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Report
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 
Bandpass Signal
X
sm  t  cos  2 f ct 
cos  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
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
8
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 

9
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
10
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)
11
Probability of Error
Binary PAM Baseband Signals
Consider binary PAM baseband signals s1  t    s2  t   gT  t 
where gT  t  is an arbitrary pulse which is nonzero in the
interval 0  t  T and zero elsewhere. This can be pictured
geometrically as
b
 b
s2
s1
0
Assumption: signals are equally likely and that s1 was
transmitted. Then the received signal is
r  s1  n   b  n
Decision Rule:
r
s1


s2
0
The two conditional PDFs for r are
1
 r 
f  r | s1  
e
 N0
1
 r 
f  r | s2  
e
 N0
b

b

2
/ N0
2
/ N0
12
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.
13
Probability of Error
M-ary PAM Baseband Signals
Recall baseband M-ary PAM are geometrically represented in 1D with signal point values of
sm   g Am m  1, 2,..., M
And, for symmetric signals about the origin,
Am   2m  1  M  m  1, 2,..., M
where the distance between adjacent signal points is 2  g .
Each signal has a different energies. The average is
 av
1

M



 Pav 
g
M
M

m 1
m
cos 1 
M
  2m  1  M 
2
m 1
2
 g M  M  1
M
3
 M 2  1
3
 av
T
g
M


2
3
 1  g
T
14
Demodulation and Detection
Carrier-Amplitude Modulated Signals
Demodulation of bandpass digital PAM signal
Received
Signal
r(t)
Oscillator
Transmitted Signal:
um  t   Am gT  t  cos  2 f ct  0  t  T
Received Signal:
r  t   Am gT  t  cos  2 f ct   n  t 
0t T
where n  t   nc  t  cos  2 f c t   ns  t  sin  2 f c t 
Crosscorrelation
 r  t   t  dt  A
T
m
0
 Am
Optimum Detector
2
g
g
2

T
0
g
2
T
 t  cos  2 f c t  dt  0 n  t   t  dt
2
T
n
D  r , sm    r  s m 
2
or C  r , sm   2 r sm  s
2
m
15
Two-Dimensional Signal Waveforms
Baseband Signals
• Are these orthogonal?
• Calculate ξ.
• Find basis functions of (b).
16
Problem 7.22
In an additive white Gaussian noise channel with noise powerN
spectral density of 0 , two equiprobable messages are
2
transmitted by
 At
s1  t  
 , 0t T
T

 0, otherwise
 
t
 A 1 
s2  t     T
0,


' 0  t  T

otherwise
1. Determine the structure of the optimal receiver
2. Determine the probability of error.
17
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
18
Demodulation and Detection
Carrier-Phase Modulated Signals
r  t   um  t   n  t 
The received signal:
 [ Amc gT  t   nc  t ]cos  2 f ct 
[ Ams gT  t   ns  t ]sin  2 f ct 
where m  0,1,... M  1
Giving basis vectors as
 1 t  
 2 t   
2
g
gT  t  cos 2 f ct
2
g
gT  t  sin 2 f c t
Outputs of correlators:
r  sm  n


 s cos 2 m / M  nc ,
 s sin 2 m / M  ns

19
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
20
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
21
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
22
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.
23
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

24
Demodulation and Detection
Carrier-Amplitude Modulated Signals
Demodulation of bandpass digital PAM signal
Received
Signal
r(t)
Oscillator
Transmitted Signal:
um  t   Am gT  t  cos  2 f ct  0  t  T
Received Signal:
r  t   Am gT  t  cos  2 f ct   n  t 
0t T
where n  t   nc  t  cos  2 f c t   ns  t  sin  2 f c t 
Crosscorrelation
 r  t   t  dt  A
T
m
0
 Am
Optimum Detector
2
g
g
2

T
0
g
2
T
 t  cos  2 f c t  dt  0 n  t   t  dt
2
T
n
D  r , sm    r  s m 
2
or C  r , sm   2 r sm  s
2
m
25

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