### MS PowerPoint

```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
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
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
• 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.
13
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
15
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
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
18
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
```