# Chapter 3. Continuous-wave modulation by 7BvvE875

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```									Lecture on Communication Theory

Chapter 3. Continuous-wave modulation

3.1 Introduction
m(t)
                m(t)cos(wct)
baseband                               passband
cos(wct)
modulation: the process by which some characteristic of a
carrier is varied in accordance with a modulating
wave(signal)

3.2 Amplitude Modulation

1. Am
1) Sinusoidal carrier wave
c(t)=Ac cos(2fct)
carrier frequency
carrier amplitude

2) AM signal
s(t) =Ac [1+ka m(t)] cos(2fct)
message signal
amplitude sensitivity

CNU   Dept. of Electronics              1                        D. J. Kim
Lecture on Communication Theory

3) s(t) 의 envelop 이 m(t)와 똑같은 shape이 될 조건
a) | Kam(t) | < 1          for all t
b) fc >> W                 where W is message BW

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4) 주파수상에서의 표현
A                                    k A
S(f)  c [  ( f  f c )   ( f  f c )]  a c [M( f  f c )  M( f  f c )]
2                                     2

BT=2W

ex1) Single-Tone Modulator
message                m(t)=Am cos(2f m t )
AM                     s(t)=Ac [1+cos(2f m t)]cos(2fc t)
Where         = kaAm ; Modulation factor
100 = 100 kaAm ; percentage Modulation

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1 2
Carrier power      Ac
2
1 2 2
Upper - side - frequency power     Ac
8
1
Lower - side - frequency power   2 A c2

8

total sideband power

total power
2      1
         
2  2
3
if   1  100% modulation

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2. Switching Modulation

m(t)

v1(t) = Accos(2(t)) + m(t))

If  m(t)   Ac
v2(t)        v1(t), c(t) > 0
0,     c(t) <0

BPF[v2(t)]

 v2(t)  [AC cos(2(t)) + m(t))] gTo(t)
where T0=1/fc

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Diode function

on   off

By Fourier series

1 2   1
n 1

gTo (t)                cos2f c t 2 n  1
2  n 1 2 n  1

 1 2   1n 1                        
 v 2 ( t )   Ac cos 2f c t   m( t )                cos2f c t 2 n  1
 2  n 1 2 n  1                       
A          4                          1
 c 1            m( t ) cos( 2f c t )  m( t )
2       Ac                          2
                 
 ( )cos(4 f c t )  ( )cos(6 f c t )

             
 BPFv 2 (t) f  f 
Ac        4
1     m( t ) cos( 2f c t )
c
2       Ac       
AM signal

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Charging time constant = (f + RS ) C
For Rapid charge (f + RS )C << 1/fc
Discharging time constant =Rl C
1             1
 Rl C 
fc            W where W  message BW

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CNU   Dept. of Electronics       8   D. J. Kim
Lecture on Communication Theory

3.3 Virtues, Limitations , and Modifications
of AM
1. Virtues
1) easy    modulator: switching mod, square-law modulator
demodulator: envelop detector, square-law detector
2) relatively cheap

2. Limitations
1) Wasteful of power           carrier power
2) Wasteful of BW              1/2로 줄일 수 있다.
LSB와 USB가 symmetry.

3. Modifications of AM
1) DSB-SC modulation : no carrier
2) VSB modulation : BW를 약 1/2로
3) SSB modulation : BW를 1/2로

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3.4 DSB - SC Modulation

1. DSB - SC signal

s( t )  c( t )m( t )
 Ac cos( 2f c t )m( t )

S( f )  Ac M ( f  f c )  M ( f  f c )
1
2

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2. Ring Modulator

+        -

c(t) > 0                         s(t)=m(t)

c(t) < 0                         s(t)= -m(t)

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Lecture on Communication Theory

M(f)

w

-3fc   -fc          fc           3fc

S(f)

-3fc                                 2W
2W

4 ( 1 )n1
cos[ 2f ct 2n  1 ]

c(t) = 
 n1 2n  1

BPF[s(t)] =BPF[c(t)m(t)] f=fc
= m(t) 4 cos2f t 
 c

(주의점) Transformers are perfectly balanced and diodes are identical
 no leakage of modulation frequency into modulator output

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3. Coherent detection or synchronous demodulation
v( t )
s( t )                                    LPF                     v0 (t )

c( t )  Ac ' cos( 2f c ' t   )
Local Osc

v( t )  Ac ' cos( 2f c ' t   ) Ac cos( 2f c t   )m( t )

= Ac Ac ' cos2 ( f c  f c ' )t   m( t )
1
2
+ 2 Ac Ac ' cos2 ( f c  f c ' )t   m( t )
1

1
v0 ( t )  LPF [ v( t )]  Ac Ac ' cos[ 2 ( f c  f c ' )t   ] m( t )
2

frequency coherent detection

f c  f c'
1
v0 ( t )  Ac Ac ' cos[  ] m( t )
2
1
if   0 v0 ( t )  Ac Ac ' m( t )
2
1         1
if   45 0 v0 ( t )  Ac Ac '     m( t )
2          2
if   90 0 v0 ( t )  0 ; quadrature null

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frequency & phase coherent : f c  f c ' ,   0
1
v0 ( t )      Ac Ac ' m( t )
2
V(f)

-2fc                 2fc      f
2w
2w
Coherent Detection 특징 : perfect demodulation
but 복잡       cost

m(t) : real
PhaseDiscrimina 구현 근거:   0
tor                              I  m( t )
Q0
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sin 



cos φ  1 Q channel sin  m( t ) sin 
for small  ,            ,                           
sinφ  φ I channel cos  m( t )    1

s(t)

OSC

빠른 주파수
450

- 450

늦은 주파수          정상주파수

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s( t )  Ac m1 ( t ) cos( 2f c t )  Ac m2 ( t ) sin( 2f c t )

CNU   Dept. of Electronics                       16                      D. J. Kim
Lecture on Communication Theory

Key points> Correct phase & frequency
 Send a pilot signal outside the passband of the modulated signal
Pilot : Low power sinusoidal tone whose frequency and phase are
related to c(t)
Add pilot signal of small carrier

r( t )  s( t )  cos( 2f c t )
 m1 ( t ) cos( 2f c t ) cos( 2f c t )  m2 ( t ) sin( 2f c t ) cos( 2f c t )
1
m1 ( t )  m( t ) cos( 4f c t )  1 m2 ( t ) sin( 4f c t )
2                                      2
 LPFr1 ( t )  m1 ( t )
1
2
LPFr2 ( t )  m2 ( t )
1
2

3.5 Filtering of side-bands

<Band pass filtering>

CNU   Dept. of Electronics                           17                                    D. J. Kim
Lecture on Communication Theory

1.BPF의 LSB와 USB가 symmetric 할 경우
H(f)

f
-fc                            fc
0

m(t)                 LPF                                 s(t)

Ac cos( 2f ct )

2. BPF의 LSB와 USB가 unsymmetric 할 경우

-fc                        fc

FILTER
Hi(T)

m(t)

FILTER
HQ(T)

CNU   Dept. of Electronics               18                             D. J. Kim
Lecture on Communication Theory

H(f)와 HI(f) HQ(f) 간의 관계는?

s( t )  sI ( t ) cos( 2f ct )  sQ ( t ) sin( 2f ct )
BPF식

S( f ) 
Ac
M ( f  f c )  M ( f  f c )H ( f )
2

LPF식

S( f ) 
1
2
                               
S I ( f  f c )  S I ( f  f c )  1 S Q ( f  f c )  S Q ( f  f c )
2j

S( f  f c ) 
1
2                                   2j

S I ( f  2 f c )  S I ( f )  1 S Q ( f  2 f c )  S Q ( f )         
S ( f  f c )  S I ( f )  S I ( f  2 f c ) 
1
2
1
2j

SQ ( f )  SQ ( f  2 f c )          
 S I ( f )  S ( f  f c )  S ( f  f c ),                             f W

M ( f )H ( f  f c )  H ( f  f c ),
Ac
                                                            f W
2
H ( f )  H ( f  f c )  H ( f  f c )
I
또한
S Q ( f )  j [ S ( f  f c )  S ( f  f c )],                       f W

M ( f )H ( f  f c )  H ( f  f c ),
Ac
 j                                                         f W
2
 H ( f )  j H ( f  f c )  H ( f  f c )
Q

CNU    Dept. of Electronics                       19                                 D. J. Kim
Lecture on Communication Theory

HI(f) : symmetric component

                         

-fc                                             fc

+jHQ(f)


-fc                                             fc

jHQ ( f )  H ( f  f c )  H ( f  f c ) ; anti - symmetric
component

SSB & VSB
HI(f)

jHQ(f)
fc

CNU   Dept. of Electronics                   20                              D. J. Kim
Lecture on Communication Theory

3.6 Vestigial Side-Band Modulation

1. Filtering
HI

1
HQ
0.5
f
fc-fv
fc fc+fv fc+W

HI

1

0.5
f    HQ
fc-W      fc-fv f fc+fv
c

1) fc를 중심으로 odd-symmetry
 LSB(or USB)의 vestige만 보냄.
2) USB or LSB

CNU       Dept. of Electronics                   21         D. J. Kim
Lecture on Communication Theory

2.Television Signals
1)TV 신호의 특징
a)Video 신호가 large BW
 대부분의 energy가 low-frequency에
b)수신기가 간단, cheap  use envelope detection
2)주파수 특성

3.58 MHz

color

CNU   Dept. of Electronics       22                  D. J. Kim
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3. 0.75 MHz (25%)의 LSB를 full scale로 보내는 이유
Envelope detection에서 waveform distortion을 줄이기 위해
due to cheap

1) Waveform distortion
 1                             1
s( t )  Ac 1  k a m( t ) cos( 2f c t )  k a Ac m' ( t ) sin( 2f c t )
 2                             2
add carrier component to apply envelope detector

Envelop detector output
1
 1                                  2
2                  2
 1              
a( t )  Ac 1  k a m( t )   k a m' ( t ) 
 2             2               
1
  1                        
2

2

 1              k a m' ( t )            

 Ac 1  k a m( t ) 1   2                    
 2               1  1 k m( t )          
  2 a
                           


2) m’(t)에 의한 distortion을 줄이는 방법
a) ka를 줄인다.
b) Vestigial sideband의 폭을 늘인다.

3) TV 신호에서의 해법
a) 100% percentage modulation               negligible
b) 0.75MHz의 LSB                             distortion

CNU   Dept. of Electronics                 23                                 D. J. Kim
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3.7 SSB

1. 방법 : USB 나 LSB만 전송
2. 이유 : Message m(t)가 실수  M(f)는 conjugate symmetric
3. 문제점 : LSB나 USB가 붙어 있을 경우 filtering이 어렵다.
4. 적용분야 : Message spectrum이 origin에서 energy gap이 있을 때
ex) Voice (-3400 ~ -300)         (300 ~ 3400Hz)

H( f )

f

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5. Pass-band에서 BPF로 구현

m(t)                                 BPF

cos( 2f c t )
USB

f
0        fc         fc+W

BPF : highly selective filters ( 예: Crystal resonator)

Multiple modulation으로 구현

-f1     0         f1
easy BPF

-f2                                               f2       f2+f1

CNU   Dept. of Electronics                    25                               D. J. Kim
Lecture on Communication Theory

6. Time Domain Description of SSB (Baseband에서 구현)

HI(f)

jH Q(f)                                        즉, HQ ( f )   j sgn( f )

1                            1
USB s( t )    Ac m( t ) cos( 2f c t )  Ac m( t ) sin( 2f c t )
ˆ
2                            2
1                            1
LSB s( t )  Ac m( t ) cos( 2f c t )  Ac m( t ) sin( 2f c t )
ˆ
2                            2

Hartley modulator


m(t)                                        cos( 2f c t )

OSC
              s(t)
-900

sin( 2f ct )
H.T
ˆ
m(t)


CNU   Dept. of Electronics                 26                                 D. J. Kim
Lecture on Communication Theory

7. Demodulation of SSB signal
1) 구현
                       LPF         m(t)
cos( 2f c t )

s(t)
-900

sin( 2f c t )
                       LPF         ˆ
m (t)

2) Coherent detector : both in phase & in frequency
a) Low-power 의 pilot carrier를 전송
b) Highly stable oscillator 사용 still phase error

3) Phase error 가 있을 경우

Ac Ac ' m( t ) cos   m( t ) sin  
1
v0 ( t )                          ˆ
4
1
4
               ˆ
V0 ( f )  Ac Ac ' M ( f ) cos   M ( f ) sin             
ˆ
여기서 M ( f )   j sgn( f )M ( f )
1                                                   
 4 Ac Ac ' M ( f ) exp(  j ),               f 0 
V0 ( f )                                                      
1
 Ac Ac ' M ( f ) exp( j ),                    f 0 
4                                                   

Phase distortion
Voice  insensitive to phase error  Donald Duck voice effect
Music or Video  unacceptable

CNU    Dept. of Electronics                    27                             D. J. Kim
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3.8 Frequency Translation

f3
f1                                                   f5

f2                 f4

f3 = f 2 + f 1
f5 = f4 + f3= f4  (f2  f1)
= f4(f2-f1)
f4(f2+f1)

Upward
Downward

ex) f1=      0M         f2=    44M  f3 = 44M
f4=    66M  f5 = 110M,    22M        TV

ex) f1=110M             f2= 1030M  f3 = 920M, 1140M
f4= 876M  f5 = 44M, 1796M            DTV
f6= 44M  f7 = 0M,       88M

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3.9 Frequency-Division Multiplexing

w1                            w1

w2                            w2

wn                            wn

w1        w2   w3          wn       f

ex2) Voice BW= 4kHz, SSB
Basic Groups=12 Voice, fc=60+4nkHz.  n=1~12
Super Groups= 5Basic Groups, fc=372+48nkHz. n=1~5
Master Group
Very Large Group

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<HW #3> 3.4, 3.6, 3.8, 3.16, 3.21

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3.10 Angle Modulation

1. 장점 : better discrimination against noise and
interference than AM
단점 : increased BW

2. Basic Definitions

Let  i ( t ) : angle, a function of m(t)
Angle modulated wave s( t )  Ac cos[  i ( t )]
1 d i ( t )
instantane ous frequency f i ( t ) 
2 dt

1) PM
 i ( t )  2f c t  k p m( t )
phase sensitivit  y

 phase - modulatedsignal,s( t )  Ac cos 2f c t  k p m( t )            
2) FM

f ( t )  f c  k f m( t )
i

y
frequency sensitivit
 i ( t )  2f c t  2k f 0t m( t )dt

 frequency - modulatedsignal,s( t )  Ac cos 2f c t  2k f 0t m( t )dt      
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3) AM과 다른점
a) AM은 zero crossing이 주기적, PM과 FM은 비주기적
b) AM 은 envelope이 변화, PM과 FM은 constant

carrier

m(t)

AM

PM

FM

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4) PM과 FM의 관계 : 미적분의 관계

4.11 Frequency Modulation

1. FM signal
1) 특징 : nonlinear modulation
analysis is more difficult than AM

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2) FM signal

considerm(t)  Am cos( 2f m t )
instantaneous frequency

f i ( t )  f c  k f Am cos( 2f m t )
 f c  f cos( 2f m t )
where f  k f Am ( frequency deviation)
m ax : f c  f
m in : f - f
        c

phase  i ( t )  2 0t f i ( t )dt
Δf
 2πf c t           sin ( 2πf m t)
fm
 2πf c t   sin ( 2πf m t)
Δf
where               ; modulationindex
fm
; maximumdeparture of the angle i ( t ) from the angle 2πf c t
max : 2f c t  
 min: 2f t  
         c

FM signal
s( t )  Ac cos2f c t   sin( 2f m t )
 narrow - band FM :   1 radian
 wide - band FM :   1 radian

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2. Narrow-Band Frequency Modulation
1) s( t )  Ac cos2f c t   sin( 2f m t )
 Ac cos 2f c t  cos sin( 2f m t )  Ac sin2f c t  sin sin( 2f m t )

180
Narrow - Band :   1 radians. (  57 0                  )
π
 cos sin( 2f m t )  1
sin sin( 2f m t )   sin( 2f m t )
 s( t )  Ac cos 2f c t   Ac sin2f c t  sin( 2f m t )                             (1)
 Ac cos 2f c t  
1
2
                           
Ac cos 2 f  f m t  cos 2 f  f m t               (2)

AM case

s AM ( t )  Ac cos 2f c t  
1
2
                             
Ac cos 2 f c  f m t  cos 2 f  f m t          
(3)
2) < BW of Narrow band FM >  < BW of AM > = 2fm

3) 식(1)의 구현

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Lecture on Communication Theory

4) (2) (3)식의 그림상에서 비교
문제: envelope이 변한다
  0.3 radians  negligible

3. Wide-band FM
1) FM wave
s (t )  ReAc exp  j 2f c t  j sin 2f m t 
 Re~ (t ) exp  j 2f t 
s                c

where ~ (t )  Ac exp  j sin 2f m t 
s
~ (t ) : periodic function with f
s                                    m

CNU   Dept. of Electronics                36                   D. J. Kim
Lecture on Communication Theory

Expand ~( t ) by complex Fourier series
s

~( t ) 
s           c       n   exp j 2nf m t 
n  
1
~( t ) exp  j 2nf t dt
where cn  f                 m 

2 fm
1
2 fm
s                   m

exp j sin 2f m t   j 2nf m t dt
1
 fm A        c
2 fm
1
2 fm

let x  2f t         m

Ac 
cn    exp j  sin x  nx dx
2 
 nth order Bessel function of the first kind 
                  1                             
      Jn (  )       exp j  sin x  nx dx 
                 2                            
 cn  Ac Jn (  )

 ~ (t)  A c  Jn (  )exp  j 2nf m t 
s
n- 

FM wave                             J (  ) exp j 2  f  nf t 
s( t )  Ac Re  n
 n  

c    m



 Ac       J (  ) cos2  f
n  
n                       c    nf m t 

F .T

 J n (  )  f  f c  nf m     f  f c  nf m 
Ac 
S( f ) 
2 n  

CNU   Dept. of Electronics                                     37                              D. J. Kim
Lecture on Communication Theory

2) Properties of Bessel function.
a) For n even
J n (  )  J n (  )
For n odd
J n (  )   J n (  )
 J n (  )  ( 1 ) n J  n (  )         for all n

b) For n odd               J0 (  )  1
            
 J1(  ) 
            2
 Jn (  )  0 , n  2

 Jn (  )  1
2
c)
n  

CNU   Dept. of Electronics                    38                    D. J. Kim
Lecture on Communication Theory

3) Observations.
a) Spectrum, fcnfm, n=0,1,2,…...
b) for small , spectrum at fc, fm  narrow-band FM
c) Amplitude of carrier component J0() varies with 
1 2              1 2
P      Ac  J n (  )  Ac
2

2    n         2
example 3.
Fixed freq (fm) & varying amplitude (i, e, f)
Varing freq(fm) & fixed amplitude (i, e, f)

f
        : moduiation index
fm

CNU   Dept. of Electronics                 39               D. J. Kim
Lecture on Communication Theory

4. Transmission Bandwidth of FM signals.

1) BW of FM의 개념.
실제 FM: infinite number of side freq.
Effectively finite number of side freq.
Single tone FM case.
Narrow band : BW  order of 2fm
Wide band : BW  order of 2f

2) Carson’s rule
Approximate BW of FM by single tone fm

    1
BT  2 f  2 f m  2 f  1  

    

 2(   1 ) f m

def
3) BW of FM                  the separation between the two freq beyond
which none of the frequencies is greater than
1% of the unmodulated carrier amplitude
= 2nmax fm
where nmax=largest value of integer n that satisfies the requirement
J n (  )  0.01

           0.1 0.3 0.5 1.0 2.0 5.0 10.0 20.0 30.0
2nmax        2    4   4   6   8 16    28 50     70
Carson’s rule     2.2 2.6     3     4        6   12   22   42    62

CNU    Dept. of Electronics                40                         D. J. Kim
Lecture on Communication Theory

Universal curve

BT          B
BT  (      )f   ( T ) f m
f          f

4) General case
Highest frequency W  worst case tone fm
Deviation ratio D : maximum possible amplitude  
  Carson’s rule
                  사이에서 결정
universalcurve

Ex4) FM radio in US f=75KHz
W= 15KHz
 D= 75 / 15 = 5
By carson’s rule BT=2(75+15)=180KHz
200KHz 사용
By universal curve BT= 3.275=240KHz

CNU   Dept. of Electronics           41                          D. J. Kim
Lecture on Communication Theory

5. Genetation of FM signals
1) Indirect FM

a) Crystal controlled OSC : to provide frequency stability

b) Frequency multiplier

c) 식.

s( t )  Ac cos 2f c t  2k f 0 m( t )dt
t

v( t )  a1 s( t )  a2 s 2 ( t )  a3 s 3 ( t )  ......  an s n ( t )

s' ( t )  Ac ' cos 2nf c t  2nk f 0 m( t )dt
t

CNU   Dept. of Electronics                          42                              D. J. Kim
Lecture on Communication Theory

d) Freq. Multiplier 2개를 사용한 예

Ex5). (목적) fc=100MHz, minimum of f = 75kHz
m(t) : 100Hz~15KHz audio
100Hz f1=20Hz
15KHz f1=3KHz
To make minimum f=75KHz
f       75000
n1n 2                             3750      
min f 1 20 Hz           20
그리고 f c   f 2  n1 f 1 n 2
100  9.5  0.1n1 n 2                      
By solving & n1=75
n2=50

2) Direct FM

a) FM fi(t)=fc+kfm(t)
VCO로 구성(voltage controlled oscillator)
m(t)          VCO          fi(t)=fc+ kfm(t)

CNU   Dept. of Electronics               43          D. J. Kim
Lecture on Communication Theory

b) Oscillator의 구현 예

c(t) : (varactor or varicap) + fixed capacitance

ex) p-n junction diode in reverse bias
the larger the reverse voltage
 the smaller the capacitance

1
f i (t ) 
2   L1    L2 c(t )
c(t )  c0  c cos 2f m t 
1

    c               
cos 2f m t 
2
 f i (t )  f 0 1 
    c0               
                                                      c                
cos 2f m t  
 where f (t )                1
 f 0 1 
                    2 L1  L2 c0                                      
0
                                                      2c0                 
 f i (t )  f 0  f cos 2f m t 
       c      f 
 where
                
       2c 0    f0 


c) VCO를 이용한 wide-band FM

CNU   Dept. of Electronics                      44                             D. J. Kim
Lecture on Communication Theory

d) VCO를 이용한 FM에서 주파수 안정화를 위한 feedback scheme

가정     m(t) is zero mean
LPF는 f0 만 control 할 수 있도록 Narrow-band로 구현
(m(t)의 BW에 비해 Narrow하게)

6. Demodulation of FM signals
1) Direct Method frequency discriminator
= slope circuit + envelope detector

slope circuit
                B             B              B
 j 2a f  f c  T   , f c  T  f  f c  T
                 2             2              2
                B             B                 B
H 1 ( f )   j 2a f  f c  T   ,  f c  T  f   f c  T
                 2             2                2
            0         ,         otherwise



CNU   Dept. of Electronics                     45                           D. J. Kim
Lecture on Communication Theory

~ (t )
s1
s1(t)

s(t)                             s2(t)                                    so(t)
~ (t )
s2

< Balanced frequency discriminator >


s( t )  Ac cos 2f c t  2k f 0 m( t )dt
t

com plex envelope of s(t)
~( t )  A exp j 2k
s            c           f
t
0 m( t )dt   
           B                    BT       B
~          j 4a f  T                       f  T
H1( f )              2                    2        2

          0                       otherwis e

~            1 ~         ~
 S1 ( f )  H 1 ( f ) S ( f )
2
             B  ~                       BT      B
 j 2a f  T  S ( f ) ,                    f  T
               2                          2       2

             0            ,               otherwis e
 d~( t )
s                   
 ~1 ( t )  a 
s                       jBT ~( t )
s
 dt                   

 jBT aAc 1 
2k f       

m( t ) exp j 2k f 0 m( t )dt
t

     BT        
~( t ) exp( j 2f t )
 s 1 ( t )  Res                c

    2k f                                         
 BT aAc 1       m( t ) cos 2f c t  2k f 0 m( t )dt  
t

    BT                                           2

CNU   Dept. of Electronics                  46                                 D. J. Kim
Lecture on Communication Theory

2k f
If          m(t )  1   for all t, we may use envelope detector
BT

~ (t )  B aA 1  2 k f m(t )
s1         T  c               
    BT         

~            ~
H 2 ( f )  H 1(  f )
~ ( t )  B aA 1  2k f m( t )
c               
s2            T
     BT        
 s0 ( t )  ~1 ( t )  ~2 ( t )  4k f aAc m( t )
s          s

CNU   Dept. of Electronics                    47                        D. J. Kim
Lecture on Communication Theory

2) Circuit diagram으로 구현

Resonant freq. (fc )
Q factor                       for RLC circuit
3dB BW
Maximum energy stored in the circuit during one cycle
 2
energy dissipated per cycle

각각의 Resonator의 3-dB BW=2B 일 때 3B separation이 ideal.
위 회로의 distortion factor
a) s(t)의 spectrum이 BW=BT밖에서 완전히 0이 아니다.
b) Tuned filter가 완전히 band limit되어 있지 않다.
c) Tuned filter특성이 모든 FM 대역에서 완전히 linear하지 않다

CNU   Dept. of Electronics             48                        D. J. Kim
Lecture on Communication Theory

7. FM Stereo Multiplexing
1) FM stereo의 조건
a) The Tx has to operate within the allocated FM channels
2) Multiplexed signal
m(t)=[ml(t)+mr(t)]+[ml(t)-mr(t)]cos(4fct)+Kcos(2 fct)
where fc=19KHz
pilot=19KHz: 8~9% of the peak freq. deviation.
ml+mr or ml-mr : DSB-SC
3) 구조

CNU   Dept. of Electronics        49                          D. J. Kim
Lecture on Communication Theory

3.12 PLL

1.용도 : Synchronization, frequency division / multiplication
indirect frequency demodulation.

2. PLL의 구조

Locking 조건  주파수 동일

 Phase 는 90 차이
0

s( t )  Ac sin 2f c t  1 ( t )
where 1 ( t )  2k f 0 m( t )dt
t

r( t )  Av cos2f c t   2 ( t )
where  2 ( t )  2k v 0 v( t )dt
t

kf
if  1 ( t )   2 ( t )  v( t )  m( t )
kv

다른 응용 : Coherent detection용 clock generation

CNU   Dept. of Electronics                      50            D. J. Kim
Lecture on Communication Theory

3. Nonlinear Model of PLL
e( t )  k m Ac Av sin e ( t )
W here k m ;multiplier gain
 e ( t )  1 ( t )   2 ( t )
 1 ( t )  2k v 0 v( t )dt
t

v( t )  e( t )  h( t ) wher e h( t ) : loop filter
d ( t ) d1 ( t )
 2K 0 sin e (  )h( t   )d

 e          
dt           dt
where k 0  k m k v Ac Av : loop - gain parameter

여기서 sin( ) : nonlinear function  difficult to analyze

4. Linear Model of the PLL
Near phase-lock : 즉 e(t)<0.5 radians.
sin[e(t)]  e(t)

d e (t )                             d (t )
                 2K 0  e (t )  h(t )  1
dt                                    dt

CNU    Dept. of Electronics                            51             D. J. Kim
Lecture on Communication Theory

1
e( f )             1( f )
1  L( f )
H( f )
where L(f)  K 0        ; open - loop transfer function of PLL
jf

K0                   jf
output V ( f )            H ( f ) e ( f )  L( f ) e ( f )
kv                   kv
jf L( f )
                  1( f )
k v 1  L( f )
jf
If     L( f )  1          V( f )       1( f )
kv
1 d1 ( f ) k f
v( t )                 m( t )
2k v dt     kv

1
2kv

d
1(t)                                            v(t)
dt

BW of h(t)=BW of m(t)

CNU   Dept. of Electronics                            52                        D. J. Kim
Lecture on Communication Theory

5. PLL 의 BW와 Lock Range
1
e ( f )               1 ( f )
1  L( f )
H( f )
where L( f )  K 0
jf
s
 e (s)                      1 ( s)         K L  2K 0
s  K L H (s)
1
 2 ( s )   e ( s )2K L H ( s )
2s
 2 (s)   K L H (s)
             
 1 (s) s  K L H (s)

1) 1st-order                  H(s)=1                                        2 ( s )
20 log
1 ( s )

stable K L  0
 2 ( s)   KL               

 1 ( s) s  K L            Lock Range W0   K L                                               20dB / decade
       L

log K L           log(  )

2) 2nd-order H ( s)  s  K 1
s  K2
K1
Lock Range w  K L H (0)  K L
K2
 2 ( s)      K L s  K L K1
 2
 1 ( s) s  K L  K 2 s  K L K 1
stable if             K 2  K L , K L K 1  0

CNU   Dept. of Electronics                            53                                        D. J. Kim
Lecture on Communication Theory
(a) |z| < |p|
  (s) 
20 log  2 
  (s) 
 1 

log( w )
log K1   log K L K1

(b) |z| = |p|

log K L K1 log K1
log    K L K1

즉 BW            K L K1      과 Lock Range   K K 1  을 별도 조절 가능

     L

K2 
          

Computer Experiment II Acquisition Mode

a) acquisition
tracking

50                1
K0       Hz ,    fn     Hz
2               2
  0.3, 0.707 , 1.0
for a freq step of 0.125Hz
   0.707 이 best

CNU   Dept. of Electronics                      54                          D. J. Kim
Lecture on Communication Theory

b) Variations in the instantaneous frequency of the PLL’s VCO
for varying frequency step f.

7                2
( a )f  0.325Hz ( b )f  0.5 ( c )f         ( d )f 
12               3

(b), (c), (d)의 경우  Cycle slipping : Phase error of 2 radians
a slip by one cycle

(a)                                       (b)

(c)                                       (d)

CNU   Dept. of Electronics              55                             D. J. Kim
Lecture on Communication Theory

3.13 Nonlinear Effect in FM system

1. Nonlinearties
1) Strong nonlinearity : square-law modulators, limiters,
frequency multiplier
2) Weak nonlinearity : due to imperfections.

2. Weak Nonlinearity 의 경우
v (t )  a v (t )  a v (t )  a v (t )               2                       3

0                1     i                 2       i                   3   i

input v (t )  A cos2f t   (t )
i                   c                           c

where  (t )  2k  m(t )dt
t

f   0

                 
 v (t )  a A   a A  a A  cos2f t   (t )  a A cos4f t  2 (t )
1                 3           2          1                                        3                        2

                 
0                     2       c                   1       c               3       c                c   2   c           c
2                 4                      2

 a A cos6f t  3 (t )
1                                3

3       c                           c
4
주파수 조건 2f  2f  w   f  f  w                  c                                       c

 f  3f  2w
c

               3 
  a1 Ac  a 3 Ac  cos 2f c t   (t )
3
 v0 (t )            f  fc
BW  2 f  2 w                   4        

(결론) FM은 Channel로 전송 중 생기는 Amplitude Nonlinearity에 의한
영향이 없다.
 Microwave radio, satellite communication system에 사용.
이 채널에서는 highly nonlinear Amp와 power transmitter를 사용한다
왜냐하면 maximum power을 내는 것이 중요하기 때문.
(단점) Extremely sensitive to phase nonlinearities

CNU       Dept. of Electronics                                                                      56                   D. J. Kim
Lecture on Communication Theory

1) Carrier-frequency tuning
2) Filtering
3) Amplification

2. Superheterodyne : RF  IFDetection(Demodulation)

f  f  f or f  f                  LO
IF    RF    LO      LO   RF

     ch2                                     ch69      
                                                       
ex) TV ; RF   54   ~   60                            800    ~   806 
                                                       
    55.25                                   801.25     
IF  44MHz
Channel Tuning
 ch2                    ch69 
LO 
11.25M                       
                      757.25M 


IF
BPF1                             BPF2                LPF
50~860M                       LO        44M
44M
55.25 - 11.25 = 44
image frequency  37.25 + 11.25 = 44

<HW #4> 3.28, 3.30, 3.45

CNU   Dept. of Electronics                      57                   D. J. Kim

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