1. Error Correction Code (2)
Fire Tom Wada
Professor, Information Engineering, Univ. of the Ryukyus
2019/4/25
System Arch (Fire Tom Wada)

2. Two major FEC technologies
Reed Solomon code (Block Code)
Convolutional code
Serially concatenated code
Interleaving
Reed Solomon
Code
Convolution Code
ConcatenatedCoded
Source Information
Concatenated
Goes to Storage, Transmission
2019/4/25
System Arch (Fire Tom Wada)

3. System Arch 2008 (Fire Tom Wada)
(n, k) Block Code
BLOCK
it-2
k bit
it-1 it
Information
code
wt-2
n bit
wt-1 wt
ENCODE
Time t information block it is coded to time t code wt.
k : information length
n : code length
Code Rate : R=k/n
2019/4/25
System Arch (Fire Tom Wada)

4. System Arch 2008 (Fire Tom Wada)
Convolutional Code
it-2
k bit
it-1 it
Information
code
wt-2
n bit
wt-1 wt
Constraint length K
Time t code wt is determined by past K information.
K : Constraint length
Code Rate : R=k/n
2019/4/25
System Arch (Fire Tom Wada)

5. Simple Convolutional Coder
　D is delay operator
　c1t, c2t depends on not only current it bu also past it-1, it-2
2019/4/25
System Arch (Fire Tom Wada)

6. Simple Convolutional Coder(2)
Time t 1 2 3 4 5 it
F1=it-1
F2=it-2
c1t
c2t
If input information is “110010” then
Output code is “ ”.
Constraint length K=3, R=1/2
2019/4/25
System Arch (Fire Tom Wada)

7. State Transition Diagram
Number of State: 4
Time t 1 2 3 4 5 it
F1=it-1
F2=it-2
c1t
c2t
2019/4/25
System Arch (Fire Tom Wada)

8. State Transition Diagram(2)
Time t 1 2 3 4 5 it
F1=it-1
F2=it-2
c1t
c2t
t=0
t=5
t=1
t=3
t=2
In each time t, a traveler stays in one state.
And move to different state cycle by cycle.
2019/4/25
System Arch (Fire Tom Wada)

9. System Arch 2008 (Fire Tom Wada)
Trellis Diagram
Convert the State transition diagram to 2D diagram
Vertical axis : states
Horizontal axis : time
states
time 00 01 10 11
00/0
11/1
11/0
00/1
01/0
10/1
10/0
01/1
2019/4/25
System Arch (Fire Tom Wada)

10. System Arch 2008 (Fire Tom Wada)
Encoding in Trellis
Time t 1 2 3 4 5 it
F1=it-1
F2=it-2
c1t
c2t
Input information determines a path in Trellis
Each Branch outputs 2bit code.
t=0
t=1
t=2
t=3
t=4
t=5 00 01 10 11
00/0
11/1
11/0
00/1
01/0
10/1
10/0
01/1 11 11 11 01 10 10
2019/4/25
System Arch (Fire Tom Wada)

11. Punctured Convolutional Code
The example Encoder has Code Rate R=1/2
Punctured Convolutional Code means that one or some code output is removed.
Then Code Rate can be modified
2019/4/25
System Arch (Fire Tom Wada)

12. Merit of Punctured Code
Larger code rate is better.
Using Punctured technology,
When communication channel condition is good, weak error correction but high code rate can be chosen.
When communication channel condition is bud, strong error correction but low code rate can be chosen.
This capability can be supported by small circuit change. One coder or decoder can be used for several code rate addaptively
2019/4/25
System Arch (Fire Tom Wada)

13. System Arch 2008 (Fire Tom Wada)
R=2/3 example
2019/4/25
System Arch (Fire Tom Wada)

14. System Arch 2008 (Fire Tom Wada)
Viterbi Decode
Viterbi Decode is one method of Maximum likelihood decoding for Convolutional code.
Maximum likelihood decoding
Likelihood function is P(xi|r):
Probability of seding xi under the condition of receiving r
N message x1, x2, x3, …, xN x? r
Sender take one message and send it out!
Receiver find the maximum conditional Probability
2019/4/25
System Arch (Fire Tom Wada)

15. System Arch 2008 (Fire Tom Wada)
Viterbi Decode(2)
Branch metric is the Likelihood function of each branch.
-Ln{p(xk|ri)}
High possibility  small value
Example : Hamming distance
Path metric is the sum of branch metrics along the possible TRELLIS PATH.
2019/4/25
System Arch (Fire Tom Wada)

16. Viterbi Decoding Example
R=1/2, Receive
Calculate Each Branch metric (This time Hamming distance) 00 01 10 11
00(2)
11(0)
01(1)
10(1)
00(1)
11(1)
01(0)
10(2)
01(2)
10(0)
t=0
t=1
t=2
t=3
t=4
t=5
t=6
Rece- ived
2019/4/25
System Arch (Fire Tom Wada)

17. Viterbi Decoding Example(2)
Calculate Path metric in order to find minimum path metric path.
Until t=2
Lower path has smaller path metric,
Then Take it!
t=0
t=1
t=2
t=3
t=4
t=5
t=6 5 3 00
00(2) 00
00(1) 00
00(2) 00
00(1) 00
00(2) 00
00(1) 00 2
11(0)
11(1)
11(0)
11(1)
11(0)
11(1) 3 01 01
11(0) 01
11(1) 01
11(0) 01
11(1) 01 2
00(2)
00(1)
00(2)
00(1)
01(2)
01(1)
01(0)
01(1)
01(0) 10 10 10 10 10
10(0)
10(1)
10(2)
10(1)
10(2)
10(1)
10(2)
10(1)
10(2) 11 11 11 11 11
01(1)
01(0)
01(1)
01(0)
2019/4/25
System Arch (Fire Tom Wada)

18. Viterbi Decoding Example(3)
Calculate Path metric in order to find minimum path metric path.
Until t=6
t=0
t=1
t=2
t=3
t=4
t=5
t=6 3 2 2 3 3 00
00(2) 00
00(1) 00
00(2) 00
00(1) 00
00(2) 00
00(1) 00
11(0)
11(1)
11(0) 3
11(1)
11(0)
11(1) 3 3 2 2 01 01
11(0) 01
11(1) 01
11(0) 01
11(1) 01 2
00(2)
00(1)
00(2)
00(1) 2 3
01(2)
01(1)
01(0)
01(1)
01(0) 10 10 10 10 10
10(0)
10(1) 1
10(2)
10(1)
10(2) 2
10(1)
10(2)
10(1)
10(2) 11 11 11 11 11
01(1)
01(0)
01(1)
01(0) 1 1 2 2
2019/4/25
System Arch (Fire Tom Wada)

19. Viterbi Decoding Example(4)
Select Minimum Path Metric and get original information
In this example, two minimum path
Upper path :
Lower path :
If we increase the time, we might find ONLY ONE MINIMUM PATH. 00 01 10 11
00(2)
11(0)
01(1)
10(1)
00(1)
11(1)
01(0)
10(2)
01(2)
10(0)
t=0
t=1
t=2
t=3
t=4
t=5
t=6 1 2 1 1 2 2
2019/4/25
System Arch (Fire Tom Wada)

20. Received signal has many level
In the previous example, we have assumed the received sequence is
Usually, received signal is analog (Many Levels) such as
SIGNAL LEVEL 1 2 3 4 5 6 7
2019/4/25
System Arch (Fire Tom Wada)

21. System Arch 2008 (Fire Tom Wada)
Hard Decision
HARD DECISION LINE
SIGNAL LEVEL 1 2 3 4 5 6 7
Loosing Reliability Information by Hard Decision!
HARD
DECISION
OUTPUTS 1 1 1 1 1 1 1 1 1
Highly reliable 1
Low reliable 1
We have to distinguish!
2019/4/25
System Arch (Fire Tom Wada)

22. System Arch 2008 (Fire Tom Wada)
Soft Decision
Use soft decision metric
One Example
LEVEL 1 2 3 4 5 6 7
Branch Metric for ‘0’
Branch Metric for ‘1’
Difference -3 -2 -1
Reliable ‘1’
Reliable ‘0’ 00
00(branch metric = = 3)
LEVEL= 65
HOW TO COMPUTE BRANCH METRIC
No effect on Viterbi decoding!
Looks like Erased or Punctured!
2019/4/25
System Arch (Fire Tom Wada)

23. System Arch 2008 (Fire Tom Wada)
Soft Decision Viterbi
Calculate Branch Metric based on Soft-Table 00 01 10 11
00(1)
11(0)
01(1)
10(0)
00(2)
01(0)
10(2)
01(2)
00(3)
00(4)
11(1)
10(4)
t=0
t=1
t=2
t=3
t=4
t=5
t=6 65
LEVEL 63 54 36 66 27
LEVEL 1 2 3 4 5 6 7
Branch Metric for ‘0’
Branch Metric for ‘1’
2019/4/25
System Arch (Fire Tom Wada)

24. Soft Decision Viterbi(2)
Calculate Path metric in order to find minimum path metric path.
Until t=6 00 01 10 11
00(1)
11(0)
01(1)
10(0)
00(2)
01(0)
10(2)
01(2)
00(3)
00(4)
11(1)
10(4)
t=0
t=1
t=2
t=3
t=4
t=5
t=6 65
LEVEL 63 54 36 66 27 3 5 2 3 4 3 3 2 4 2 3 3 1 1 3 3
2019/4/25
System Arch (Fire Tom Wada)

25. Soft Decision Viterbi(3)
Select Minimum Path Metric and get original information
In this example : 00 01 10 11
00(1)
11(0)
01(1)
10(0)
00(2)
01(0)
10(2)
01(2)
00(3)
00(4)
11(1)
10(4)
t=0
t=1
t=2
t=3
t=4
t=5
t=6 65
LEVEL 63 54 36 66 27
2019/4/25
System Arch (Fire Tom Wada)

26. System Arch 2008 (Fire Tom Wada)
Summary
2 types of FEC
Block code such as RS
Convolutional Code
Code Rate
Punctured
Viterbi Decoder : Maximum likelihood decoding
Trellis
Hard Decision vs. Soft Decision
Branch Metric, Path Metric
2019/4/25
System Arch (Fire Tom Wada)