Error Correction Code (2)

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Presentation transcript:

Error Correction Code (2) Fire Tom Wada Professor, Information Engineering, Univ. of the Ryukyus 2018/11/27 System Arch 2007 (Fire Tom Wada)

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 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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 2018/11/27 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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 2018/11/27 System Arch 2007 (Fire Tom Wada)

Simple Convolutional Coder 　D is delay operator 　c1t, c2t depends on not only current it bu also past it-1, it-2 2018/11/27 System Arch 2007 (Fire Tom Wada)

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 “11 10 10 11 11 01”. Constraint length K=3, R=1/2 2018/11/27 System Arch 2007 (Fire Tom Wada)

State Transition Diagram Number of State: 4 Time t 1 2 3 4 5 it F1=it-1 F2=it-2 c1t c2t 2018/11/27 System Arch 2007 (Fire Tom Wada)

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 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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 2018/11/27 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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 2018/11/27 System Arch 2007 (Fire Tom Wada)

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 2018/11/27 System Arch 2007 (Fire Tom Wada)

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 2018/11/27 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (Fire Tom Wada) R=2/3 example 2018/11/27 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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 2018/11/27 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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. 2018/11/27 System Arch 2007 (Fire Tom Wada)

Viterbi Decoding Example R=1/2, Receive 11 10 11 01 11 01 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 2018/11/27 System Arch 2007 (Fire Tom Wada)

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) 2018/11/27 System Arch 2007 (Fire Tom Wada)

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 System Arch 2007 (Fire Tom Wada)

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. 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 2018/11/27 System Arch 2007 (Fire Tom Wada)

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 SIGNAL LEVEL 1 2 3 4 5 6 7 2018/11/27 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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! 2018/11/27 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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 = 1 + 2 = 3) LEVEL= 65 HOW TO COMPUTE BRANCH METRIC No effect on Viterbi decoding! Looks like Erased or Punctured! 2018/11/27 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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’ 2018/11/27 System Arch 2007 (Fire Tom Wada)

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 System Arch 2007 (Fire Tom Wada)

Soft Decision Viterbi(3) Select Minimum Path Metric and get original information In this example : 1 1 0 0 1 0 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 System Arch 2007 (Fire Tom Wada)

System Arch 2007 (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 2018/11/27 System Arch 2007 (Fire Tom Wada)