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Error Correction Code (2)

Error Correction Code (2). Fire Tom Wada Professor, Information Engineering, Univ. of the Ryukyus. Two major FEC technologies. Reed Solomon code (Block Code) Convolutional code Serially concatenated code. Interleaving. Reed Solomon Code. Convolution Code. ConcatenatedCoded.

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Error Correction Code (2)

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  1. Error Correction Code (2) Fire Tom Wada Professor, Information Engineering, Univ. of the Ryukyus System Arch 2008 (Fire Tom Wada)

  2. Two major FEC technologies • Reed Solomon code (Block Code) • Convolutional code • Serially concatenated code Interleaving ReedSolomon Code ConvolutionCode ConcatenatedCoded SourceInformation Concatenated Goes to Storage, Transmission System Arch 2008 (Fire Tom Wada)

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

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

  6. Simple Convolutional Coder(2) • If input information is “110010” then • Output code is “11 10 10 11 11 01”. • Constraint length K=3, R=1/2 System Arch 2008 (Fire Tom Wada)

  7. State Transition Diagram • Number of State: 4 System Arch 2008 (Fire Tom Wada)

  8. State Transition Diagram(2) t=4 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. System Arch 2008 (Fire Tom Wada)

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

  10. 00 00/0 00 00/0 00 00/0 00 00/0 00 00/0 00 00/0 00 11/1 11/1 11/1 11/1 11/1 11/1 01 01 01 01 01 01 11/0 11/0 11/0 11/0 00/1 00/1 00/1 00/1 01/0 01/0 01/0 01/0 01/0 10 10 10 10 10 10/1 10/1 10/1 10/1 10/1 10/0 10/0 10/0 10/0 11 11 11 11 11 01/1 01/1 01/1 01/1 Encoding in Trellis • Input information determinesa path in Trellis • Each Branch outputs 2bit code. t=0 t=1 t=2 t=3 t=4 t=5 11 11 11 01 10 10 System Arch 2008 (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 System Arch 2008 (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 System Arch 2008 (Fire Tom Wada)

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

  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 messagex1, x2, x3, …, xN x? r Sender take one message and send it out! Receiver find the maximum conditional Probability System Arch 2008 (Fire Tom Wada)

  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. System Arch 2008 (Fire Tom Wada)

  16. t=0 t=1 t=2 t=3 t=4 t=5 t=6 Rece-ived 11 10 11 01 11 01 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) 11(1) 11(0) 11(1) 01 01 01 01 01 01 11(0) 11(1) 11(0) 11(1) 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) Viterbi Decoding Example • R=1/2, Receive 11 10 11 01 11 01 • Calculate Each Branch metric (This time Hamming distance) System Arch 2008 (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 01 01 01 01 11(0) 11(1) 11(0) 11(1) 00(2) 00(1) 00(2) 00(1) 2 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) 0 System Arch 2008 (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 3 2 2 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) 11(1) 11(0) 11(1) 3 3 3 2 2 01 01 01 01 01 01 11(0) 11(1) 11(0) 11(1) 00(2) 00(1) 00(2) 00(1) 2 2 3 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) 1 2 10(1) 10(2) 10(1) 10(2) 11 11 11 11 11 01(1) 01(0) 01(1) 01(0) 0 2 1 1 2 System Arch 2008 (Fire Tom Wada)

  19. t=0 t=1 t=2 t=3 t=4 t=5 t=6 2 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) 11(1) 11(0) 11(1) 2 01 01 01 01 01 01 11(0) 11(1) 11(0) 11(1) 00(2) 00(1) 00(2) 00(1) 2 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) 1 10(1) 10(2) 10(1) 10(2) 11 11 11 11 11 01(1) 01(0) 01(1) 01(0) 0 Viterbi Decoding Example(4) • Select Minimum Path Metric and get original information • In this example, two minimum path • Upper path : 1 1 0 0 1 0 • Lower path : 1 1 1 1 1 1 • If we increase the time, we might find ONLY ONE MINIMUM PATH. 2 1 1 2 System Arch 2008 (Fire Tom Wada)

  20. SIGNAL LEVEL 6 5 6 3 5 4 3 6 6 6 2 7 7 6 5 4 3 2 1 0 Received signal has many level • In the previous example, we have assumed the received sequence is • 11 10 11 01 11 01 • Usually, received signal is analog (Many Levels) such as System Arch 2008 (Fire Tom Wada)

  21. SIGNAL LEVEL 6 5 6 3 5 4 3 6 6 6 2 7 7 6 5 4 3 2 1 0 Hard Decision HARD DECISION LINE Loosing Reliability Information by Hard Decision! HARD DECISION OUTPUTS 1 1 1 0 1 1 0 1 1 1 0 1 Highly reliable 1 Low reliable 1 We have to distinguish! System Arch 2008 (Fire Tom Wada)

  22. HOW TO COMPUTE BRANCH METRIC LEVEL= 65 00 00(branch metric = 1 + 2 = 3) Soft Decision • Use soft decision metric • One Example Reliable ‘1’ Reliable ‘0’ No effect on Viterbi decoding! Looks like Erased or Punctured! System Arch 2008 (Fire Tom Wada)

  23. t=0 t=1 t=2 t=3 t=4 t=5 t=6 LEVEL 65 63 54 36 66 27 00 00(3) 00 00(2) 00 00(1) 00 00(2) 00 00(4) 00 00(3) 00 11(0) 11(0) 11(0) 11(0) 11(0) 11(1) 01 01 01 01 01 01 11(0) 11(0) 11(0) 11(1) 00(1) 00(2) 00(4) 00(3) 01(2) 01(1) 01(0) 01(2) 01(0) 10 10 10 10 10 10(0) 10(0) 10(2) 10(2) 10(4) 10(0) 10(2) 10(2) 10(4) 11 11 11 11 11 01(1) 01(0) 01(2) 01(0) Soft Decision Viterbi • Calculate Branch Metric based on Soft-Table System Arch 2008 (Fire Tom Wada)

  24. t=0 t=1 t=2 t=3 t=4 t=5 t=6 LEVEL 65 63 54 36 66 27 00 00(3) 00 00(2) 00 00(1) 00 00(2) 00 00(4) 00 00(3) 00 11(0) 11(0) 11(0) 11(0) 11(0) 11(1) 01 01 01 01 01 01 11(0) 11(0) 11(0) 11(1) 00(1) 00(2) 00(4) 00(3) 01(2) 01(1) 01(0) 01(2) 01(0) 10 10 10 10 10 10(0) 10(0) 10(2) 10(2) 10(4) 10(0) 10(2) 10(2) 10(4) 11 11 11 11 11 01(1) 01(0) 01(2) 01(0) Soft Decision Viterbi(2) • Calculate Path metric in order to find minimum path metric path. • Until t=6 3 4 3 5 2 0 3 3 2 0 4 0 2 0 3 3 0 0 1 1 3 3 System Arch 2008 (Fire Tom Wada)

  25. t=0 t=1 t=2 t=3 t=4 t=5 t=6 LEVEL 65 63 54 36 66 27 00 00(3) 00 00(2) 00 00(1) 00 00(2) 00 00(4) 00 00(3) 00 11(0) 11(0) 11(0) 11(0) 11(0) 11(1) 01 01 01 01 01 01 11(0) 11(0) 11(0) 11(1) 00(1) 00(2) 00(4) 00(3) 01(2) 01(1) 01(0) 01(2) 01(0) 10 10 10 10 10 10(0) 10(0) 10(2) 10(2) 10(4) 10(0) 10(2) 10(2) 10(4) 11 11 11 11 11 01(1) 01(0) 01(2) 01(0) Soft Decision Viterbi(3) • Select Minimum Path Metric and get original information • In this example : 1 1 0 0 1 0 0 0 0 0 0 0 System Arch 2008 (Fire Tom Wada)

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

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