1. II. Modulation & Coding

2. Design Goals of Communication Systems
Maximize transmission bit rate
Minimize bit error probability
Minimize required transmission power
Minimize required system bandwidth
Minimize system complexity, computational load & system cost
Maximize system utilization

3. Some Tradeoffs in M-PSK Modulaion2 1
m=4
m=3
m=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

4. Shannon-Hartley Capacity Theorem
System Capacity for communication over of an AWGN Channel is given by:
C: System Capacity (bits/s) W: Bandwidth of Communication (Hz) S: Signal Power (Watt) N: Noise Power (Watt)

5. Shannon-Hartley Capacity Theorem
Unattainable
Region
Practical
Systems

6. Shannon Capacity in terms of Eb/N0
Consider transmission of a symbol over an AWGN channel

7. Shannon Limit
Let

8. Shannon Limit
Shannon Limit=-1.6 dB

9. 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. Shannon Limit Room for improvement by channel coding
16 PSK Uncoded
Pb=10-5
8 PSK Uncoded
Pb=10-5
QPSK Uncoded Pb = 10-5
Normalized Channel Capacity b/s/Hz
BPSK Uncoded
Pb = 10-5
Shannon Limit=-1.6 dB
Eb/N0 10

11. 1/3 Repetition Code BPSK Is this really purely a gain?
Coding Gain= 3.2 dB
Is this really purely a gain?
No! We have lost one third of the information transmitted rate

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

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

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

15. Hard Decision: Example 1/3 Repetition Code BPSK
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] y r
Channel Encoder
Waveform
Generator
Waveform
Detection
Channel Decoder
Channel
0 0 0
1 0 1 1
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

16. Soft Decision: Example 1/3 Repetition Code BPSK
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 Encoder
Waveform
Generator
Channel Decoder
Channel
0 0 0
Accumulated Voltage = =-0.7<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

17. 1/3 Repetition Code BPSK Soft Decision
Channel Coding
(1/3 Repetition Code)
Waveform
Representation
Channel r
Soft Decision
Decoding
Important Note

18. 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

19. BER Performance Soft Decision 1/3 Repetition Code BPSK
Select b*=0 if

20. BER Performance Soft Decision 1/3 Repetition Code BPSK
where
n is Gaussian distributed with mean 0 and variance 3N0/2

21. Hard Vs Soft Decision: 1/3 Repetition Code BPSK
Coding Gain= 4.7 dB

22. 1/3 Repetition Code 8 PSK Hard Decision1 2 3 4 5 6 7 8 9 10 -6 -5 -4 -3 -2 -1 E b /N P
BPSK Uncoded
8PSK 1/3 Repetition Code Hard Decision
8PSK 1/3 Repetition Code Soft Decision
Coding Gain= 1.5 dB 22

23. Shannon Limit and BER Performance
8 PSK Uncoded
Pb=10-5
16 PSK Uncoded
Pb=10-5
QPSK Uncoded Pb = 10-5
8PSK 1/3 Rep. Code Soft Decision
Pb = 10-5
8PSK 1/3 Rep. Code Hard Decision
Pb = 10-5
Normalized Channel Capacity b/s/Hz
BPSK Uncoded
Pb = 10-5
1/3
BPSK 1/3 Rep. Code Sodt Decision
Pb = 10-5
BPSK 1/3 Rep. Code Hard Decision
Pb = 10-5
Shannon Limit=-1.6 dB
Eb/N0 23