# Quadrature Amplitude Modulation (QAM) Transmitter

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```					EE345S Real-Time Digital Signal Processing Lab   Fall 2009

(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

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
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

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