1. Error Correction Code (2)
Fire Tom Wada
Professor, Information Engineering, Univ. of the Ryukyus
2018/11/27

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
2018/11/27

3. (n, k) Block Code it-2 it-1 it wt-2 wt-1 wt BLOCK k bit Information
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
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4. Convolutional Code it-2 it-1 it wt-2 wt-1 wt k bit Information code
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
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5. Simple Convolutional Coder
　D is delay operator
　c1t, c2t depends on not only current it bu also past it-1, it-2
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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
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7. State Transition Diagram
Number of State: 4
Time t 1 2 3 4 5 it
F1=it-1
F2=it-2
c1t
c2t
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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.
2018/11/27

9. 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
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10. Encoding in Trellis Input information determines a path 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
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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
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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
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13. R=2/3 example
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14. 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
2018/11/27

15. 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.
2018/11/27

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
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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)
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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
2018/11/27

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
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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
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21. 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!
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22. Soft Decision Use soft decision metric One Example LEVEL 1 2 3 4 5 6 71 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!
2018/11/27

23. Soft Decision Viterbi Calculate Branch Metric based on Soft-Table00 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’
2018/11/27

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
2018/11/27

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
2018/11/27

26. Summary 2 types of FEC Viterbi Decoder : Maximum likelihood decoding
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
2018/11/27