Tc

Tc

Tc

Bff

Bf

Bffaj

Bff

Bf

Bffaj

fH

(2.60)

33

34

1 1( ) 2 ( ) , 0c

H f f H f f

2 2( ) 2 ( ) , 0c

H f f H f f

Appendix 2.3 Hilbert Transform

Fourier Transform-frequency-selective

Hilbert Transform-phase-selective

(±900shift)

Let g(t)G(f)

Denote the Hilbert transform of g(t) as

(A2.32) )(ˆ1

)(

ansformHilbert tr inverse The

(A2.31) )(1

)(ˆ

dt

gtg

dt

gtg

35

j ft

f

f f

f

g t

G f j f G f

(A2.33)

(A2.34)

The Fourier transform of is

(A2.35)

1sgn( )

1 0

sgn( ) 0 0

1 0

( )

ˆ( ) sgn( ) ( )

H(f) g(t) )(ˆ tg

36

Properties of the Hilbert Transform

(time domain operation)

If g(t) is real

)(ˆ)g( 0)(g)g(3.

)( is )(ˆ of transform2.Hilbert

spectrum magnitude same thehave )( and )(ˆ.1

- tgtdttt

tgtg

tgtg

37

(take H.F of and

compare with A2.32)

( )g t

For a band-pass system , we consider

x(t) X( f )

X( f ) is limited within ± W Hz

W fc

(A2.48)

The complex evelope of x(t) is

(A2.49)

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

( ) ( ) ( )

I c Q c

I Q

x t x t f t x t f t

x t x t j x t

band pass

system, fc

x(t) y(t)

H(f)

fc

f

2B

(A2.50)( ) ( )cos(2 ) ( )sin(2 )I c Q c

h t h t f t h t f t 38

)'(2)'(~

, )( from )(~

obtain can We

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

with tolimited is )(H~

and

)()(*

real is )( Since

(A2.54) )(*~

)(~

)(2

(A2.53) toansformFourier trApply

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

)2exp()(~

)(2

)*2 , ( have we(A2.52) From

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

(A2.52) )2exp()(~

Re)(

)( oftion representacomplex The

(A2.51) )()()(~

response implusecomplex theDefine

c

c

c

cc

cc

QI

c

QI

ffHfHfHfH

ffHffH

fBBff

fHfH

th

ffHffHfH

t fjtht fjthth

zzvjuvz

tthth

t fjthth

th

tj hthth

39

band-pass h(t)

system

x(t) y(t)

(A2.57)

(A2.58)

Define the pre-envelope of as

( ) Re ( )exp( 2 )

( ) ( )

( )

( ) ( ) ( ),

( ) ( ) sgn( ) ( )

2 ( )

( ) (0)

0

cy t y t j f t

h x t d

h t

h t h t j h t

H f H f f H f

H f f

H f H

A

0

0 ( 2.37)

0

f

f

:)(th

Hilbert T. of )(th

40 A A2.59)2.58 ( ) Re ( ) Re ( ) (y t h x t d

1 1

-

ˆRecall ( ) ( )

( ) Re ( )

( ) Re ( )

Re ( ) ( )

ˆ ˆRe [ ( ) ( )][ ( ) ( )]

ˆ ˆ( ) ( ) ( ) ( )

h( )x(t- )d

h t h t jh t

h t h t

x t x t

h x t d

h jh x t jx t d

h x t d h x t d

1 1

ˆ( ) ( ) ,

ˆ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 ( ) ( )

2 Re[ ( )]Re[ ( )]

u

t u

h u x t d du

h x t d x d h u du

h x t d h u x t u du

h x t d

h x t d

dd

t

, t

To prove (A2.60)

41

)(~)(~

)2exp(Re2

1

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

Re2

1

)()(Re2

1

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

becomes (A2.58)

dtxhtfj

dtfjtxfjh

dtxh

dtxhty

c

cc

42

)(

}{ (1)

)(2

222

c

tnffj

tfjtnfjtnfj

nff

dte

dteeeF

c

ccc

)()(1

0),(1

0),(1

0,1

0,1

0,

0,

, 令, }{ (2)

)(2

)(2

22

22

222

cc

c

c

kfn

fj

kfn

fj

kn

fj

kfj

kn

fj

kfj

tfjtnfjtnfj

nfffn

f

n

nfn

f

n

nfn

f

n

ndken

ndken

nn

dkee

nn

dkee

n

dkdtkntdteeeF

c

c

c

c

ccc

{

{

{ =

=

=

43

)2exp(factor he without t(t)h~ and

(t)y~ (t),x~ functions lowpass equivalent by the

systems and signals bandpassrepresent can We

(A2.63) )(~*)(~

)(~2or

(A2.62) )(~)(~

)(~2

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

tfj

txthty

dtxhty

c

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

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

(A2.66) )(~)(~)(~let

(A2.65) )()()()(

)()()()(

(A2.64) )()()()()(~2

Q

I

txthtxth

txthtxth

tyjtyty

txthtxthj

txthtxth

tjxtxtjhthty

QIIQ

QQII

QI

QIIQ

QQII

QIQI

44

45

Procedure for evaluating the response

of a band-pass system

)2exp()(~ Re)( .4

)(~*)(~

)(~2Obtain .3

)2exp()(~

Re)( .2

)2exp()(~ Re)(

)(~by )( Replace 1.

tfjtyty

txthty

tfjthth

tfjtxtx

txtx

c

c

c

46

To simplify the analysis

1. shift to the right by to align to the band-pass frequency

2. set , for (2.61)

Recall

1

1 1

1

( )

( ) 2 ( ) 0

2 ( )2

( )

c

c

Tc

H f f

H f f H f f

Bj πa f f

H f j

(2.60)

elsewhere

From (2.60) and (2.61), we get

(2.62)

elsewhere

1

2 2

2 ( )2 2 2

0

4 ( )( ) 2 2 2

0

T Tc c

T T Tc c c

T T T

B Bf f f

B B Bπa f f f f f

B B Bj a f f

H f

47

48

t

cc f

t

c f

s t

s t A f t k m d

s t A j k m d

s t

1

Recall FM signal ( )

The complex envelope is

(2.63)

Let denote the complex envelop

0

0

( ) cos 2 2 ( )

( ) exp 2 ( )

( )

T T T

y t h t x t

S f H f S f

B B Bj a f S f f

e of the slope ckt. response output.

Recall (A2.63) 2 , we have

upper arm of Fig 2.30 in text)

elsewhere

11

( ) ( ) ( )

1( ) ( ) ( ) (

2

2 ( ) ( )2 2 2

0

T

tf

T c f

T

d s ts t a j B s t

dt

ks t j B aA m t j k m d

B

(2.64)

(2.65)

From (2.63) and (2.65) , we have

(2.66)

1

10

( )( ) ( )

2( ) 1 ( ) exp 2 ( )

(2.67) 2

)(22cos)(2

1

)2exp()(~Re)(

0

11

t

fc

T

f

cT

c

dm kt ftmB

kaA B

t fjtsts

0sin 2 2 ( )

t

cf

f t k m d

is a hybrid-modulated signal (amplitude , frequency)

However, provided that we choose 1, for all

using an envelope detector, we have

1

1

( )

2( )

2( ) 1 ( )

f

T

f

T c

T

s t

km t t

B

ks t B aA m t

B

(2.68)

The bias term can be removed by a second frequency

discriminator with ( ) , where ( )2 2 1( ).

T c B aA

H f H f H f

49

(2.71) )(4

)(~)(~)(

(2.70) )(2

1)(~

(2.69) )(~

)(~

210

2

12

tmaA k

tststs

tmB

kaA Bts

fHfH

cf

T

f

cT

Balanced Frequency Discriminator

Let the transfer function of the second branch of Fig 2.30

be (complementary slope circuit)

50

FM Stereo Multiplexing

Two factors which influence FM stereo standards

1.Operation within the allocated FM channels.

2.Compatible with monophonic radio receiver.

(2.72) )2cos()4cos()()()()()( t fKt ftmtmtmtmtm ccrlrl 51

Figure 9-40. FM stereo generation block diagram.

Stereo FM

51-1

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. This 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-SC.

Stereo FM

51-2

Stereo FM transmitter using frequency-division multiplexing.

Stereo FM Transmitter

51-3

Stereo FM transmitter: (a) block diagram; (b) resulting spectrum.

SAC: Subsidiary Communication Authorization

Stereo FM Transmitter

51-4

The stereo receiver will need a frequency-coherent 38-kHz

reference signal to demodulate the DSB-SC.

To simplify the receiver, a frequency- and phase-coherent

signal is derived from the subcarrier oscillator by frequency

division (÷2) to produce a pilot.

The 19-kHz pilot fits nicely between the L+R and DSB-SC L-

R signals in the baseband frequency spectrum.

Stereo FM

51-5

As indicated by its relative amplitude in the baseband

composite signal, the pilot is made small enough so that

its 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

oscillator, 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 used

to illustrate the formation of the composite stereo (without

pilot) in Figure 9-41.

Stereo FM

51-6

Figure 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.

Stereo FM

51-7

Spectrum of stereo FM signal. SCA: Subsidiary communication authorization (commercial-free program)

Stereo FM

51-8

51-9

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

2.8 Nonlinear Effects in FM Systems

1.Strong nonlinearity, e.g., square-law modulators ,

hard limiter, frequency multipliers.

2.Weak nonlinearity, e.g., imperfections

Nonlinear input-output relation

(2.73) )()()()(

32

3210 tvatvatvatv iii

Nonlinear

Channel (device)

vi(t) v0(t)

52

(2.75) )(36cos4

1

)(24cos2

1

)(2cos)4

3(

2

1

(2.74) )(2cos

)(2cos)(2cos)(

)(2)(

)(2cos)(

signal FMFor

3

3

2

2

3

31

2

2

33

3

22

210

0

tt fAa

tt fAa

tt fAaAaAa

tt fAa

tt fAatt fAatv

dm kt

tt fAtv

cc

cc

cccc

cc

cccc

t

f

cci

53

WfffB mT 2222 , rule sCarson'

W W W W f4f2

fc 2fc

In order to seperate the desired FM signal from the second

harmonic , we have

2

(2.76)

The output of the band-pass fil

(2 )

3 2

c c

c

f f W f f W

f f W

ter is

no effect to

An FM system is extremely sensitive to phase nonlinearities.

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

3

0 1 3

3'( ) ( )cos 2 ( ) ( ( ))

4c c c

v t a A a A f t t m t

54

2.9 Super Heterodyne Receiver

(Carrier-frequency tuning , filtering , amplification , and demodulation)

fIF=fLO-fRF (2.78)

A FM system may use a limiter to remove amplitude variations.

55

AM radio receiver

Commercial FM Broadcast、

Allocations and Sidebands

56

2.10 Noise in CW modulation System

1.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 .

20N

57

(2.81) (SNR)

(SNR)merit of Figure

output at the noise ofpower average

signal ddemodulate theofpower average)SNR(

ratio noise-to-signaloutput The

)( ofpower average

)( ofpower average)SNR(

ratio noise-to-signal channel The

(2.80) )()()(

ison demodulatifor signal filtered The

(2.79) )2sin()()2cos()()(

:tionrepresenta noise narrowbandin noise filtered The

C

O

O

C

tn

ts

tntstx

t ftnt ftntn cQcI

58

2.11 Noise in Linear Receiver Using Coherent Detection

The DSB-SC system

C,DSB

(2.83)

(SNR)

(baseband) (2.84)

C:system dependent scaling factor

2 2

0

2 2

0

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

( )

2

2

c c M

W

MW

c

c

s t CA f t m t m t S f

P S f df

PC A

WN

C A P

WN

59

detector.coherent by the rejected completely is )( 2.

output.receiver at the additive are )( and )( 1.

:indicates (2.86)

(2.86) )(2

1)(

2

1)(filter pass-Low

componentsfrequency high

)4sin()(2

1)4cos()()(

2

1

)(2

1)(

2

1

)2()cos()(

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

)()()(

tn

tntm

tntmCAty

t ftnt ftntmCA

tntmCA

t ftxtv

t ftnt ftntmt fCA

tntstx

Q

I

Ic

cQcIc

Ic

c

cQcIcc

60

problem! Serious

bandwidth. and eperformancbetween off- tradeNo 2.

SC-DSB ofmerit of figure same thehas SSBCoherent 1.

(2.88) 1(SNR)

(SNR)

(2.87) 2

2

4)SNR(

2

12)

2

1(power ))(

2

1( noise average The

2Let

4power ))(

2

1( signaloutput average The

SC-DSBC

O

0

22

0

22

SCDSBO,

002

22

WN

PAC

NW

PAC

WNWNtn

WB

PACtmCA

cc

I

T

cc

61

2.12 Noise in AM Receivers Using Envelope Detection

)2cos()()2cos(

(2.89) )2cos()(1)(

t ftmkAt fA

t ftmkAts

caccc

cac

(2.91) )2sin()()2cos()()(

)()()(

:filter theofoutput At the

(2.90) 2

)1()SNR(

0

22

AMC,

t ftnt ftntmkAA

tntstx

WN

PkA

cQcIacc

ac

62

(2.92) )()()(

)( of envelope)(

21 22

c tntntmkAA

txty

QIac

(2.95) 1(SNR)

(SNR)

(2.94) 2

(SNR)

1 2.

2

.1

)()()(

)( )()( Assume

2

2

C

O

0

22

AMO,

0

2

Pk

Pk

WN

PkA

k

WNA

tntmkAAty

tntntmkAA

a

a

AM

ac

a

c

Iacc

QIacc

63

(carrier power > noise power)

Define the pre-demodulation SNR as

The average power of the modulated signal

SNR pre-de = The average noise power at the input of the demodulator

modulated Band pass

signal m(t)

s(t) filter

n(t) (SNR)pre-de (SNR)o

The Bandwidth of the bandpass filter is then the average noise power at the input of

the demodulator is

f

demodulator

(f)SN

20N

For an AM system SNR

If =2W SNR

To

ac

To

ac

BN

pkA

BN

pkA

2

)1(2)1( 2222

depre

AM

depre

AM

WN

pkA

o

ac

4

)1(22

Supplements

TB TB

TB

To BNTB

Supplements For a DSB-SC system,

SNR

為與課本一致加

depre

SCDSB

WN

pAC

BN

PAC

o

c

To

c

4

222

22

2C

For an FM system

SNR

If using Carson’s rule, we have

=2Δf+2fm>> fm =w

depre

FM

To

c

To

c

BN

A

BN

A

2

2 22

TB

For the purpose of comparing different CW modulation systems, we define

The average power of the modulated signal

(SNR)c= The average power of channel noise in the message band

Message signal with LP filter

the same power as output

modulated wave

noise

n(t)

The equivalent baseband transmission model.

with bandwidth w

Supplements

More precisely, we may express the DSB-SC

as m(t) S‘(t)

cos(2πfc t+θ)

θ is uniformly distributed over ﹝0, 2π﹞

S'(t)=Ac m(t) cos(2πfc t+θ)

At the receiver we may write

S(t)=C Ac m(t) cos(2πfc t+θ)

w

wmm

cmc

cc

cc

x

ss

dffSPR

PACRAC

tmEtfEAC

tftmCAE

dffS

RtSEP

)()0(

22)0(

)()2(cos

))2cos()((

)(

)0()(

2222

2222

2

2

The average noise power in –w<f<w

w

won WNdf

NP

2

0

For convenience we write the modulated signal

as θ不出現

Since is ergodic and we take as a sample function

Supplements

SNRc=

=

= =

The average power of the modulated signal

The average power of channel noise in the message band

The average power of S(t)

The average power of channel noise in the message band

Ps

Pn WN

PAC

o

c

2

22

)2cos()()( tftmCAtS cc

)2cos( tfc )2cos( tfc

2

)0(

22

22

PAC

RACP

c

mcs

[time average of [ ]] )2(cos 2 tfc

WN

PAC

WN

PACSNR cc

c

0

22

0

22

2

2

64

Threshold Effect

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

(2.130) )()2(cos)()( tt ftrtn c

(2.131) )()()( is envelope The 21

22

tntntr QI

(2.132) )(

)(tan)( is phase The 1

tn

tnt

I

Q

(1.114) 2 0 ,2

1)(

(1.115) 0 ),2

exp()(

.2over ddistribute uniform is )( and d,distributeRayleigh is )( where

2

2

2R

Ψf

rrr

rf

tΨtr

66

The incoming FM signal s(t) is defined by

(2.133) )(22cos)(0

t

fcc dm kt fAts

(2.134) )(2)( where

0 dm kt

t

f

(2.135) )(2cos tt fA cc

(2.136) )(2cos)()(2cos

)()()(

outputfilter bandpass At the

tt ftrtt fA

tntstx

ccc

)137.2( )()(cos)(

)()(sin)( tan)()(

)( where

1

tttrA

tttrtt

trA

c

c

67

r

cA

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

(2.141) )()( sin)( 2

1)(

where

tttrdt

d

Atn

c

d

)138.2( )()(sin)(

)()(

)( Because

ttA

trtt

trA

c

c

(2.139) )()(sin)(

)(20

ttA

trdm k

c

t

f

noise additivemessage

(2.140) )()(

)(

2

1)(

2.40) (Fig isoutput tor discrimina The

tntmk

dt

tdtv

df

68

(2.142) )(sin)(2

1)( ttr

dt

d

Atn

c

d

(2.143) )(sin)()(

have we, )( and )( of definition From

ttrtn

ttr

Q

(2.144) )(

2

1)(

dt

tdn

Atn

Q

c

d

as )(simplify may We

signal. message oft independen is )(then

),2 (0,over ddistributeuniformly is )()( Assume

tn

tn

tt

d

d

The quadrature component appears

69

From (2.140)

The average output signal power = kf2P

Recall

noise is enhanced at high frequency

fjdt

d TF

2.

nQ(t) nd(t)

dt

d

Ac2

1

)( fSQN )( fS

dN

(2.145) )()(2

2

fSA

ffS

Qd N

c

N

70

Assume that nQ(t) has ideal low-pass characteristic

with bandwidth BT

(2.146) 2

, )(2

2

0 T

c

N

Bf

A

fNfS

d

(2.147) , )(

output receiver At the

2

If

2

2

0

0Wf

A

fNfS

WB

c

N

T

71

effect quieting noise 1

(2.148) 3

2

)( ofpower Average

2

2

3

0

2

2

0

0

c

c

W

Wc

A

A

WN

dffA

Ntn

(2.149) 2

3)SNR(

3

0

22

FM,WN

PkA fc

O

,FM

The average power of is ,

the average noise power in message bandwidth is

SNR (2.150)

2

0

2

0

2

,

( )2

( )2

(2.29) ( ) ( )

c

cC

f m o FM

As t

WN

A

WN

f k A SNR f

(2.151) 3

)SNR(

)SNR(2

2

FMW

Pk f

C

O

72

when increasing carrier power

)2sin(2cos)( t f

f

ft fAts m

m

cc

2.4) Example (from 3

1

)SNR(

)NR( , AM tocompare

C

O

AM

S

)2cos()( sideboth

)2sin()(2 , may write We0

t fk

ftm

dt

d

t ff

fdm k

m

f

m

m

t

f

2

2

2

)( is load) 1 (across )( ofpower average The

fk

fPtm

,FM

0

3 3From (2.149), SNR

2 2 2 2

3

0

( )( ) ,

4 4c c

OA f A f

N W N W W

(2.152) 2

3)(

2

3

)SNR(

)SNR( 22

FM

W

f

C

O

FM. widebandand FM narrowbandbetween n transitio theas 5.0 Define

471.03

2

e.performancbetter has FM , 3

1

2

3When 2

Example 2.5 Single-Tone Modulation

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)

( ) tan

Occasionally,

1

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

( )

( )

cc I c Q

Q

c I

x t A n t f t n t f t

n tt

A n t

'

may sweep around the origin , ( >

(t) increases or decreases 2

The discriminator output is equal to

1 ( ) )

( )

2

cP r t A

t

nQ(t)

r(t)

x(t)

Ac

P1

0 P2

nI(t) 74

Figure 2.44 Illustrating impulselike components in (t)

d (t)/dt produced by changes of 2 in (t); (a) and (b)

are graphs of (t) and (t), respectively.

75

A positive-going click occurs , when

, , 0

A negative-going click occurs when

, , 0

The carrier-to -noise ratio is defin

( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

c

c

d tr t A t t d t

dt

d tr t A t t d t

dt

ed by

(2.154)

The output signal-to-noise ratio is calculated as

1. The average output signal power is calculated assuming

a sinusoidal modulation which produces . (noise free)

2.

02

2

c

T

T

A

B N

Bf

The average output noise power is calculated when no

signal is present (The carrier is unmodulated). 76

2

avoided bemay effects threshold

(2.155),202

or 202

When 0

2

0

2

NBA

NB

AT

c

T

c

Figure 2.45 Dependence of

output signal-to-noise ratio on

input carrier-to-noise ratio for

FM receiver. In curve I, the

average output noise power is

calculated assuming an

unmodulated carrier. In curve

II, the average output noise

power is calculated assuming

a sinusoidally modulated

carrier. Both curves I and II

are calculated from theory.

77

The procedure to calculate minimum

1. Given and W, determine

(using Figure 2.26 or Carson's rule)

2. Given , we have 20

Capture Effect:

The receiv

2

0 0

( 20)

2

c

T

cT

A

B

AN B N

er locks onto the stronger signal

and suppresses the weaker one.

78

FM Threshold Reduction (tracking filter)

• FM demodulator with negative feedback (FMFB)

• Phase locked loop

Figure 2.46

FM threshold extension.

Figure 2.47

FM demodulator with

negative feedback.

79

Pre-emphasis and De-emphasis on FM

Figure 2.48 (a) Power spectral density of noise at FM receiver output.

(b) Power spectral density of a typical message signal.

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

(2.162) )(3

2

is factor t improvemen The

(2.158) )(emphsis-de power with

noiseoutput Average

(2.157) 2

, )()()(

(2.146) 2

, )(

isoutput tor discrimina at the PSD The

(2.156) , )(

1)(

w

w-

22

3

2

de

2

2

0

2

de2

202

de

2

2

0

pe

de

dffHf

WI

I

dffHfA

N

BffH

A

fNfSfH

Bf

A

fNfS

WfWfH

fH

de

W

Wc

T

cN

T

c

N

d

d

81

(2.161)

)(tan)( 3

)(

)(13

2

1

1)(

is responsefilter emphsis-deA

1)(

is responsefilter emphsis-pre simpleA

0

1

0

3

0

2

0

2

3

0

de

0

pe

fW

fW

fW

ff

dff

WI

fj f

fH

f

j ffH

W

W

Example 2.6

Figure 2.50 (a) Pre-emphasis filter.

(b) De-emphasis filter.

82

The main difference between FM and PM is in the

relationship between frequency and phase.

f = (1/2).d/dt.

A PM detector has a flat noise power (and voltage) output

versus frequency (power spectral density). This is

illustrated in Figure 9-38a.

However, an FM detector has a parabolic noise power

spectrum, as shown in Figure 9-38b. The output noise

voltage increases linearly with frequency.

If 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.

Preemphasis for FM

83

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

Preemphasis for FM

84

A preemphasis network at the modulator input

provides a constant increase of modulation index mf

for high-frequency audio signals.

Such a network and its frequency response are

illustrated in Figure 9-39.

Preemphasis for FM

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

85

With the RC network chosen to give = R1C = 75s in North America (150s in Europe), a constant input audio signal will result 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.

At the receiver demodulator output, a low-pass RC network

with = RC = 75s 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-

kHz audio baseband and a noise performance improvement of about 12dB over no preemphasis. Phase modulation systems do not require emphasis.

Preemphasis for FM

86

Preemphasis and deemphasis: (a) schematic diagrams; (b) attenuation curves

Pre-emphasis and De-emphasis on FM

87

Example of S/N without preemphasis and deemphasis.

Pre-emphasis and De-emphasis on FM

88

Example of S/N with preemphasis and deemphasis.

Pre-emphasis and De-emphasis on FM

89

Dolby dynamic preemphasis

90

Figure 2.55 Comparison of the noise performance of various CW modulation

systems. Curve I: Full AM, = 1. Curve II: DSB-SC, SSB. Curve III: FM, = 2.

Curve IV: FM, = 5. (Curves III and IV include 13-dB pre-emphasis, de-

emphasis improvement.) 91

In making the comparison, it is informative to keep in

mind the transmission bandwidth requirement of the

modulation systems in question. Therefore, we define

normalized transmission bandwidth as

W

BB T

n

Table 2.4 Values of Bn for various CW modulation schemes

FM

AM, DSB-SC SSB

Bn

2 5

2 1 8 16

92

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