Quadrature Amplitude Modulation (QAM) Transmitter

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					EE345S Real-Time Digital Signal Processing Lab   Fall 2009


Quadrature Amplitude Modulation
      (QAM) Transmitter


               Prof. Brian L. Evans
  Dept. of Electrical and Computer Engineering
        The University of Texas at Austin

                         Lecture 15
                    Introduction
• Digital Pulse Amplitude Modulation (PAM)
  Modulates digital information onto amplitude of pulse
  May be later upconverted (e.g. by sinusoidal modulation)
• Digital Quadrature Amplitude Modulation (QAM)
  Two-dimensional extension of digital PAM
  Requires sinusoidal modulation
• Digital QAM modulates digital information onto
  pulses that are modulated onto
  Amplitudes of a sine and a cosine, or equivalently
  Amplitude and phase of single sinusoid
                                                         15 - 2
                                    Review

 Amplitude Modulation by Cosine
• Example: y(t) = f(t) cos(wc t)
  Assume f(t) is an ideal lowpass signal with bandwidth w1
  Assume w1 < wc
  Y(w) is real-valued if F(w) is real-valued
F(w)                  ½F(w + wc)      Y(w)     ½F(w - wc)
          1
                                                       ½

                   w                                                                    w
-w1   0       w1       -wc - w1         -wc + w1   0       wc - w1        wc + w1
                                  -wc                                wc
 Baseband signal                            Upconverted signal
• Demodulation: modulation then lowpass filtering
• Similar for modulation with sin(w0 t) shown next
                                                                               15 - 3
                                    Review

   Amplitude Modulation by Sine
• Example: y(t) = f(t) sin(wc t)
   Assume f(t) is an ideal lowpass signal with bandwidth w1
   Assume w1 < wc
   Y(w) is imaginary-valued if F(w) is real-valued
 F(w)                        j ½F(w + wc)            Y(w)              -j ½F(w - wc)
            1
                                                       j   ½
                                                                              wc
                                                                    wc - w1        wc + w1
                     w                                                                        w
  -w1   0       w1       -wc - w1         -wc + w1
                                    -wc
                                                           -j   ½
 Baseband signal                             Upconverted signal
• Demodulation: modulation then lowpass filtering
                                                                                     15 - 4
                  Digital QAM Modulator
                                                 bn                         b*(t)
   d[n]                                               Impulse modulator
          Serial/            Map to 2-D
  bit 1   Parallel    J      constellation
                                                      Impulse modulator
stream         J bits/symbol         an                                                            Q
                                                                      a*(t)
                           in-phase (I)                                                   d
                                                      Pulse shaping gT(t)
                            component
                                                                                     -d                 d
                     Matched                                                                                I
  s(t)        +       Delay                                                               -d
          +                          Local
                                                                                    4-level QAM
              –                     Oscillator
                                                                                    Constellation
                       90o


                               quadrature (Q)
                                              Pulse shaping gT(t)
                                 component
  Matched Delay matches delay through 90o phase shifter
                                                                                               15 - 5
  (required but often omitted on block diagrams)
         Phase Shift by 90 Degrees
• 90o phase shift performed by Hilbert transformer
                                           1               1
  cosine => sine           cos(2 f 0 t )   ( f + f 0 ) +  ( f - f 0 )
                                           2               2
                                            j               j
  sine => – cosine         sin( 2 f 0 t )   ( f + f 0 ) -  ( f - f 0 )
                                            2               2


• Frequency response of ideal
                                                1 if x  0
  Hilbert transformer:                         
                                               
  H ( f )  - j sgn( f )             sgn( x)   0 if x  0
                                               
                                               - 1 if x  0
                                                                   15 - 6
                Hilbert Transformer
• Magnitude response                  • Phase response
   All pass except at origin               Piecewise constant
           | H( f )|                               H ( f )
                                                          90o
                               f                                        f
                        H ( f )  - j sgn( f )     -90o


• For fc > 0                          • For fc < 0
                                                                       
  cos( 2f ct + )  sin(2f ct )       cos(2f ct - )  cos(-(2f ct + ))
               2                                   2                     2
                                                           
                                        cos(2 (- f c )t + )  sin( 2 (- f c )t )
                                                           2          15 - 7
                  Hilbert Transformer
• Continuous-time ideal                • Discrete-time ideal
  Hilbert transformer                    Hilbert transformer
  H ( f )  - j sgn( f )                 H (w )  - j sgn(w )
               1/( t) if t  0                     2 sin 2 (n / 2)   if n0
  h(t) =                                                    n
                                         h[n] =
                  0     if t = 0                           0           if n=0

        h(t)                                 h[n]

                                   t                                    n
                                                            Even-indexed
                                                           samples are8 zero
                                                                   15 -
 Discrete-Time Hilbert Transformer
• Approximate by odd-length linear phase FIR filter
  Truncate response to 2 L + 1 samples: L samples left of
     origin, L samples right of origin, and origin
  Shift truncated impulse response by L samples to right to
     make it causal
  L is odd because every other sample of impulse response is 0
• Linear phase FIR filter of length N has same phase
  response as an ideal delay of length (N-1)/2
  (N-1)/2 is an integer when N is odd (here N = 2 L + 1)
• Matched delay block on slide 15-5 would be an
  ideal delay of L samples
                                                           15 - 9
                                 Review

     Performance Analysis of PAM
• If we sample matched filter output at correct time
  instances, nTsym, without any ISI, received signal
   x(nTsym )  s(nTsym ) + v(nTsym )      v(nT) ~ N(0; 2/Tsym)
   where the signal component is
   s(nTsym )  an  (2i -1)d for i = -M/2+1, …, M/2                  3d

   v(t) output of matched filter Gr(w) for input of                  d
      channel additive white Gaussian noise N(0; 2)             -d
   Gr(w) passes frequencies from -wsym/2 to wsym/2 ,
                                                                 -3 d
      where wsym = 2  fsym = 2 / Tsym
• Matched filter has impulse response gr(t) 4-level PAM
                                                       Constellation
                                                           15 - 10
                                Review

        Performance Analysis of PAM
  • Decision error                                          d        
                           PI (e)  P( v(nTsym )  d )  2 Q    Tsym 
    for inner points                                                
  • Decision error                                          d        
                           PO- (e)  P(v(nTsym )  d )  Q      Tsym 
    for outer points                                                
                                                          d        
    PO+ (e)  P(v(nTsym )  -d )  P(v(nTsym )  d )  Q      Tsym 
                                                                  
  • Symbol error probability
         M -2          1           1           2( M - 1)  d      
P ( e)       PI (e) +   PO+ (e) +   PO- (e)           Q   Tsym 
          M            M           M              M             
                   O-     I     I        I    I     I     I     O+
  8-level PAM
  Constellation   -7d    -5d   -3d       -d   d    3d     5d     7d
                                                               15 - 11
     Performance Analysis of QAM
                                                                Q
• Received QAM signal
                                                        d
  x(nTsym )  s(nTsym ) + v(nTsym )
                                                   -d               d
• Information signal s(nTsym)                           -d
                                                                        I

  s(nTsym )  an + j bn  (2i -1)d + j (2k -1)d
                                                  4-level QAM
   where i,k  { -1, 0, 1, 2 } for 16-QAM         Constellation
• Noise, vI(nTsym) and vQ(nTsym) are independent
  Gaussian random variables ~ N(0; 2/Tsym)
  v(nTsym )  vI (nTsym ) + j vQ (nTsym )
• Decision regions must span entire I-Q plane
                                                             15 - 12
   Performance Analysis of 16-QAM
                                                                                   Q
• Type 1 correct detection                                                 3      2     2      3


P (c)  P( vI (nTsym )  d & vQ (nTsym )  d )
 1                                                                         2
                                                                                  1     1
                                                                                               2
                                                                                                     I

           (                      ) (
      P vI (nTsym )  d P vQ (nTsym )  d              )                  2
                                                                                  1     1
                                                                                               2




       (       (                        ))(   (
      1 - P vI (nTsym )  d 1 - P vQ (nTsym )  d                ))       3      2     2      3

                                                                                       16-QAM
                         d                              d
                   2Q(       T)                   2Q(       T)
                                                       
                                    2
             d      
      1 - 2Q
                Tsym  
                        
                                                                1 - interior decision region
                                                                 2 - edge region but not corner
                                                                                3 - corner -region
                                                                                         15 13
    Performance Analysis of 16-QAM
                                                                  Q
 • Type 2 correct detection                               3      2     2      3


P2 (c)  P(vI (nTsym )  d & vQ (nTsym )  d )                   1     1
                                                          2                   2
                                                                                  I
       P(vI (nTsym )  d ) P( vQ (nTsym )  d )
                                                          2                   2
              d             d                            1     1
       1 - Q 
                 Tsym  1 - 2Q
                                  Tsym  
                                           
                                                3                   3
                                                                 2     2


 • Type 3 correct detection                                           16-QAM

P3 (c)  P(vI (nTsym )  d & vQ (nTsym )  -d )
       P(vI (nTsym )  d ) P(vQ (nTsym )  -d )
                            2
             d                                     1 - interior decision region
       1 - Q
                Tsym  
                        
                                              2 - edge region but not corner
                                                                  3 - corner -region
                                                                           15 14
  Performance Analysis of 16-QAM
• Probability of correct detection
                              2                       2
         4     d         4     d      
  P(c)  1 - 2Q  Tsym   + 1 - Q  Tsym  
        16 
               
                              
                          16           
                                              

             8       d             d      
        +      1 - Q   Tsym  1 - 2Q  Tsym  
            16 
                     
                                
                                           
                                                  

              d       9 d         
       1 - 3Q  Tsym  + Q 2  Tsym 
                     4          

• Symbol error probability
                      d       9 d         
  P(e)  1 - P(c)  3Q  Tsym  - Q 2  Tsym 
                             4          

• What about other QAM constellations?
                                                          15 - 15
          Average Power Analysis
                                                                      3d
• Assume each symbol is equally likely                                d
• Assume energy in pulse shape is 1                                   -d
• 4-PAM constellation
                                                                      -3 d
  Amplitudes are in set { -3d, -d, d, 3d }
  Total power 9 d2 + d2 + d2 + 9 d2 = 20 d2                4-level PAM
  Average power per symbol 5 d2                            Constellation
• 4-QAM constellation points                                          Q
  Points are in set { -d – jd, -d + jd, d + jd, d – jd }         d
  Total power 2d2 + 2d2 + 2d2 + 2d2 = 8d2
                                                            -d             d
  Average power per symbol 2d2                                                 I
                                                                 -d
                                                           4-level QAM
                                                                 15 - 16
                                                           Constellation