II. Modulation & Coding
Design Goals of Communication Systems
1. Maximize transmission bit rate
2. Minimize bit error probability
3. Minimize required transmission power
4. Minimize required system bandwidth
5. Minimize system complexity, computational load & system
cost
6. Maximize system utilization
© Tallal Elshabrawy
2
Some Tradeoffs in M-PSK Modulaion
0
10
BPSK,QPSK
8 PSK
16 PSK
2
-1
10
1
-2
Pb
10
m=4
-3
10
m=3
m=1, 2
3
-4
10
0
2
4
6
8
10
12
14
16
18
Eb/N0
1
2
3
Trades off BER and Energy per Bit
Trades off BER and Normalized Rate in b/s/Hz
Trades off Normalized Rate in b/s/Hz and Energy per Bit
© Tallal Elshabrawy
3
Shannon-Hartley Capacity Theorem
System Capacity for communication over of an
AWGN Channel is given by:
 S
C  W  log2  1  
 N
C:
W:
S:
N:
© Tallal Elshabrawy
System Capacity (bits/s)
Bandwidth of Communication (Hz)
Signal Power (Watt)
Noise Power (Watt)
4
Normalized Channel Capacity C/W (b/s/Hz)
Shannon-Hartley Capacity Theorem
16
8
Unattainable
Region
4
2
Practical
Systems
1
1/4
1/8
-10
© Tallal Elshabrawy
0
10
20
SNR
30
40
50
5
Shannon Capacity in terms of Eb/N0
Consider transmission of a symbol over an AWGN channel
S
E
ER
T
S
S
S
S
NN W
0
 E R 
C  W  log 1 

 N W 
S
S
2
0
E R  mE R  E C
S
S
b
S
b
 E C 
C

 log 1 

W
 N W 
b
2
0
© Tallal Elshabrawy
6
Shannon Limit
 E C 
C
 log 1 

W
 N W 
b
2
0
E C 
Let x   
N W 
b
0
x
 log 1  x
E
N


E 1
log 1  x
N x

2
b
1

0
b
2

0
E
1
log 1  x
N
b
2


1
x
0
1
x
x  0  1  x   e
E
1

 0.693  1.6dB
N log e
b
0
© Tallal Elshabrawy
7
Shannon Limit
Mormalized Channel Capacity b/s/Hz
16
8
4
2
1
1/2
1/4
1/8
1/16
-2
0
2
4
Shannon Limit=-1.6 dB
© Tallal Elshabrawy
6
8
10
12
14
16
18 20
Eb/N0
22
24
26
28
30
32
34
36
38
40
Shannon Limit
No matter how much/how smart you decrease the
rate by using channel coding, it is impossible to
achieve communications with very low bit error
rate if Eb/N0 falls below -1.6 dB
© Tallal Elshabrawy
Shannon Limit
Room for improvement by channel coding
Normalized Channel Capacity b/s/Hz
16
8
4
8 PSK
Uncoded
Pb=10-5
QPSK
Uncoded
Pb = 10-5
2
1
16 PSK
Uncoded
Pb=10-5
BPSK
Uncoded
Pb = 10-5
1/2
1/4
1/8
1/16
-4
-2
0
2
4
Shannon Limit=-1.6 dB
© Tallal Elshabrawy
6
8
10
12
14
16
18
20
Eb/N0
22
24
26
28
30
32
34
36
38
40
1/3 Repetition Code BPSK
-1
10
BPSK Uncoded
BPSK 1/3Repetition Code
-2
10
-3
Pb
10
-4
10
Coding Gain= 3.2 dB
-5
10
-6
10
0
1
2
3
4
5
Eb/N0
6
7
8
9
10
Is this really purely a gain?
No! We have lost one third of the information transmitted rate
© Tallal Elshabrawy
11
1/3 Repetition Code 8 PSK
-1
10
BPSK Uncoded
8 PSK 1/3 Repitition Code
-2
10
-3
Pb
10
-4
10
-5
10
Coding Gain= -0.5 dB
-6
10
0
1
2
3
4
5
Eb /N0
6
7
8
9
10
When we don’t sacrifice information rate 1/3 repetition codes did not help
us
© Tallal Elshabrawy
12
Hard Decision Decoding
v
v = [v1 v2 … vi … vn]
e = [e1 e2 … ei … en]
r = [r1 r2 … ri … rn]
x = [x1 x2 … xi … xn]
y = [y1 y2 … yi … yn]
r
Channel
e
v
Channel
Encoder
Waveform
Generator
x
Channel
y
Waveform r
Detection
+1 V.
vi
vi=1
vi=0
0
T
0
T
xi
-1 V.



+
zi  ]-∞, ∞[
yi
0
yi>0
yi<0
Channel
Decoder
ri=1
ri=0
The waveform generator converts binary data to voltage levels (1 V., -1 V.)
The channel has an effect of altering the voltage that was transmitted
Waveform detection performs a HARD DECISION by mapping received
voltage back to binary values based on decision zones
© Tallal Elshabrawy
ri
Soft Decision Decoding
v
Channel
v = [v1 v2 … vi … vn]
e = [e1 e2 … ei … en]
r = [r1 r2 … ri … rn]
x = [x1 x2 … xi … xn]
r
e
v
Channel
Encoder
Waveform
Generator
x
r
Channel
Channel
Decoder
+1 V.
vi
vi=1
vi=0
0
T
0
T
xi
-1 V.



+
ri
zi  ]-∞, ∞[
The waveform generator converts binary data to voltage levels (1 V., -1 V.)
The channel has an effect of altering the voltage that was transmitted
The input to the channel decoder is a vector of voltages rather than a vector
of binary values
© Tallal Elshabrawy
Hard Decision: Example 1/3 Repetition Code BPSK
v
v = [v1 v2 … vi … vn]
e = [e1 e2 … ei … en]
r = [r1 r2 … ri … rn]
x = [x1 x2 … xi … xn]
y = [y1 y2 … yi … yn]
r
Channel
e
Channel
Encoder
0
Waveform
Generator
000
-1 -1 -1
Channel
y
Waveform r
Detection
0.1 -0.9 0.1
Channel
Decoder
101
Hard Decision
Each received bit is detected individually
If the voltage is greater than 0 detected bit is 1
If the voltage is smaller than 0 detected bit is 0
Detection information of neighbor bits within the same codeword is
lost
© Tallal Elshabrawy
1
Soft Decision: Example 1/3 Repetition Code BPSK
v
Channel
v = [v1 v2 … vi … vn]
e = [e1 e2 … ei … en]
r = [r1 r2 … ri … rn]
x = [x1 x2 … xi … xn]
y = [y1 y2 … yi … yn]
r
e
Channel
Encoder
0
Waveform
Generator
000
-1 -1 -1
r
Channel
0.1 -0.9 0.1
Channel
Decoder
Accumulated Voltage =
0.1-0.9+0.1=-0.7<0
0
Soft Decision
If the accumulated voltage within the codeword is greater than 0
detected bit is 1
If the accumulated voltage within the codeword is smaller than 0
detected bit is 0
Information of neighbor bits within the same codeword contributes to
the channel decoding process
© Tallal Elshabrawy
1/3 Repetition Code BPSK Soft Decision
b ∈ {0,1}



 

c  000,111 s    Eb ,  Eb ,  Eb  ,  Eb , Eb , Eb 
Channel Coding
(1/3 Repetition Code)
Waveform
Representation
n = [n1 , n2 , n3 ]
Channel
Soft Decision
Decoding
b*
Important Note
Eb
N0
1 / 3 Re p. C ode =
© Tallal Elshabrawy
3Eb Eb
=
3N0 N0
Uncoded
r
BER Performance Soft Decision 1/3 Repetition Code BPSK
Select b*=0 if
f R b  0   f R b  1
f r0
r1 r2  b  0   f r0
r1 r2  b  1
Note that r0 r1 and r2 are independent and identically distributed. In other words
f  ri b  0  
1
2πσ 2
r 

i
e
Eb

2
2σ 2
Therefore
 1 
f  ri b  0   

2
2πσ


3
2
e
r 

i
Eb

2σ 2
i0
Similarly
 1 
f  ri b  1  

2
2πσ


© Tallal Elshabrawy
3
2
e
i0
r 

i
Eb
2σ 2

2
2
BER Performance Soft Decision 1/3 Repetition Code BPSK
Select b*=0 if
f (R b = 0) > f (R b = 1)
 1 


2
 2πσ 
3
2
e

r +
i
2
Eb

 ri + Eb
i0
2
r  0
i0

2
Eb

2
2σ 2
i
© Tallal Elshabrawy
2
 
r 
i 0

2
 1 


2
 2πσ 

 2
ri  Eb 

  ln  e 2σ2



 i 0


2
2σ 2
i 0
i
i0
 2
ri  Eb 

2
ln   e 2σ
 i0


2
r 

2

i
Eb
2σ 2
  ri  Eb
i0

2

2
2





3
2
e
i0
r 

i
Eb
2σ 2

2
BER Performance Soft Decision 1/3 Repetition Code BPSK
 2

Pr error b = 0  = Pr  ri > 0 b = 0 
 i 0



 2

Pr error b = 0  = Pr   Eb  ni  0 
 i0

 2

Pr error b = 0  = Pr  ni  3 Eb 
 i 0

Pr error b = 0  = Pr n  3 Eb 


where
2
n   ni
i0
n is Gaussian distributed with mean 0 and variance 3N0/2
 3 Eb 
Pr error b = 0  = Q 

 3N / 2 
0


© Tallal Elshabrawy
Pr error b  0  
 3Eb 
1
erfc 
 N 
2
0 

Hard Vs Soft Decision: 1/3 Repetition Code BPSK
0
10
BPSK Uncoded
BPSK 1/3 Repitition Code Hard Decision
BPSK 1/3 Repetition Code Soft Decision
-1
10
-2
Pb
10
-3
10
-4
10
Coding Gain= 4.7 dB
-5
10
-6
10
0
© Tallal Elshabrawy
1
2
3
4
5
Eb/N0
6
7
8
9
10
1/3 Repetition Code 8 PSK Hard Decision
0
10
BPSK Uncoded
8PSK 1/3 Repetition Code Hard Decision
8PSK 1/3 Repetition Code Soft Decision
-1
10
-2
Pb
10
-3
10
-4
10
Coding Gain= 1.5 dB
-5
10
-6
10
0
© Tallal Elshabrawy
1
2
3
4
5
Eb /N0
6
7
8
9
10
22
Shannon Limit and BER Performance
Normalized Channel Capacity b/s/Hz
16
8
4
QPSK Uncoded
Pb = 10-5
8 PSK Uncoded 16 PSK Uncoded
Pb=10-5
Pb=10-5
8PSK 1/3 Rep.
Code Soft Decision
Pb = 10-5
2
1
8PSK 1/3 Rep. Code
Hard Decision
Pb = 10-5
BPSK Uncoded
Pb = 10-5
1/2
1/3
1/4
1/8
BPSK 1/3 Rep. Code
Sodt Decision
Pb = 10-5
1/16
-4
-2
0
2
4
Shannon Limit=-1.6 dB
© Tallal Elshabrawy
6
BPSK 1/3 Rep. Code
Hard Decision
Pb = 10-5
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
Eb/N0
23