1. 資訊理論 授課老師 : 陳建源 Email:cychen07@nuk.edu.tw 研究室 : 法 401 網站 http://www.csie.nuk.edu.tw/~cychen/ Ch5: Error Correction

2. 5. 1 Introduction N blocks SourceCoderChannelDecoderRecipient Input source radio, optical fibre Code wordThe probability of correct decoding

3. Ch 5: Error Correction 5. 2 Upper bound for inputs of equal probability SourceCoderChannelDecoderRecipient Change probability p radio, optical fibre If equal probability

4. Ch 5: Error Correction 5. 2 Upper bound for inputs of equal probability When p=0.5 When p=0.1 When p=0.01

5. Ch 5: Error Correction 5. 5 Error correcting code codeword000 Check parity New codeword codeword001 Check parity New codeword

6. Ch 5: Error Correction 5. 5 Error correcting code codewordX=001 Check parity New codeword Parity-check code

7. Ch 5: Error Correction 5. 5 Error correcting code F Systematic parity-check matrix

8. Ch 5: Error Correction 5. 5 Error correcting code Error correcting algorithm Step 1. 針對 F 建立可能 error e i 的 syndrome s i =Fe i 之資料庫 已知 F ，收到 message y ，擬找到對應的 codeword x Step 2. 針對 y ，計算 s=Fy Step 3. 針對 s ，查資料庫得到 e i Step 4. 計算 codeword

9. Ch 5: Error Correction 5. 6 Error sequence 輸入 y=010011 ，如何找到其正確 codeword syndrome Step 1. 針對 F 建立可能 error e i 的 syndrome s i =Fe i 之資料庫

10. Ch 5: Error Correction 5. 6 Error sequence 已知 y=010011 ，如何找到其正確 codeword Codeword Step 2. 針對 y ，計算 s=Fy Step 3. 針對 s ，查資料庫得到 e i Step 4. 計算 codeword

11. Ch 5: Error Correction 5. 7 Hamming codes If the columns of F are all different and non-zero, the syndromes associated with 0 and e k are all distinct from one another Hamming codes are those for which If the code has n-r check digits there are n-r rows in F There are 2 n-r different columns available for F 扣掉 0 ，最多有 2 n-r -1 個 non-zero columns

12. Ch 5: Error Correction 5. 7 Hamming codes Hamming codes