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




 CNU   Dept. of Electronics               2           D. J. Kim
Lecture on Communication Theory

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



 CNU   Dept. of Electronics                  3                         D. J. Kim
Lecture on Communication Theory




                         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


 CNU   Dept. of Electronics                4       D. J. Kim
 Lecture on Communication Theory

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




  CNU   Dept. of Electronics        5                     D. J. Kim
Lecture on Communication Theory




   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



 CNU   Dept. of Electronics                    6                         D. J. Kim
 Lecture on Communication Theory


3. Envelope Detector : AM radio receiver




    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




  CNU   Dept. of Electronics       7              D. J. Kim
Lecture on Communication Theory




 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




  CNU   Dept. of Electronics          9                  D. J. Kim
 Lecture on Communication Theory

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




  CNU   Dept. of Electronics                   10   D. J. Kim
 Lecture on Communication Theory

2. Ring Modulator




                                   +        -




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




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




  CNU   Dept. of Electronics           11             D. J. Kim
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

 CNU   Dept. of Electronics            12                           D. J. Kim
 Lecture on Communication Theory

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


  CNU    Dept. of Electronics                    13                            D. J. Kim
 Lecture on Communication Theory

 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


4. Costas Receiver




           m(t) : real
           PhaseDiscrimina 구현 근거:   0
                         tor                              I  m( t )
                                                          Q0
  CNU   Dept. of Electronics                    14                     D. J. Kim
Lecture on Communication Theory

                              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


 CNU   Dept. of Electronics              15                       D. J. Kim
 Lecture on Communication Theory

5. Quadrature-Carrier Multiplexing or QAM
        s( t )  Ac m1 ( t ) cos( 2f c t )  Ac m2 ( t ) sin( 2f c t )
                           In-phase                        Quadrature




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

 Key points> Correct phase & frequency
     Costas Receiver 사용
     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
 Lecture on Communication Theory

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
 Lecture on Communication Theory


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



  CNU   Dept. of Electronics            24          D. J. Kim
Lecture on Communication Theory

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
Lecture on Communication Theory


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




 CNU    Dept. of Electronics                 28                    D. J. Kim
 Lecture on Communication Theory


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


  CNU   Dept. of Electronics          29                  D. J. Kim
Lecture on Communication Theory




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

 CNU   Dept. of Electronics       30   D. J. Kim
 Lecture on Communication Theory


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      
  CNU   Dept. of Electronics                     31                          D. J. Kim
Lecture on Communication Theory

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




 CNU   Dept. of Electronics       32         D. J. Kim
 Lecture on Communication Theory

 4) PM과 FM의 관계 : 미적분의 관계




4.11 Frequency Modulation

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



  CNU   Dept. of Electronics       33         D. J. Kim
Lecture on Communication Theory

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
                    
 CNU    Dept. of Electronics                         34           D. J. Kim
 Lecture on Communication Theory

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




  CNU   Dept. of Electronics                        35                                 D. J. Kim
 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
                f1=0.1MHz, 1=0.2 radians.
             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
    b) Compatible with monophonic radio receivers
 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
                              BW  K Radian
                                     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


3.14. The Superheterodyne Receiver
 1. Tasks of receiver
   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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