ECE 6640
Digital Communications
Dr. Bradley J. Bazuin
Assistant Professor
Department of Electrical and Computer Engineering
College of Engineering and Applied Sciences
Chapter 4
Chapter 4:
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
ECE 6640
Optimum Receivers for AWGN Channels
Waveform and Vector Channel Models
Waveform and Vector AWGN Channels
Optimal Detection and Error Probability for
Band-Limited Signaling
Optimal Detection and Error Probability for
Power-Limited Signaling
Optimal Detection in Presence of Uncertainty:
Noncoherent Detection
A Comparison of Digital Signaling Methods
Lattices and Constellations Based on Lattices
Detection of Signaling Schemes with Memory
Optimum Receiver for CPM Signals
Performance Analysis for Wireline and Radio
Communication Systems
Bibliographical Notes and References
Problems
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
160
160
167
188
203
210
226
230
242
246
259
265
266
2
Waveform and Vector Channel
Models
• The additive white Gaussian noise (AWGN) channel
model is a channel whose sole effect is addition of a white
Gaussian noise process to the transmitted signal.
r t   sm t   nt 
vl t  
ECE 6640

I
m  
m
 g t  m  T 
r t  

I
m  
m
 g t  m  T   nt 
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
3
Symbol Detection
• The receiver observes the received signal r (t) and, based
on this observation, makes the optimal decision about
which message m, 1 ≤ m ≤ M, was transmitted.
• An optimal decision rule results in minimum error
probability or “disagreement” between the transmitted
message m and the detected message .
Pe  Pmˆ  m
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
4
Optimal Detection Assumption
• Oservation prompts us to view the waveform channel
r(t)=sm(t)+n(t) in the vector form r = sm + n where all
vectors are N-dimensional and components of n are i.i.d.
zero-mean Gaussian random variables with variance N0/2.
r t   sm t   nt 
• If we assume optimal signal processing and sampling
z t   sm t   hc t   hs t   nt   hs t 
r t   s m t   hc t   n t 
z n  T   z t    t  n  T 
s m t 
r  sm  n
n t 
r  sm  n
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
5
Baseband Vector
• The mathematical model for the AWGN vector channel is
r  sm  n
given by
• We study a more general vector channel model in this
section which is not limited to the AWGN channel model.
• In our model, vectors sm are selected from a set of possible
signal vectors {sm, 1 ≤ m ≤ M} according to prior or a
priori probabilities Pm and transmitted over the channel.
The received vector r depends statistically on the
transmitted vector through the conditional probability
density functions p(r|sm).
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
6
Decision Function
• The receiver observes r and based on this observation
decides which message was transmitted.
• The decision function determines and estimate of the
symbol m.
g r   mˆ
• The probability of a correct decision becomes.
Pcorrect _ decision | r   Pmˆ sent | r 
Pcorrect _ decision   Pmˆ sent | r  pr   dr
• Our goal is to design an optimal detector that minimizes
the error probability or, equivalently, maximizes P [correct
decision].
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
7
Maximization
Pcorrect _ decision   Pmˆ sent | r  pr   dr
• Maximization is equivalent to maximizing the probability
term
Pmˆ sent | r 
mˆ  g opt r   arg max Pm | r 
1 m  M
• The optimal detection scheme described simply looks
among all P [m |r ] for 1 ≤ m ≤ M and selects the m that
maximizes P [m |r ]. The detector then declares this
maximizing m as its best decision.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
8
Equivalent Maximum
• Note that since transmitting message m is equivalent to
transmitting sm, the optimal decision rule can be written as
either
mˆ  g opt r   arg max Pm | r 
1 m  M
mˆ  g opt r   arg max Psm | r 
1 m  M
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
9
MAP and ML Receivers
• Detection methods based on probability and estimation of
the symbol sent
– MAP: maximum a-posteriori probability
– ML: maximum likelihood
• MAP detection
mˆ  g opt r   arg max
1 m  M
Pm p r | sm 
pr 
mˆ  g opt r   arg max Pm pr | sm 
1 m  M
– as p(r) is independent of m, the second function is valid
– Pm is the a-priori probability of m
• For equally probably messages:
mˆ  g opt r   arg max p r | sm 
1 m  M
– this is referred to as the maximum-likelihood
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
10
Decision Regions
• Any detector partitions the output space RN into M regions
denoted by D1, D2, . . . , DM such that if r ∈ Dm, then
= g(r) = m, i.e., the detector makes a decision in favor of m.
• The region Dm, 1 ≤ m ≤ M, is called the decision region for
message m; and Dm is the set of all outputs of the channel that
are mapped into message m by the detector.
• If a MAP detector is employed, then the Dm’s constitute the
optimal decision regions resulting in the minimum possible
error probability.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
11
Error Probability
• An error occurs when the received r is not in Dm.
• The symbol error probability is given by
M
M
m 1
m 1
Pe   Pm Pr  Dm | sm    Pm Pe|m
M
Pe   Pm
m 1
  pr | s  dr
1 m ' M Dm '
m ' m
m
• This equation gives the probability that an error occurs in
transmission of a symbol or a message and is called
symbol error probability or message error probability.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
12
Preprocessing
• The concepts defined are applicable whether
“preprocessing” is performed or not.
– Preprocessing in fact is usually, if not always, performed. The
signal will always be filtered based on the required signal
bandwidth prior to detection.
ECE 6640
13
Preprocessing (cont)
• Preprocessing forms
– Noise power reduction filtering. Goals: (1) reduce noise power, (2)
minimize any signal effects.
– Matched filter for signal of interest. Goal: (1) maximize signal
power received for detection, (2) filter the noise power.
– Whitening filter. Goal: (1) “Whiten” the signal spectrum to remove
interference (make the noise power flat)
• Critical filter considerations
– How does the filter change the spectral response of the signal-ofinterest? Does it cause ISI or signal power loss?
– What is the noise-equivalent bandwidth of the filter?
N0Beq noise power.
ECE 6640
14
Developing Error Rates
• The following slides are going to follow an alternate
methodology and derivations.
• It looks more specifically at bit detection of a binary signal
in Gaussian noise.
– A non-vector approach prior to the more theoretical vector
approach taken in the textbook.
– The following can be reviews in either Carlson or Sklar textbooks.
ECE 6640
15
Baseband Binary Receiver
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 11.2-1


yt    a k  pt  kT  h t   n in t   h t 
k

y t k   ak  n t k 
• Synchronous Time sampling of maximum filter output
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
16
Regeneration of a unipolar signal
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) signal plus noise (b) S/H output (c) comparator output: Figure 11.2-2
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
17
Unipolar NRZ Binary Error
Probability
• Hypothesis Testing using a voltage threshold
– Hypothesis 0
• The conditional probability distribution expected if a 0 was sent
pY  yk | H 0   pY ak  n t k  | ak  0  pY n t k 
pY  yk | H 0   p N  yk 
– Hypothesis 1
• The conditional probability distribution expected if a 1 was sent
pY  yk | H1   pY ak  n t k  | ak  A  pY A  n t k 
pY  yk | H1   p N  yk - A 
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
18
Decision Threshold and
Error Probabilities
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Conditional PDFs Figure 11.2-3
y t k   ak  n t k 
V
Pe1  PY  V    p Y y | H1   dy


Pe 0  PY  V    p Y y | H 0   dy
V
• Use Hypothesis to establish a decision rule
– Use threshold to determine the probability of correctly and
incorrectly detecting the input binary value
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
19
Average Error Probability
• Using the two error conditions:
– Detect 1 when 0 sent
– Detect 0 when 1 sent
Perror  PH 0   Pe 0  PH1   Pe1
• Selecting an Optimal Threshold
PH   p V | H   PH   p V
0
Y
opt
0
1
Y
opt
| H1 
• For equally likely binary values
PH 0   PH1  
1
2
Perror 
1
 Pe 0  Pe1 
2
pY Vopt | H 0   pY Vopt | H1 
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
20
Threshold regions for conditional PDFs
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 11.2-4
P H 0   P H1  
1
2
Vopt 
A
2
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
21
For AWGN
• The pdf is Gaussian
 y2 

 exp 
p Y y | H 0   p N y  
2 
2
2  
 2 
1
for

 2 
1
Q x  
  exp    d
2 x
 2

V
 A 
Pe 0  PY  V    p N y   dy  Q   Q

2







V
V
for
AV
 A 
Pe1  PY  V    p N y  A   dy  Q
  Q

2








P H 0   P H 1  
1
2
Vopt 
A
2
 A V 
 A 
Pe1  Q
  Q
  Pe 2
  
 2  
22
Modification for
Polar NRZ Signals (+/- A/2)
• Hypothesis Testing using a voltage threshold
– Hypothesis 0
• The conditional probability distribution expected if a 0 was
sent
A

 A

pY  yk | H 0   pY  ak  n t k  | ak     pY    n t k 
2

 2

A

pY  yk | H 0   p N  yk  
2

– Hypothesis 1
• The conditional probability distribution expected if a 1 was
sent
A

A

pY  yk | H1   pY  ak  n t k  | ak    pY   n t k 
2

2

A

pY  yk | H1   p N  yk - 
2

Vopt 
A A
 0
2 2
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
23
Modification for
Polar NRZ Signals (+/- A/2)
• Determining the error probability
 A V
A

Pe 0  PY  V    p N  y    dy  Q 2
 
2

V


 A V
A

Pe1  PY  V    p N  y    dy  Q 2
 
2



V

  Q A 



2






  Q A 



2





• Notice that the error is the same as Unipolar NRZ
– The distance between the expected signal values is the
same
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
24
Modification for
Bipolar NRZ Signals (+/- A)
• Hypothesis Testing using a voltage threshold
– Hypothesis 0
• The conditional probability distribution expected if a 0 was
sent
pY  yk | H 0   pY ak  n t k  | ak   A  pY  A  n t k 
pY  yk | H 0   p N  yk  A
– Hypothesis 1
• The conditional probability distribution expected if a 1 was
sent
pY  yk | H1   pY ak  n t k  | ak  A  pY  A  n t k 
pY  yk | H1   p N  yk - A 
Vopt  A  A  0
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
25
Modification for Bipolar NRZ Signals
• Determining the error probability

AV
A
Pe 0  PY  V    p N y  A   dy  Q
  Q 
  

V
V
AV
A
Pe1  PY  V    p N y  A   dy  Q
  Q 
  


• Notice that the error has been reduced
– The distance between the expected signal values may
be twice as large as the unipolar case (using +/- A)
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
26
Relationship to signal power
• Defining the average received signal power
– Unipolar NRZ
– Polar NRZ
SR 
SR 
– Bipolar NRZ
1 2
A ,
2
1 2
A ,
4
S R  A2 ,
0, A
 A A
 2 , 2 
  T2

2
 1

S R  E  lim   xc t   dt  
T 

  T T 2

 A, A
• In terms of SNR
1  S 
2
2  N 
2
A
  R
 A 


 
4  N R  S 
 2 
 
 N  R
for Unipolar
for Polar
2
A 2  S 
A
  
  

N
 
R
 N  R
for Bipolar
27
Probability of error
• The probability of detecting a transmitted symbol
correctly is dependent upon the received signal-to1
noise ratio …. assuming
PH   PH  
0
– Unipolar NRZ (orthogonal)
 1 S 
 A 
Pe  Q
  
  Q

 2  
 2  N R 
1
2
2
A2
1 S
 A 
  

 
4  NR 2  N R
 2  
– Polar NRZ (antipodal)
 S 
 A 
Pe  Q
  Q   
 2  
  N R 
2
A2
 A 
S
 

 
4  N R  N R
 2  
– Bipolar NRZ (antipodal)
 S 
 A
Pe  Q   Q   
  N R 
 


2
A2  S 
 A
 
  
N R  N R
 
28
Power Ratio vs. Bit Energy
• For continuous time signals, power is a normal
way to describe the signal.
• For a discrete symbol, the “power” is 0 but the
energy is non-zero
– Therefore, we would like to describe symbols in terms
of energy not power
• For digital transmissions how to we go from
power to energy?
– Power is energy per time, but we know the time
duration of a bit. Noise has a bandwidth.
1
S R  Eb 
Tb
N R  N 0 W
SR
S

?
 
 N R N R
29
SNR to Eb/No
• For the Signal to Noise Ratio
– SNR relates the average signal power and average noise
power (Tb is bit period, W is filter bandwidth)
1
 Eb  R b
Tb  E b 
1
S


 


 
 

 N  N 0  W  N 0  Tb  W  N 0  W
Eb 
– Eb/No relates the energy per bit to the noise energy
(equal to S/N times a time-bandwidth product)
 Eb   S  W  S 

    
    Tb  W 
 N0   N  R b  N 
30
Relationship to Eb/No
• Defining the energy per bit to noise power ratio
for a time-bandwidth product of W  T  R  T  1
b
b
2
b
2
2
– Unipolar
 Eb 
A2
1 S
 A 

     

 
4  N R 2  N R  N0 
 2 
– Polar
 2  Eb 
A2
 A 
S

    

 
4  N R  N R  N0 
 2 
– Bipolar
2
2
 2  Eb 
A2  S 
A





 
 
N R  N R  N0 

31
Relationship to Bit Error Probability
• Defining the binary bit error probability
1
for a time-bandwidth product
PH   PH  
0
– Unipolar
1
2
 Eb 
A

Perror  Q   Q

 2 
 N0 
 2  Eb
A
 Q   Q
 2 
 N0
– Polar
Perror
– Bipolar
 2  Eb
A
Perror  Q   Q

 N0








32
Bit Error Rate Plot
Classical Bit Error Rates
0.5
Orthogonal
Antipodal
0.45
0.4
Bit Error Rate
0.35 EbNo=(0:10000)'/1000;
0.3
% Q(x)=0.5*erfc(x/sqrt(2))
0.25
Ortho=0.5*erfc(sqrt(EbNo)/sqrt(2));
0.2 Antipodal=0.5*erfc(sqrt(2*EbNo)/sqrt(2));
0.15
semilogx(EbNo,[Ortho Antipodal])
0.1 ylabel('Bit Error Rate')
xlabel('Eb/No')
0.05 title('Classical Bit Error Rates')
legend('Orthogonal','Antipodal')
0
-3
10
10
-2
-1
10
Eb/No
10
0
10
1
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
33
BER Performance, Classical Curves
log-log plot
10
10
Bit Error Rate
10
10
10
10
10
10
Classical Bit Error Rates
0
Orthogonal
Antipodal
-1
-2
-3
-4
-5
-6
-7
-1
0
1
2
3
4
5
6
7
8
Eb/No
9
10
11
12
13
14
15
34
Antipodal and Orthogonal Signals
• Antipodal
– Distance is twice “signal voltage”
– Only works for one-dimensional signals
d  2  Eb
T
 1
1
zij    si t   s j t   dt  
E 0
 1
for i  j
for i  j
• Orthogonal
– Orthogonal symbol set
– Works for 2 to N dimensional signals
d  2  Eb
T
1
1
z ij    s i t   s j t   dt  
E 0
0
for i  j
for i  j
35
Returning to Textbook Notes
ECE 6640
36
4.2 AWGN Channels
• This sections proves that the noise contribution to the
sampled signal vector has the expected properties
• The noise is zero mean (Eq. 4.2-10)
• The Noise covariance is described as (Eq. 4.2-11)
 N 0
Cov ni  n j   2
0


for i  j
for i  j
• Noise is uncorrelated with other noise components
(Eq. 4.2-12)
Covni  n2 t   0
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
Optimal Detection
• Based on the MAP detector and AWGN noise
mˆ  arg maxPm  pr | sm 
1 m  M
for
r  sm  n
mˆ  arg maxPm  pn r  sm 
1 m  M

 1

mˆ  arg max Pm  
  N
1 m  M 
0


  r  sm

  exp


N0


N
2




This function should be computed for all M symbols
and the “arg max” determined in order to make a
detection decision.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
38
Optimal Detection (cont.)

 1
mˆ  arg max  Pm  
  N
1 m  M 
0


  r  sm

  exp


N0


N
2




• As there are elements of this computation that are identical
for every potential symbol, the text “drops” the common
elements to focus on what is unique for different symbols.
2

r  sm 
mˆ  arg max ln Pm  

N 0 
1 m  M 

1
2
N
mˆ  arg max  0  ln Pm    r  sm 
2
1 m  M  2

1

N
mˆ  arg max  0  ln Pm    Em  r  sm 
2
1 m  M  2

ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
39
Detection Interpretations
1
 N0

mˆ  arg max   ln Pm    Em  r  sm 
2
1 m  M  2

• A symbol decision region based on terms related to noise
and symbols
N0
1
nm 
 lnPm    Em
2
2
• For equally probably symbols, a simplified detection
A nearest-neighbor or
criteria
mˆ  arg min r  sm
1 m  M

minimum distance
detection
– the smallest distance between the received values one of M
possible symbols
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
40
Decision Regions
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
41
Binary Antipodal Signaling
• In a binary antipodal signaling scheme s1(t) = s(t) and
s2(t) = −s(t).
• The probabilities of messages 1 and 2 are p and 1 − p,
respectively.
s1  Es
s2   E s
• Es is energy in each signal and is equal to Eb.
• The decision regions based on MAP becomes
N
N
1
1


D1  r : r  Eb  0  ln p    Eb  r  Eb  0  ln 1  p    Eb 
2
2
2
2



 1  p 
N 0  1  p  
N0
  r : r 

 ln
 ln
D1  r : r  2  Eb 
2
4  Eb
 p  
 p 

ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
42
Binary Antipodal Signaling
• Defining a detection threshold
rth 
D1  r : r  rth 
1 p 
N0

 ln
4  Eb
 p 
and
D2  r : r  rth 
• For equally probable symbols, p=0.5 and
ECE 6640
rth  0
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
43
Bit Error Probability
• To derive the error probability for this system, use Eq. 4.1–15.
M
Pe   Pm
m 1
  pr | s  dr
2
Pe   Pm
m 1

m
1 m ' M Dm '
m ' m
  pr | s  dr
m
1 m ' 2 Dm '
m ' m



Pe  p   p r | sm  Eb  dr  1  p    p r | sm   Eb  dr
D2
rth
Pe  p 
 pr | s
D1
m





 Eb  dr  1  p    p r | sm   Eb  dr
rth
• Assuming AWGN
 
 
N
N


Pe  p  P  N  Eb , 0  rth   1  p   P  N   Eb , 0  rth 
2
2


 
 
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
44
Bit Error Probability (cont)
 
 
N
N


Pe  p  P  N  Eb , 0  rth   1  p   P  N   Eb , 0  rth 
2
2


 
 
• Recognizing the tail of the Gaussians










E
r
E
r
b
th 
b
th 
 1  p   Q
Pe  p  Q


N0 
N0 




2
2




• for equally probably symbols p=-.5
 2  Eb
Pe  Q
 N0
ECE 6640




45
Error Based on Symbol Distance
• For equiprobable symbols, the regions are
separated by the perpendicular bisector of s1 and s2.
• By symmetry, error probabilities are equal,
therefore Pb = P[error |s1 sent ].
d12  s2  s1
n  r  s1
ECE 6640
 n  s2  s1  d12 


Pb P 

2 
d12

 d12 2 


2
 d 2

d12 
12

  Q
2
Pb  P n  s2  s1  
  Q
 2  N0
2 
N0 


 d12 

2 





Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
46
Error Based on Symbol Distance
(cont)
• For the distance defined as
d12  s2  s1
2

d12 
 s2 t   s1 t 
2

2
d12 

 s t 

 dt 
2
2

 2  s1 t   s2 t   s1 t   dt
2





 s2 t   dt 2   s1 t  s2 t  dt   s1 t   dt
2

2
d12  E1  E2  2  s1 t , s2 t 
2
• Based on equal symbol energy and the cross correlation
d12  2  Eb  2  Eb    2  Eb  1   
2
1    1
ECE 6640
47
Error Based on Symbol Distance
(cont)
• With distance defined as
d12  2  Eb  1   
2
 d 2
12
Pb  Q
 2  N0


  Q Eb  1    



N0



– For antipodal signals =-1
– For orthogonal signals =0
ECE 6640
 2  Eb
Pb  Q
 N0




 Eb 

Pb  Q

 N0 
48
Bit Error Rate Probability
• The reference comparisons for all
digital communication systems.
– The focus on Eb/N0
– The interest in antipodal or
orthogonal
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
49
The Correlation Receiver
• A receiver typically operates on the time domain
waveform r(t). Therefore computation of the time domain
is needed based on options.
– (1) Resolve the basis components to determine the r vector.

r j   r t    j t   dt ,
1 j  N

– (2) Directly correlate r(t) with every possible sm(t)

r , sm   r t   sm t   dt ,
1 m  M

mˆ  arg maxnm  r , sm
1 m  M
ECE 6640

50
Correlation with Basis Set

r j   r t    j t   dt ,

ECE 6640
1 j  N
mˆ  arg maxnm  r , sm
1 m  M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.

51
Correlation with Signal

r , sm   r t   sm t   dt ,

ECE 6640
1 m  M
mˆ  arg maxnm  r , sm
1 m  M
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.

52
Matched Filter
• Matched Filter
hm t   u t   sm L  T  t 
*
t
z m t   r t   hm t    r    hm t     d

• For the correct “matched” received symbol m
t
z m t    sm    sm L  T  t     d
*
0
z m L  T  
LT
 sm   sm L  T  L  T    d 
0
*
LT
 sm   sm   d 
0
*
LT

sm    d
2
0
Note: LT used for filter length.
ECE 6640
53
Optimum binary detection
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) parallel matched filters (b) correlation detector: Figure 14.2-3
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
54
Frequency Domain Matched Filter
• The equivalent noise bandwidth of the “matched filter”
must be known.
Note: Single symbol SNR,
not a detection region
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
55
Bandpass binary receiver
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.2-1
• Using superposition of the “parallel matched filters”, the
BPF is the difference of the two filters.
hBPF t   h1 t   h0 t 
• This results in an optimal binary detector
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
56
Binary Receiver
hBPF t   h1 t   h0 t 
• OOK
h1 t   K  s1 T  t 
h0 t   K  s0 T  t 
hBPF t   h1 t   K  s1 T  t   cos2  f c  T  t 
• BPSK
hBPF t   h1 t   h0 t   2  h1 t   2  K  s1 T  t   cos2  f c  T  t 
• BFSK
hBPF t   h1 t   h0 t   K  s1 T  t   K  s0 T  t 
hBPF t   cos2   f c  f d   T  t   cos2   f c  f d   T  t 
hBPF t   2  sin 2  f c  T  t   sin 2  2  f d  T  t 
57
Correlation receiver for
OOK or BPSK
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.2-4
• Since both optimal filters consist of cosine waveforms,
mix and integrate instead of filter an optimally sample.
– Note that the integrator can be a rectangular window filter that is
optimally sampled. (Provides functionality near synchronization as
well.)
Notes and figures are based on or taken the course textbook:
A. Bruce Carlson, P.B. Crilly, “Communication Systems, 5th ed.”, McGraw-Hill, 2010.
58
Optimal Parallel Matched Filter Receiver
Error Analysis
2
 z1  z0 


2



 max
T

2
E   s1 t   s0 t   dt 

 0
2  N0
• Evaluating the expected value


T

T

T
T
2
2
2
E   s1 t   s0 t   dt   E   s1 t   dt   2  E   s1 t   s0 t   dt   E   s0 t   dt 


0

0

0
0

T
2
E   s1 t   s0 t   dt   E1  2  E10  E0

0
Eb  E1  E0  2
2
2  Eb  2  E10 Eb  E10
 z1  z0 


 
2
2
N
N0




 max
0
59
Optimal Parallel Matched Filter Receiver
Error Analysis
E10    Eb 

T
Eb
 E   s1 t   s0 t   dt 
E1  E0

0
2
• OOK
• PSK
• FSK
E10  0
Eb
 z1  z0 



 2    max N 0
E10   1  Eb
2  Eb
 z1  z0 



N0
 2    max
E10  0
2
2
E
 z1  z0 

  b
 2    max N 0
60
Generalized Probability of Error
• Using the optimal BPF filter and sampling for
each symbol, the relationship will be based on:
Eb  E10 Eb  1  
 z1  z0 




N0
N0
 2    max
2
• The BER is then based on
 E  1   
z  z 
Pe  Q  1 0   Q  b

N
 2 
0


• Therefore picking arbitrary symbols is possible,
but the symbol correlation coefficient will drive
the BER performance.
61
Generalized FSK
s0 t   Ac  cos2   f c  f d   t  
s1 t   Ac  cos2   f c  f d   t  

T
E10  Ac  E   cos2   f c  f d   t  cos2   f c  f d   t   dt 

0
2 T
A
E10  c   cos2  2 f c  t   cos2  2 f d  t  dt
2 0
2
T
E 1
E10  b    expi 2  2 f d  t   exp i 2  2 f d  t  dt
T 2 0
  Eb 
Eb 1  expi 2  2 f d  T  exp i 2  2 f d  T  
 


T 2 
i 2  2 f d
i 2  2 f d

 f 
Eb sin 2  2 f d  T 

 Eb  sinc4  f d  T   Eb  sinc 4  d 
T
2  2 f d
 rb 
k
2 f d  f step 
There are multiple “orthogonal” tone separations.
2T
  Eb 
•
•
The correlation coefficient can go negative! The minimum occurs at
approximately sinc(1.22) = -0.166
62
MATLAB Coherent Receivers
• BASK example code
• BPSK example code
• BFSK example code
ECE 6640
63
4.2-3 Bounds on Probability of Error
• For multiple symbol constellation, it is important to be able
to estimate and/or bound the expected BER performance.
• The text approach is to develop a union bound of error for
maximum likelihood detection.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
64
Union Bound (cont)
•
For equally likely symbols
•
In AWGN
•
For very few constellations, decision regions Dm’ are regular enough that the
integrals in the last line of Equation 4.2–63 or Equation 4.2–62 can be
computed in a closed form.
For most constellations these integrals cannot be put in a closed form. In such
cases it is convenient to have upper bounds for the error probability.
•
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
65
Union Bound (cont)
•
•
There exist many bounds on the error probability under ML detection. The
union bound is the simplest and most widely used bound which is quite tight
particularly at high signal-to-noise ratios.
First we note that in general the decision region Dm’ under ML detection can
be expressed as
Note: m are true
symbols, m’ are
‘generalized’ symbol
(worse case?)
•
Define
•
Note that Dmm’ is the decision region for m’ in a binary equiprobable system
with signals sm and sm’. Note that the decision regions of Dm’ are a subset of
Dmm’.
•
Therefore
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
66
Union Bound (cont)
•
For
•
•
Note that the right-hand side of this equation is the error probability of a
binary equiprobable system with signals sm and sm’ when sm is transmitted.
We define the pairwise error probability, denoted by Pm→m’ as
•
The derivation proceeds as
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
67
Union Bound (cont)
•
Equations 4.2–70 is the union bound for a general communication channel.
•
In the special case of an AWGN channel, we know from Equation 4.2–37 that
the pairwise error probability is given by
•
•
For Q bounded as
This becomes
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
68
Union Bound (cont)
•
•
The derived union bound is uses the distance between symbols in the
constellation to determine the bit error bound.
While a more exact expression will use specific “enumerated” distances,
approximations may either use the minimum distance or functions base on the
enumerated distances.
Let us define dmin, the minimum distance of a constellation, as
•
Substituting in Equation 4.2–70 results in
•
Equation 4.2–78 is a looser form of the union bound in terms of the Q function
and dmin which has a very simple form.
•
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
69
Union Bound (cont)
•
The union bound clearly shows that the minimum distance of a
constellation has an important impact on the performance of the
communication system. A good constellation should be designed such
that, within the power and bandwidth constraints, it provides the
maximum possible minimum distance; i.e., the points in the
constellation should be maximally separated.
•
In Summary
•
If Q is approximated as
•
This becomed
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
70
Example 4.2-2: 16 QAM
Close observation of this constellation shows that from
a total of 16 × 15 = 240 possible distances between any
two points in the constellation, 48 are equal to dmin,
36 are equal to √2 dmin, 32 are 2dmin, 48 are √5 dmin,
16 are √ 8 dmin, 16 are 3dmin, 24 are √ 10 dmin, 16 are
√13 dmin, and finally 4 are √18 dmin. Note that each
line connecting any two points in the constellation is
counted twice.
Taking just the minimum distance (term 1 in T(X))
assuming a high enough SNR to ignore the other terms
and using the Q(x) approx,
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
71
Example 4.2-2: 16 QAM
BER vs. Eb/No
The bound derived was
The dmin bound previoulsy derived as a
simplified form of the union bound
The exact expression (derived in Example
4.3-1) is
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
72
Probability Lower Bound
• See the derivation in the textbook.
• The result suggested is
– where Nmin is the number of the points in the constellation that are
at the distance from dmin from at least one other point in the
constellation.
• If we take the conservative upper bound and this lower
bound, the probability of error is bounded as
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
73
4.3 Optimal Detection and Error
Probability for Band-Limited Signaling
• These are for “lower bandwidth”, low dimensionality
signaling types.
• This section is an excellent reference for some of the
primary signal types discussed.
• Explicit BER vs. Eb/No equations are derived based on the
previous material presented.
– An assumption of equally likely symbols is made for each
derivation.
– 4.3-1 Derives ASK or PAM
– 4.3-2 Derives MPSK
– 4.3-3 Derives QAM
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
74
MASK Summary
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
75
MASK
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
76
MASK Performance
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
77
MPSK
•
The marginal probability density
function for a symbol can be defined as
•
The pdf is a function of the average
symbol energy
•
The higher the number of symbols, the
tighter the symbol decision regions must
become and more errors can be
expected.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
78
MPSK
• In general, the integral of p(θ) does not reduce to a simple
form and must be evaluated numerically, except for M = 2
and M = 4.
• For M=2
• For M=4
• For other M (M large and SNR large)
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
79
MPSK
M=2
M=4
M other
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
80
QAM
QAM is dependent upon the symbol
constellation selected.
• Default to square constellations
of 4, 16, 64, & 256
• Numerous others are possible
with potentially better system
performance
• The optimal detector uses 2 basis
symbols to resolve the in-phase
and quadrature components
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
81
Square Constellation QAM
• This case appears as two dimensional ASK/PAM
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
82
Square Constellation QAM
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
83
Comparing QAM and MPSK
• Looking at the ratio of the Q(x) arguments
– At M=4 the systems are equivalent, but for higher M QAM has
better performance.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
84
Demodulation and Detection of BandLimited Signals
• Matched filter involve the basis form of the signals.
Note: Filter are
matched to basis,
not matched to
symbols!
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
85
4.4 Optimal Detection and Error
Probability for Power-Limited Signaling
• These are for “wider bandwidth”, higher dimensionality
signaling types.
• BER vs. Eb/No equations are derived based on the
previous material presented.
– An assumption of equally likely symbols is made for each
derivation.
– 4.4-1 Orthogonal FSK
– 4.4-2 Biorthogonal
– 4.4-3 Simplex
ECE 6640
86
Orthogonal Signals - MFSK
• For equiprobable, equal-energy orthogonal signals, the
optimum detector selects the signal resulting in the largest
cross-correlation between the received vector r and each of
the M possible transmitted signal vectors {sm}, i.e.,
d min  2  E
ECE 6640
87
Orthogonal Signals (cont)
•
The probability of correct symbol detection can be described as
•
assuming Gaussian noise elements where the elements are independent and
identically distributed (IID)
•
with an individual dimension represented as
ECE 6640
88
Orthogonal Signals (cont)
•
The integral becomes
•
The error is the complement, therefore
•
In general, Equation 4.4–10 cannot be made simpler, and the error probability
can be found numerically for different values of the SNR.
To determine bit errors, let us assume that s1 corresponds to a data sequence of
length k with a 0 at the first component. The probability of an error at this
component is the probability of detecting an sm corresponding to a sequence
with a 1 at the first component. Since there are 2k−1 such sequences, we have
•
ECE 6640
89
FSK Signaling
Another Union Bound
•
The orthogonal case is easier than the previous derivation as all symbols have
the minimum distance. Taking the result
•
For orthogonal signaling
•
Using M = 2k and Eb = E/k, we have
•
Note that if
•
Then Pe  0 as k  ∞ (not Pe  infinite like the text says!!)
ECE 6640
d min  2  E
Note: a necessary and sufficient
condition for reliable comm. that is
slightly lower is derived in Chap. 6.
It is called the Shannon limit.
90
Orthogonal Signaling
ECE 6640
91
4.4-2 Biorthogonal Signaling
d min  2  E
d other  2  E
ECE 6640
92
Biorthogonal Signaling
ECE 6640
93
ECE 6640
94
COMMENT ON SIMULATIONS
ECE 6640
95
Correlation Receiver Design
•
Maximizing Design (maximum symbol correlation)
– Create a matched filter based on every one of the symbols being
transmitted.
*
sm L  T    , for 1  m  M
– Determine the correct symbol sampling time. (Proportional to LT)
– The maximum of the sampled correlates is selected as the output symbol.
max s t   nt   sm L  T  t 
*
1 m  M
•
@ t  k  T  filter _ delay
Minimizing Design (minimum distance from signal to symbol)
– Create a matched filter based om every one of the basis symbol set.
*
rn t   s t   nt   n t  ,
for 1  n  N
– Determine the correct symbol sampling time.
– Find the symbol vector, sm, that is a minimum distance.
ECE 6640
min rn k   sm
1 m  M
for t  k  T  filter _ delay
96
Design Considerations
• If the dimensionality of the symbols is small, the
minimization approach using the basis set for the symbols
will require less hardware/processing.
– For ASK N=1
– For PSK N=2
• If the dimensionality of the symbols is large, the
maximization approach using the matched filter of the
symbols will be a likely choice
– For FSK N=M
ECE 6640
97
Matlab Minimization for MPSK
% RECEIVER ---------------------------------------------% Process the array of symbol plus noise values
% Detect the correlator output using an optimal threshold
corrc= (ccarrier/sqrt(NSampSym))'*symbol_noise;
corrs=-(scarrier/sqrt(NSampSym))'*symbol_noise;
csym = corrc + sqrt(-1)*corrs;
Note: Correlator
perfectly matched to
symbol time
% map to the closest symbol for the symbol detection/decision
% The first column is noise only
[ee ind]=min(abs(sym_map-csym(1)));
sd(:,1)= [real(sym_map(ind));imag(sym_map(ind))];
ECE 6640
98
Matlab Maximization for MFSK
% RECEIVER ---------------------------------------------Note: Correlator
% Process the array of symbol plus noise values
perfectly matched to
% Detect the correlator output using an optimal threshold
symbol time
for nnn=1:M
FcorrM=Fstart+(2*(nnn-1)*Fstep);
corrsym=sqrt(2)*(-1)^(n-1)*sin(2*pi*(FcorrM)*time_symbol)/sqrt(NSampSym);
corr(nnn,:) = corrsym'*symbol_noise;
end
[val received_sym]=max(corr);
bits_rcvd=Sym_bits(received_sym',:)';
ECE 6640
99
Matlab Matched Filter for
MASK, QAM & MPSK
%%
% Symbol plus scaled filter and SNR noise
RX_samples = TX_samples + sqrt(1/SNR)*input_noise;
%%
% Received signal baseband processing
%
% firrcos matched filter receiver
pred_samples =filter(hsqnyq,1,RX_samples);
…
% optimal symbol time sampling
pred_sym = pred_samples(Tdelay-1:expand:end);
num_pred_sym = length(pred_sym);
…
ECE 6640
%%
% Symbol and Bit Error Analysis
%
% map to the closest symbol for the symbol detection/decision
eepower = 0;
sd = zeros(num_pred_sym,2);
ygsym = zeros(num_pred_sym,1);
% minimum distance computations
for jj=1:num_pred_sym
[ee,ind]=min(abs(complex_const_map/sigma_s-pred_sym(jj)));
sd(jj,:)= [real(complex_const_map(ind)) imag(complex_const_map(ind))];
ygsym(jj) = ind-1;
eepower = eepower + ee^2;
end
100
Useful Matlab Study
• Compare BER performance based on using different
filters.
– Using different forms of matched filters.
• rectangular, square-root Nyquist, truncated sinc functions, etc.
• must have appropriate nulls to minimize ISI
– Using “unmatched” filters.
• rect TX with Nyquist filter receive
• Section 4.5 is on Non-coherent Detection
ECE 6640
101