EE445S Real-Time Digital Signal Processing Lab
Spring 2014
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. to radio frequency)
• Digital Quadrature Amplitude Modulation (QAM)
Two-dimensional extension of digital PAM
Baseband signal requires sinusoidal amplitude modulation
May be later upconverted (e.g. to radio frequency)
• 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
Y1 (w ) 
• y1(t) = x1(t) cos(wc t)
1
1
X 1 (w + wc ) + X 1 (w - wc )
2
2
Assume x1(t) is an ideal lowpass signal with bandwidth w1
Assume w1 << wc
Y1(w) is real-valued if X1(w) is real-valued
X1(w)
Y1(w) ½X1(w - wc)
½X1(w + wc)
1
½
-w1
0
w1
Baseband signal
w
-wc - w1
-wc
-wc + w1
0
wc - w1
Upconverted signal
wc
wc + w1
• Demodulation: modulation then lowpass filtering
15 - 3
w
Review
Amplitude Modulation by Sine
j
j
Y2 (w )  X 2 (w + wc ) - X 2 (w - wc )
2
2
• y2(t) = x2(t) sin(wc t)
Assume x2(t) is an ideal lowpass signal with bandwidth w2
Assume w2 << wc
Y2(w) is imaginary-valued if X2(w) is real-valued
X2(w)
Y2(w) -j ½X2(w - wc)
j ½X2(w + wc)
1
j
-w2
0
w2
Baseband signal
w
½
wc – w2
-wc – w2
-wc
wc
wc + w2
-wc + w2
-j
½
Upconverted signal
• Demodulation: modulation then lowpass filtering
15 - 4
w
Baseband Digital QAM Transmitter
• Continuous-time filtering and upconversion
Impulse
modulator
i[n]
Index
Bits
1
Serial/
parallel
converter
gT(t)
Pulse shapers
(FIR filters)
Map to 2-D
constellation
J
q[n]
Q
Impulse
modulator
s(t)
Delay
Local
Oscillator
+
90o
gT(t)
d
-d
d
-d
4-level QAM
Constellation
I
Delay matches delay through 90o phase shifter
Delay required but often omitted in diagrams
15 - 5
Phase Shift by 90 Degrees
• 90o phase shift performed by Hilbert transformer
cosine => sine
sine => – cosine
1
1
cos( 2 f 0 t )   ( f + f 0 ) +  ( f - f 0 )
2
2
j
j
sin( 2 f 0 t )   ( f + f 0 ) -  ( f - f 0 )
2
2
• Frequency response H ( f )  - j sgn( f )
Magnitude Response
Phase Response
H ( f )
| H( f )|
90o
f
All-pass except at origin
f
-90o
15 - 6
Hilbert Transformer
• Continuous-time ideal
Hilbert transformer
• Discrete-time ideal
Hilbert transformer
H (w )  - j sgn( w )
H ( f )  - j sgn( f )
1/( t) if t  0
h(t) =
h[n] =
0
if t = 0
h(t)
2 sin 2 (n / 2)

n
if n0
0
if n=0
h[n]
t
n
Even-indexed
samples are
zero
15 - 7
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 - 8
Baseband Digital QAM Transmitter
i[n]
Impulse
modulator
Index
Bits
1
Serial/
parallel
converter
J
Pulse shapers
(FIR filters)
i[n]
Index
Bits
1
Serial/
parallel
converter
J
L samples/symbol
(upsampling factor)
Impulse
modulator
L
s(t)
Delay
Local
Oscillator
Map to 2-D
constellation
q[n]
100% discrete time
gT(t)
+
90o
gT(t)
gT[m]
s[m]
Pulse shapers
(FIR filters)
Map to 2-D
constellation
q[n]
L
gT[m]
cos(w0 m)
sin(w0 m)
+
s(t)
D/A
15 - 9
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 transmitted signal 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
channel additive white Gaussian noise N(0; 2)
Gr(w) passes frequencies from -wsym/2 to wsym/2 ,
where wsym = 2  fsym = 2 / Tsym
d
-d
-3 d
4-level PAM
• Matched filter has impulse response gr(t) Constellation
15 - 10
Performance Analysis of PAM
d

PI (e)  P( v(nTsym )  d )  2 Q
Tsym 


d

PO- (e)  P(v(nTsym )  d )  Q
Tsym 


d

PO+ (e)  P(v(nTsym )  -d )  P(v(nTsym )  d )  Q
Tsym 


• Decision error
for inner points
• Decision error
for outer points
• 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


8-level PAM
Constellation
O-
I
I
I
I
I
I
-7d
-5d
-3d
-d
d
3d
5d
O+
7d
15 - 11
Performance Analysis of QAM
• If we sample matched filter outputs at correct time
instances, nTsym, without any ISI, received signal
x(nTsym )  s (nTsym ) + v(nTsym )
Q
d
• Transmitted signal
s(nTsym )  an + j bn  (2i - 1)d + j (2k - 1)d
where i,k  { -1, 0, 1, 2 } for 16-QAM
-d
d
I
-d
4-level QAM
• Noise v(nTsym )  vI (nTsym ) + j vQ (nTsym )
Constellation
For error probability analysis, assume noise terms independent
and each term is Gaussian random variable ~ N(0; 2/Tsym)
In reality, noise terms have common source of additive noise in
channel
15 - 12
Performance Analysis of 16-QAM
Q
• Type 1 correct detection
3
P1 (c)  P( vI (nTsym )  d & vQ (nTsym )  d )
(
) (
 P vI (nTsym )  d P vQ (nTsym )  d
(
(
)) (
d

T)
)

d

 1 - 2Q
Tsym  



d

1
1
))
3
3
2
2
(
2Q (
2
2
 1 - P vI (nTsym )  d 1 - P vQ (nTsym )  d
2Q (
2
I
2
1
1
2
2
3
16-QAM
T)
2
1 - interior decision region
2 - edge region but not corner
3 - corner
15 -region
13
Performance Analysis of 16-QAM
• Type 2 correct detection
P2 (c)  P(vI (nTsym )  d & vQ (nTsym )  d )
 P(vI (nTsym )  d ) P( vQ (nTsym )  d )

d
 
d

 1 - Q
Tsym  1 - 2Q
Tsym  

 



• Type 3 correct detection
Q
3
2
2
1
1
2
2
2
3
3
I
2
1
1
2
2
3
16-QAM
P3 (c)  P(vI (nTsym )  d & vQ (nTsym )  -d )
 P(vI (nTsym )  d ) P(vQ (nTsym )  -d )

d

 1 - Q
Tsym  



2
1 - interior decision region
2 - edge region but not corner
3 - corner
15 -region
14
Performance Analysis of 16-QAM
• Probability of correct detection
2
4
4
d

d

P(c)  1 - 2Q
Tsym   + 1 - Q
Tsym  
16 

  16 


+
2
8
d
 
d

1 - Q
Tsym  1 - 2Q
Tsym  
16 

 


d
 9 d

 1 - 3Q
Tsym  + Q 2 
Tsym 

 4 

• Symbol error probability (lower bound)
d
 9 2 d

P(e)  1 - P(c)  3Q
Tsym  - Q 
Tsym 

 4 

• What about other QAM constellations?
15 - 15
Average Power Analysis
3d
• Assume each symbol is equally likely
• Assume energy in pulse shape is 1
• 4-PAM constellation
Amplitudes are in set { -3d, -d, d, 3d }
Total power 9 d2 + d2 + d2 + 9 d2 = 20 d2
Average power per symbol 5 d2
d
-d
-3 d
4-level PAM
Constellation
• 4-QAM constellation points
Points are in set { -d – jd, -d + jd, d + jd, d – jd }
Total power 2d2 + 2d2 + 2d2 + 2d2 = 8d2
Average power per symbol 2d2
Q
d
-d
d
-d
4-level QAM
15 - 16
Constellation
I