Digital Transmission through the
AWGN Channel
ECE460
Spring, 2012
Geometric Representation
Orthogonal Basis
1. Orthogonalization (Gram-Schmidt)
2. Pulse Amplitude Modulation
a. Baseband
b. Bandpass
c. Geometric Representation
3. 2-D Signals
a. Baseband
b. Bandpass
1) Carrier Phase Modulation (All have same energy)
1) Phase-Shift Keying
2) Two Quadrature Carriers
2) Quadrature Amplitude Modulation
4. Multidimensional
a. Orthogonal
1) Baseband
2) Bandpass
b. Biorthogonal
1) Baseband
2) Bandpass
2
Geometric Representation
Gram-Schmidt Orthogonalization
1. Begin with first waveform, s1(t) with energy ξ1:
 1 t  
s1  t 
1
2. Second waveform
a. Determine projection, c21, onto ψ1

c21   s2  t   1  t  dt

b. Subtract projection from s2(t)
d 2  t   s2  t   c21 1  t 
c.
Normalize
 2 t  
d2 t 
2

where  2   d 22  t  dt

3. Repeat

cki   sk  t   i  t  dt

k 1
d k  t   sk  t    cki i  t 
i 1
 k t  
dk t 
k

where  k   d k2  t  dt

3
Example 7.1
4
Pulse Amplitude Modulation
Baseband Signals
Binary PAM
•
•
Bit 1 – Amplitude + A
Bit 0 – Amplitude - A
M  2k
M-ary PAM
M-ary PAM
sm  t   Am gT  t 
 m   sm2  t  dt
T
A

2
m T
g
2
T
Binary PAM
 t  dt
 Am2  g
Fixed Rb 
1
k

Tb kT
5
Pulse Amplitude Modulation
Bandpass Signals
Baseband Signal
sm  t 
X
Bandpass Signal
sm  t  cos  2 f ct 
2cos  2 f ct 
um  t   Am gT  t  cos  2 f ct  m  1, 2, ... , M
Um  f  
Am
GT  f  f c   GT  f  f c  
2 
What type of Amplitude Modulation signal does this appear to be?
m 

2
u
m
  t  dt




Am2  t  gT2  t  cos 2  2 f c t  dt

Am2

2

Am2
2
 gT  t  dt  2

 g  t  cos  4 f t  dt
2
T
c

6
PAM Signals
Geometric Representation
M-ary PAM waveforms are one-dimensional
sm  t   sm  t  m  1, 2,..., M
 t  
where
1
g
gT  t  0  t  T
sm   g Am
d
d
d
m  1, 2,..., M
d
d
d = Euclidean distance
between two points
0
7
Two-Dimensional Signal Waveforms
Baseband Signals
• Are these orthogonal?
• Calculate ξ.
• Find basis functions of (b).
8
Two-Dimensional Bandpass Signals
Carrier-Phase Modulation
1. Given M-two-dimensional signal waveforms sm  t  , m  1, 2,..., M
um  t   sm  t  cos  2 f ct  0  t  T
2. Constrain bandpass waveforms to have same energy
T
 m   um2  t  dt
0
T
  sm2  t  cos 2  2 f c t  dt
0
T
T
1
1
  sm2  t  dt   sm2  t  cos  4 f ct  dt
20
20
 s
m
9
Two-Dimensional Bandpass Signals
Quadrature Amplitude Modulation
um  t   Amc gT  t  cos  2 f ct   Ams gT  t  sin  2 f ct  m  1, 2,..., M
10
Multidimensional Signal Waveforms
Orthogonal
Multidimensional means
multiple basis vectors
Baseband Signals
• Overlapping
(Hadamard Sequence)
• Non-Overlapping
o Pulse Position Mod.
(PPM)
sm  t   A gT  t   m  1 T / M 
where
m  1, 2,..., M
 m  1 T / M  t  mT / M
11
Multidimensional Signal Waveforms
Orthogonal
Bandpass Signals
As before, we can create bandpass signals by simply multiplying
a baseband signal by a sinusoid:
um t   sm t  cos  2 fct  0  t  T
Carrier-frequency modulation: Frequency-Shift Keying (FSK)
2 b
cos  2 f ct  2 m f t  m  0,1,..., M , 0  t  T
T
um  t  
 mn 

1
s
T
 u  t  u  t  dt
m
n
0
sin 2  m  n  f T
2  m  n  f T
12
Multidimensional Signal Waveforms
Biorthogonal
Baseband
Begin with M/2 orthogonal vectors in N = M/2 dimensions.
  , 0, 0,..., 0 
  0,  , 0,..., 0 
s1 
s2
s
s

s M /2  0, 0, 0,...,  s

Then append their negatives
sM
2
1

   s , 0, 0,..., 0

s M  0, 0, 0,...,   s


Bandpass
As before, multiply the baseband signals by a sinusoid.
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Multidimensional Signal Waveforms
Simplex
Subtract the average of M orthogonal waveforms
1
sm  t   sm  t  
M
T
M
 s t 
k 1


k
1 

 s    sm  t  dt  1    s
M
2
0
In geometric form (e.g., vector)
1
sm  s m 
M
M
s
k 1
k
Where the mean-signal vector is
1
s
M
M
s
k 1
k
Has the effect of moving the origin to s reducing the energy per
symbol
2
 s  sm
 sm  s
2
1

 1 
 M

s

14
Multidimensional Signal Waveforms
Binary-Coded
M binary code words
c m   cm1 , cm 2 ,..., cmN  m  1, 2,..., M
where cmj  0 or 1 for all m and j. For example:
c1  1 1 1 1 0
c 2  1 1 0 0 1
c3  1 0 1 0 1
c 4   0 1 0 1 0
In vector form:
sm   sm1 , sm 2 ,..., smN  m  1, 2,..., M
where
smj    s / N  m and j
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Optimum Receivers
Start with the transmission of any one of the M-ary signal
waveforms:
g M  2k symbols having k -bits sm  t  , m  1, 2,..., M 
g Transmitted within timeslot 0  t  T
g Corrupted with AWGN: r  t   sm  t   n  t 
r  t   sm  t   n  t 
Demodulator
Detector
Sampler
r  t   r   r1 , r2 ,..., rN 
Output
Decision
r  sm  t 
1. Demodulators
a. Correlation-Type
b. Matched-Filter-Type
2. Optimum Detector
3. Special Cases (Demodulation and Detection)
a.
b.
c.
d.
Carrier-Amplitude Modulated Signals
Carrier-Phase Modulation Signals
Quadrature Amplitude Modulated Signals
Frequency-Modulated Signals
16
Demodulators
Correlation-Type
k  1, 2,..., N
rk   r  t  k  t  dt
T
0
   sm  t   n  t   k  t  dt
0
T
  sm  t  k  t  dt   n  t  k  t  dt
T
T
0
0
 smk  nk
r  sm  n
Next, obtain the joint conditional PDF
f  r | sm  

1
 N 0 
N /2
1
 N 0 
N /2
 N

2
exp    rk  smk  / N 0 
 k 1

exp   r  s m

2
m  1, 2,..., M
/ N0 

17
Demodulators
Matched-Filter Type
Instead of using a bank of
correlators to generate {rk},
use a bank of N linear filters.
The Matched Filter
Key Property: if a signal s(t) is corrupted
by AGWN, the filter with impulse response
matched to s(t) maximizes the output SNR
Demodulator
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Optimum Detector
Decision based on transmitted signal
in each signal interval based on the
observation of the vector r.
Maximum a Posterior Probabilities (MAP)
P  signal s m was transmitted | r  m  1, 2,..., M
P  sm | r  
f  r | sm  P  s m 
N
 f r | s  P s 
m 1
m
m
If equal a priori probabilities, i.e., P  sm   1/ M for all M and
the denominator is a constant for all M, this reduces to
maximizing f  r | sm  called maximum-likelihood (ML) criterion.
N
D  r, s m     rk  smk 
2
minimum distance detection
k 1
D  r, s m   2r  s m  s m
C  r, s m   2r  s m  s m
2
2
minimize
maximize (correlation metric)
19
Example 7.5.3
Consider the case of binary PAM signals in which two possible
signal points are s1   s2   b where  b is the energy per bit.
The prior probabilities are P  s1   p and P  s2   1  p. Determine
the metrics for the optimum MAP detector when the
transmitted signal is corrupted with AWGN.
20