© 2010 The McGraw-Hill Companies Communication Systems, 5e Chapter 5: Angle CW Modulation A. Bruce Carlson Paul B. Crilly (modified by J. H. Cho using

© 2010 The McGraw-Hill Companies

Communication Systems, 5e

Chapter 5: Angle CW Modulation

A. Bruce CarlsonPaul B. Crilly

(modified by J. H. Cho using Prof. W.J. Song’s lecture note)


Chapter 5: Angle CW Modulatation

Phase and frequency modulation

Transmission bandwidth and distortion

Generation and detection of FM and PM

Interference


5.1 Phase and frequency modulation

Terms 5.1-1


• Total instantaneous angle• Angle modulation = exponential modulation

• Phase modulation (PM)• Phase modulation index = phase deviation

• Instantaneous frequency vs. spectral frequency• Frequency modulation (FM)• Frequency deviation

• Zero-crossing rate


Exponential Modulation ( 각변조 )

Phase modulation

).()(where),(cos)(

ttwttAtv

c

c

indexmodulationphase:

])(cos[)(

1)(for.shiftphasemax.:180)()(

txtwAtv

txtxt

ccph

Frequency Modulation (FM)

Phase Modulation (PM)

PM & FM Signals



Def ) 순간 주파수 (instantaneous frequency) 의 정의

Frequency modulation

)(21)(

21)( t

dtdft

dtdtf c

oo

t

t

cc

tttdxft

txftdtd

fftxfftf

o

)()(2)(

)()(21

)()(



PM signals

0Δ

CW signal:

( ) cos[ ( )]

PM: ( ) cos[ ( )]

where 180 , ( ) 1, and phase modulation index or phase deviation

c c c

c c c

x t A t t

x t A t x t

x t


CW signal:

( ) cos[ ( )]

FM:

( ) cos 2 ( )

where ( ) 1, and frequency deviation

c c c

t

c c c

x t A t t

x t A t f x d

x tf


FM and PM Signals

21Power: 2T cS A

Power is constant, and not a function of messagepower

0For PM ( ) 180t


FM and PM Message content resides in zero crossings not amplitude Modulated waveform does not resemble message waveform Amplitude is constant we can use more efficient

nonlinear amplifiers

Note: FM and PM, ( ) , or effective message power increases with larger

values of , or

x t f

f


Illustrative AM, FM, and PM waveforms

Terms 5.1-2


• Narrowband PM and FM (NBPM and NBFM)

• Single-tone modulation• Bessel function of the first kind of order n and argument beta

• Multitone modulation • Periodic modulation


Narrowband PM & FM

])(!3

1)([

)(sin

])(!2

11[

)(cos)(

sin)(sincos)(cos)](cos[)(

3

2

ttA

tAv

tA

tAtv

twtAtwtAttwAtv

c

cq

c

ci

cccc

cc


Narrowband Condition

따라서

.radian1)( t

)()()(

tAtvAtv

cq

ci

FM:)(PM:)(

)]([)(

0)(2

)(21)(

ffXjf

fXtFf

fffAjffAfV cccc

WBT 2Note this is true only when is small.)(t

Then


Narrowband PM and FM

2

3

Quadrature-carrier form:

( ) ( )cos ( )sin1 where: ( ) cos ( ) 1 ( ) ...2

1 ( ) sin ( ) ( ) ...3!

with the narrowband case (

c ci c cq c

ci c c

cq c c

x t x t t x t t

x t A t A t

x t A t A t

t) 1 rad ( ) ( ) (t)ci c cq cx t A x t A


The NBFM/NBPM spectra becomes:

1 ( ) ( ) ( )2 2

where ( ) ( )

c c c c cjX f A f f A f f

f t

The spectrum of narrowband FM and PM looks like that of AM!



Tone Modulation

Let sin PM ( ) cos FM

with ( ) cos[ ( )]

( ) sin 2

PM/ FM

m m

m m

c c c

m

m

m m

A tx t A t

x t A t t

t f t

AA f f


Consider FM

( ) cos[ ( )]

( ) cos 2 cos2

( ) 2 cos2 sin 2

with /

c c ct

cFM c c m m

t

m m m

m m

x t A t t

x t A t f A f d

t f A f d f t

A f f


Narrowband modulation

1

( ) cos sin sin

= cos cos( ) cos( )2 2

looks like a tone modulated AM signal

c c c c m c

c cc c c m c m

x t A t A t t

A AA t t t


NBFM with tone modulation (a) Line spectrum; (b) Phasor diagram


Let

)FM(cos)PM(sin

twAtwA

mm

mm

FM

PMsin)(

ffA

Atwt

m

m

mm

Then

]sin)sinsin(cos)sin[cos(]sin)(sincos)([cos

)](cos[)(

twtwtwtwAtwttwtA

twAtv

cmcmc

ccc

cc

일반적으로 임의의 에 대한 FM 의 대역폭을 구하는 것은 불가능하다 .)(tx최고 주파수 mf

)(tx


FM/PM spectra with an arbitrary index value

m

In quadrature-carrier form,

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

with (t)= sin

( ) [cos cos cos sin sin sin ]

c c c c

c c m c m c

x t A t t t t

t

x t A t t t t


Note)

Trigono. Fourier Series

deJ

n

tnwJtw

tnwJJtw

njn

moddn

nm

mevenn

nom

)sin(

21)(

0

sin)(2)sinsin(

cos)(2)()sincos(

Bessel Function of the First Kind of order and argumentn .




)()1()(})cos{()(

])cos())[cos((

])cos())[cos((

cos)()(

nn

nn

mcnc

mcevenn

mcnc

mcoddn

mcnc

coc

JJtnwwJA

tnwwtnwwJA

tnwwtnwwJA

twJAtv

)(OJ

)(3 J

)(2 J

)(1 J

)(1 J)(2 J )(3 J

cf

mf


Note) determines the magnitude of the Fourier Coeff.’s frequency spacing

)()1()( nn

n JJ

(decay rate of sideband harmonics)

mf


Infinite Sideband !

따라서

Up to N sidebands

)(2)]([2

2

1

222

2

nno

cT

cT

JJAP

AP

1)(2)(1

22

n

no JJ

)](2)([2 1

222

N

nno

cN JJAP


99% POWER

mT

mmT

T

N

wB

wNwB

NPP

2ThenNBFM,i.e.,1If

RulesCarson')1(22

199.0

Note)


0 odd

even

Given the message is a single tone and periodic, we can write( ) as a trigonometric Fourier series

cos cos ( ) ( )cos

sin sin ( )sin

Where

c

m n mn

m n mn

x t

t J J n t

t J n t

J

( ) is a Bessel function of the first kind, order

Taking advantage of ( ) ( 1) ( )

( ) ( )cos( )

n

nn n

c c n c mn

n

J J

x t A J n t


The FM/PM signal ( ) consists of a carrier with multiple sidebands that are spaced by an integer multiple of with the sideband having an amplitude of ( ).

c

m n

x t

J

Note: the lower sidebands alternate from + to -


Magnitude of tone-modulated line spectra (a) FM or PM with ƒm fixed; (b) FM with Amƒ fixed


Multitone Modulation

FM

twAtwAtx 2211 coscos)(

twfwAtwf

wAtw

dttxftwt

c

c

22

21

1

1 sinsin

)(2)(

][

}]{[

][)(

)sinsin

)sinsin(

)(

22

21

1

1

22

21

1

1

twfwA

jtwfwA

jtjw

ce

twfwA

twfwA

twj

ce

tjceFM

eeeAR

eAR

eARtv

c

c

© 2010 The McGraw-Hill Companies No superposition

])cos[()()(

])()([

}])(}{)({[

2121

)(21

21

21

21

tkwnwwJJA

eJJAR

eJeJeAR

ckn k

nc

tkwnwwjk

n knce

tjkw

kk

tjnw

nn

tjwce

c

c

21

2

1

kwnwwkwwnww

c

c

c

Intermodulation component !

because FM is a nonlinearmodulation method.






Interference


5.2 Transmission bandwidth and distortion

Bandwidth of Narrowband FM Approximate spectrum of narrowband

FM Message bandwidth vs. transmission

bandwidth


Bandwidth of Tone Modulated FM Spectrum of Single-tone modulated

FM M significant sideband pairs B=2Mf_m

M(beta) vs. beta +2 Beta = A_m f_delta/f_m=< f_delta/f_m

Deviation ratio = f_delta/W



Transmission BandwidthSeveral equations to determine , all are approximate

Let deviation ratio

1. 2 ( )

T

T

BfDW

B M D W


2 2 12. 2 1

3. Carson's rule: 2( ) 2( 1) 1, 1

3. 2( 2 ) 2( 2) 2 10

TDW f DB W D

f W D W D D

B f W D W D

• In all approximations, the transmission bandwidth is proportional to twice the transmission bandwidth.

• The proportionality constants are different.


Commercial FM radio bandwidth example

Commercial FM radio75 15 kHz, 15 kHz 55

using #3 above 2( 2) 2(5 2)15 210 kHz

using Carson's 2(

T

T

f W D

B D W

B D

1) 2(5 1)15 180 kHz

Assigned frequencies to the FM band are set up for 200 kHzT

W

B


Station engineer has set constants so BT conforms to the FCC limits dictated by their license

5.2.2 Linear and Nonlinear Distortion


Wireless channel as an LTI system FM-to-AM conversion

Controlled nonlinear distortion and filtering to remove unwanted amplitude variation

Memoryless nonlinear system


Limiter


Nonlinear processing circuits (a) Amplitude limiter; (b) frequency multiplier


Amplitude Limiter and Noise Reduction

FM signal processing using a limiter: Noiseless FM signal, (b) noisy FM signal, (c) limiter output with noisy input, (d) BPF output


Output of Memoryless Nonlinear System to FM Input

V_in(t) = A_c cos{omega_ct+phi(t)} Weierstrass Approximation Theorem For every epsilon, there exists a

polynomial such that… V_out(t) = …


Limiter for frequency multiplier

Limiter or some other nonlinear device generates harmonics

BPF selects which integer multiple of

Nonlinear device also changes frequency/phase deviation constants

'c cf nf

cf

' ' Limiter-BPF output and c cf nf f nf






Interference


5.3 Generation and detection of FM and PM


Generation of FM and PM signals

Pros. Constant envelope more power efficient

nonlinear methods can be used longer battery life

Cons. Required to have frequency vary linearly with

the message amplitude. Not straightforward.


5.3.1 Direct FM and Voltage Controlled Oscillator (VCO): Use a VCO!

0

0 1

Oscillator output frequency =

Oscillator tank circuit with resonant frequency of , , and ( )

v

f f

f f f L C C t


00

0

( ) capacitance of the varactor diode which is affected by the applied voltage ( ) ( )

1Nominal value of 2

but since the total capacitance in the tuned circuit is + ( )

v

v

v

ouput

C tC t f x t

fLC

C C C t

f f

0 0 [ ( )]f f f x t

Important: frequency change must be linear with x(t) sets a limit on maximum frequency deviation


How can we increase and still have change linearlywith ( )?

1. Different hardware, increase power supply, etc.

2. Frequency multiplier and indirect FM: recall: nonlinear frequency

f fx t

multiplier also multiplies f

Tripler0 , f f 03 , 3f f


Important: Frequency multiplication is not the sameas hetrodyning

Hetrodyning is a linear process and does not affectthe frequency or phase deviation constants


5.3.2 Phase modulators and indirect FM: Generate a NBFM, use frequency multiplier, and down-convert!

Convert a PM signal to an FM one by integrating The message signal


Phase modulators and Indirect FM

2 1 1

2

( ) ( ) PM FM

( ) ( ) ( ) ( )

multiplier also multiples frequency deviation2

Output frequency: ( ) ( )

t

c

LO

x t x d

f t nf t nf t f x t

f nT

f t f t f


5.3.4 Frequency detection

Produces output voltage that is proportional to the instantaneous frequency of the input the messagex(t).


Frequency detector = Discriminator

1. FM-to-AM conversion

2. Phase-shift discrimination

3. Zero-crossing detection

4. Frequency feedback→ phase locked loops (Chap 7)


5.3.4.1 FM to AM conversion

FM signal: ( ) cos 2 2 ( )

Taking the derivative with respect to time:

( )

( ) 2 [ ( )]sin 2 2 ( )

2 [ ( )]sin 2 ( )

( ) looks like an A

t

c c c

c

t

c c c c

c c c

c

x t A f t f x d

dx tdt

x t A f f x t f t f x d

A f f x t f t t

x t

M signal

we can recover ( ) with an envelope detectorx t


FM to AM conversion Take derivative of FM signal

Use an envelope detector


FM Detection Waveforms

(a)frequency detector with limiter and FM to AM conversion(b) waveforms


FM to AM methods - Derivative

Slope detector via a BPF

Balanced discriminator


Derivative function( ) ( )Recall ( 2 ) ( ) ( )

we can use the edge of a BPF or LPF as a differentiator

dv t dv tj f V f jKV fdt dt

Allows an AM receiver with a BPF to detect an FM signal


Balanced discriminator To get the maximum response from the BPF we combine two BPF-envelope detectors to get a balanced discriminator

(b) circuit, (c) voltage to frequency characteristic


5.3.4.2 Phase shift discriminator

1

11

1 1 1

( ) ( )( )

Assume that is small compared to variation in ( )1( ) ( ) ( )

FM wave ( ) ( ) ( ) 2 ( )

dv t v tv tdt t

t v t

v t v t v t tt

t t t t t f t x t


Phase-shift discriminator

1

1 1

1 1 1

Multiplier output: cos ( ) sin ( )

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

LPF output1 ( ) sin ( ) ( ) ( ) ( ) ( )2

c c

c

D

t t t t t

t t t t t t t

y t t t t t t t t t

( ) ( )D Dy t K f x t






Interference


5.4 Interference Occurs when another signal is received concurrently in

the receiver’s bandpass Multipath: multiple versions of the transmitted signal

with different delays can cause interference Effects can be affected by the types of modulation and

detectors used. Interference: generally not random Sometimes can be canceled out Is not the same as random noise


Interfering sinusoidsLet ( ) be a received signal consisting of the desired component at

and some interfering sinusoid at frequency with

c c i

i

v tf f f f f

f W

( ) cos2 cos[2 ( ) ]

Let , , and be the interferer's amplitude, carrier frequency, and phase respectively.

Let and ( ) 2

c c i c i i

i c i

ii i i

c

v t A f t A f f t

A f f

A t f tA


Interfering sinusoid in envelope-phase form

2

1

Putting ( ) in envelope phase form, we have:

( )cos 2 ( )

( ) 1 2 cos ( )

sin ( ) ( ) tan1 cos ( )

v c v

v c i

iv

i

v t

A t f t t

A t A t

ttt


The interfering sinusoid produces both amplitude and phasemodulation

If interference is relatively small 1

( ) 1 cos(2 ) looks like an AM signal with single tone message

( ) sin(2 ) looks like an FM or PM signal wit

v c i i

v i i

A t A f t

t f t

h single tone message

This is why nearby AM/FM signals with unsuppressed carriers generate a disproportionate amount of obnoxious background “whistles.”


1

If interference is relatively small 1

( ) 1 cos(2 ) looks like an AM signal with single tone message

( ) 2 phase corresponds to a shifted c

v i i i

v i i

A t A f t

t f t

arrier freq. plus constant


Demodulated output with interference

Assuming 1, and

(1 cos2 ) AM with envelope detector( ) sin 2 PM

sin 2 FM

i

D i

D D i

D i i

f W

K f ty t K f t

K f f t

Observe how interference level of FM depends on spacingof interference carrier frequency


Interference level as function of interference frequency spacing

Note how with FM interference is reduced if the interferencefrequency spacing is reduced.


Deemphasis and Preemphasis Filtering

We exploit the property of FM that causes the interference level to be reduced as fi ↓ by deemphasis filtering of the high frequencies at detection we preemphasize the highfrequencies at the transmitter.

at the transmitter end we predistort the signal

1( )( )pe

de

H fH f






Interference

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© 2010 The McGraw-Hill Companies Communication Systems, 5e Chapter 5: Angle CW Modulation A. Bruce Carlson Paul B. Crilly (modified by J. H. Cho using