modulation

1. II. Modulation & Coding

2. © Tallal Elshabrawy
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
2

3. © Tallal Elshabrawy
Some Tradeoffs in M-PSK Modulaion
1 Trades off BER and Energy per Bit
2 Trades off BER and Normalized Rate in b/s/Hz
3 Trades off Normalized Rate in b/s/Hz and Energy per Bit
3
0 2 4 6 8 10 12 14 16 18
10
-4
10
-3
10
-2
10
-1
10
0
Eb
/N0
P
b
BPSK,QPSK
8 PSK
16 PSK
1
2
3
m=4
m=3
m=1, 2

4. © Tallal Elshabrawy
Shannon-Hartley Capacity Theorem
C: System Capacity (bits/s)
W: Bandwidth of Communication (Hz)
S: Signal Power (Watt)
N: Noise Power (Watt)
4
2
S
C W log 1
N
 
  
 
 
System Capacity for communication over of an
AWGN Channel is given by:

5. © Tallal Elshabrawy
Shannon-Hartley Capacity Theorem
5
-10 0 10 20 30 40 50
1/8
1/4
1
2
4
8
16
SNR
Normalized
Channel
Capacity
C/W
(b/s/Hz)
Practical
Systems
Unattainable
Region

6. © Tallal Elshabrawy
Shannon Capacity in terms of Eb/N0
Consider transmission of a symbol over an AWGN channel
6
C  W log2
1
ES
RS
N0
W






S 
ES
TS
 ES
RS
N  N0
W
ES
RS
 mEb
RS
 Eb
C

C
W
 log2
1
Eb
N0
C
W







7. © Tallal Elshabrawy
Shannon Limit
7
C
W
 log2
1
Eb
N0
C
W






x
Eb
N0
 log2
1  x
 
1 
Eb
N0
1
x
log2
1  x
 
1 
Eb
N0
log2
1  x
 
1
x
x  0  
1
x
1 x e
  
Eb
N0

1
loge
 0.693  1.6dB

x 
Eb
N0
C
W






Let

8. © Tallal Elshabrawy
Shannon Limit
-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
1/4
1/2
1
2
4
8
16
1/8
1/16
Eb
/N0
Mormalized
Channel
Capacity
b/s/Hz
Shannon Limit=-1.6 dB

9. © Tallal Elshabrawy
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

10. © Tallal Elshabrawy
-4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
1/16
1/8
1/4
1/2
1
2
4
8
16
Shannon Limit
Shannon Limit=-1.6 dB
BPSK
Uncoded
Pb = 10-5
QPSK
Uncoded
Pb = 10-5
8 PSK
Uncoded
Pb=10-5
16 PSK
Uncoded
Pb=10-5
Room for improvement by channel coding
Normalized
Channel
Capacity
b/s/Hz
Eb/N0

11. © Tallal Elshabrawy
1/3 Repetition Code BPSK
Is this really purely a gain?
No! We have lost one third of the information transmitted rate
11
0 1 2 3 4 5 6 7 8 9 10
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
Eb
/N0
P
b
BPSK Uncoded
BPSK 1/3Repetition Code
Coding Gain= 3.2 dB

12. © Tallal Elshabrawy
0 1 2 3 4 5 6 7 8 9 10
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
Eb
/N0
P
b
BPSK Uncoded
8 PSK 1/3 Repitition Code
1/3 Repetition Code 8 PSK
12
Coding Gain= -0.5 dB
When we don’t sacrifice information rate 1/3 repetition codes did not help
us

13. © Tallal Elshabrawy
 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
Channel
e
r
v
Channel
Encoder
Waveform
Generator
Waveform
Detection
Channel
Decoder
Channel
v r
x y
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]
0 T
0 T
+1 V.
-1 V.
vi
vi=1
vi=0
xi
0
yi>0
yi<0
ri=1
ri=0
ri
+
zi ]-∞, ∞[
yi
Hard Decision Decoding

14. © Tallal Elshabrawy
 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
Channel
e
r
v
Channel
Encoder
Waveform
Generator
Channel
Decoder
Channel
v r
x
v = [v1 v2 … vi … vn]
e = [e1 e2 … ei … en]
r = [r1 r2 … ri … rn]
x = [x1 x2 … xi … xn]
0 T
0 T
+1 V.
-1 V.
vi
vi=1
vi=0
xi
+
zi ]-∞, ∞[
ri
Soft Decision Decoding

15. © Tallal Elshabrawy
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
Channel
e
r
v
Channel
Encoder
Waveform
Generator
Waveform
Detection
Channel
Decoder
Channel
0 0 0
r
y
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]
0 -1 -1 -1 0.1 -0.9 0.1 1 0 1 1
Hard Decision: Example 1/3 Repetition Code BPSK

16. © Tallal Elshabrawy
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
Channel
e
r
v
Channel
Encoder
Waveform
Generator
Channel
Decoder
Channel
0 0 0
r
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]
0 -1 -1 -1 0.1 -0.9 0.1 0
Accumulated Voltage =
0.1-0.9+0.1=-0.7<0
Soft Decision: Example 1/3 Repetition Code BPSK

17. © Tallal Elshabrawy
1/3 Repetition Code BPSK Soft Decision
{ }
1
,
0
b ∈
Channel Coding
(1/3 Repetition Code)
 
c 000,111

Waveform
Representation
 
b b b b b b
s E , E , E , E , E , E
   
   
   
Channel
]
n
,
n
,
n
[
=
n 3
2
1
Soft Decision
Decoding
r
*
b
Uncoded
0
b
0
b
Code
.
p
Re
3
/
1
0
b
N
E
=
N
3
E
3
=
N
E
Important Note

18. © Tallal Elshabrawy
BER Performance Soft Decision 1/3 Repetition Code BPSK
Select b*=0 if
Note that r0 r1 and r2 are independent and identically distributed. In other words
Therefore
Similarly
   
f R b 0 f R b 1
  
 
   
 
0 1 2 0 1 2
f r r r b 0 f r r r b 1
  
 
 
2
i b
2
r E
2σ
i 2
1
f r b 0 e
2πσ


 
 
 
2
i b
2
3 r E
2
2σ
i 2
i 0
1
f r b 0 e
2πσ



 
   
 

 
 
2
i b
2
3 r E
2
2σ
i 2
i 0
1
f r b 1 e
2πσ



 
   
 


19. © Tallal Elshabrawy
Select b*=0 if
( ) ( )
1
=
b
R
f
>
0
=
b
R
f
   
2 2
i b i b
2 2
3 3
r E r E
2 2
2σ 2σ
2 2
i 0 i 0
1 1
e e
2πσ 2πσ
 
 
 
   

   
   
 
   
2 2
2 2
i b i b
2 2
i 0 i 0
r + E r E
2σ 2σ
 

  
 
   
2 2
2 2
i b i b
i 0 i 0
r + E r E
 
   
 
   
2 2
i b i b
2 2
r E r E
2 2
2σ 2σ
i 0 i 0
ln e ln e
 
 
 
   
   

   
   
   
 
2
i
i 0
r 0



BER Performance Soft Decision 1/3 Repetition Code BPSK

20. © Tallal Elshabrawy
where
BER Performance Soft Decision 1/3 Repetition Code BPSK
2
i
i 0
Pr error b = 0 =Pr r > 0 b = 0

 
   
 
 

 
2
b i
i 0
Pr error b = 0 =Pr E n 0

 
    
 
 
 

b
Pr error b = 0 =Pr n 3 E
 
  
   
2
i
i 0
n n

 
n is Gaussian distributed with mean 0 and variance 3N0/2
b
0
3 E
Pr error b = 0 = Q
3N / 2
 
   
   
 
b
0
3E
1
Pr error b 0 erfc
2 N
 
 
   
   
 
2
i b
i 0
Pr error b = 0 =Pr n 3 E

 
  
 
 
 


21. © Tallal Elshabrawy
Hard Vs Soft Decision: 1/3 Repetition Code BPSK
0 1 2 3 4 5 6 7 8 9 10
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Eb
/N0
P
b
BPSK Uncoded
BPSK 1/3 Repitition Code Hard Decision
BPSK 1/3 Repetition Code Soft Decision
Coding Gain= 4.7 dB

22. © Tallal Elshabrawy
1/3 Repetition Code 8 PSK Hard Decision
22
0 1 2 3 4 5 6 7 8 9 10
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Eb
/N0
P
b
BPSK Uncoded
8PSK 1/3 Repetition Code Hard Decision
8PSK 1/3 Repetition Code Soft Decision
Coding Gain= 1.5 dB

23. © Tallal Elshabrawy
-4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
1/16
1/8
1/4
1/2
1
2
4
8
16
Shannon Limit and BER Performance
23
Shannon Limit=-1.6 dB
BPSK Uncoded
Pb = 10-5
QPSK Uncoded
Pb = 10-5
8 PSK Uncoded
Pb=10-5
16 PSK Uncoded
Pb=10-5
BPSK 1/3 Rep. Code
Hard Decision
Pb = 10-5
BPSK 1/3 Rep. Code
Sodt Decision
Pb = 10-5
Normalized
Channel
Capacity
b/s/Hz
Eb/N0
1/3
8PSK 1/3 Rep. Code
Hard Decision
Pb = 10-5
8PSK 1/3 Rep.
Code Soft Decision
Pb = 10-5