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Convolutional Coding

Convolutional Coding.

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Convolutional Coding

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  1. Convolutional Coding In telecommunication, a convolutional code is a type of error-correcting code in which m-bit information symbol to be encoded is transformed into n-bit symbol. Convolutional codes are used extensively in numerous applications in order to achieve reliable data transfer, including digital video, radio, mobile communication, and satellite communication. These codes are often implemented in concatenation with a hard-decision code, particularly Reed Solomon. Course Name: Error Correcting Codes Level : UG Author Phani Swathi Chitta Mentor Prof. Saravanan Vijayakumaran

  2. Learning Objectives After interacting with this Learning Object, the learner will be able to: • Explain the convolutional encoding and Viterbi (decoding) Algorithms

  3. Definitions of the components/Keywords: • A convolutional encoder is a finite state machine. An encoder with n registers will have 2n states. • Convolutional codes have memory that uses previous bits to encode or decode following bits • The most commonly known graphical representation of a code is the trellis representation. A code trellis diagram is simply an edge labeled directed graph in which every path represents a code sequence. • This representation has resulted in a wide range of applications of convolutional codes for error control in digital communications. • Viterbi algorithm is used for decoding a bit stream that has been encoded using forward error correction based on a convolutional code. • Viterbi decoding compares the hamming distance between the branch code and the received code. • Path producing larger hamming distance is eliminated. • In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. 1 2 3 4 5

  4. Master Layout 1 1 Part 1 – Encoder Part 2 – Decoder Place a dropdown box for encoder and decoder 0/00 + 00 1/11 2 0/11 0/01 10 01 1/00 3 1/10 0/10 11 + Encoder 1/01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State diagram 4 5 Trellis Diagram

  5. Step 1: 1 Input data: 010011101 + 0 2 0 0 0 3 + 0 4 5

  6. Step 2: 1 Input data: 010011101 + 0/00 0 00 2 0 0 10 01 3 + 0 11 4 5

  7. Step 3: 1 Input data: 10011101 + 1 2 1 0 0 3 + 1 4 5

  8. Step 4: 1 Input data: 10011101 + 0/00 1 00 1/11 2 1 0 10 01 3 + 1 11 4 5

  9. Step 5: 1 Input data: 0011101 + 0 2 0 1 0 3 + 1 4 5

  10. Step 6: 1 Input data: 0011101 + 0/00 0 00 1/11 2 0/01 0 1 10 01 3 + 1 11 4 5

  11. Step 7: 1 Input data: 011101 + 1 2 0 0 1 3 + 1 4 5

  12. Step 8: 1 Input data: 011101 + 0/00 1 00 1/11 0/11 2 0/01 0 0 10 01 3 + 1 11 4 5

  13. Step 9: 1 Input data: 11101 + 1 2 1 0 0 3 + 1 4 5

  14. Step 10: 1 Input data: 11101 + 0/00 1 00 1/11 0/11 2 0/01 1 0 10 01 3 + 1 11 4 5

  15. Step 11: 1 Input data: 1101 + 1 2 1 1 0 3 + 0 4 5

  16. Step 12: 1 Input data: 1101 + 0/00 1 00 1/11 0/11 2 0/01 1 1 10 01 3 1/10 + 0 11 4 5

  17. Step 13: 1 Input data: 101 + 0 2 1 1 1 3 + 1 4 5

  18. Step 14: 1 Input data: 101 + 0/00 0 00 1/11 0/11 2 0/01 1 1 10 01 3 1/10 + 1 11 1/01 4 5

  19. Step 15: 1 Input data: 01 + 1 2 0 1 1 3 + 0 4 5

  20. Step 16: 1 Input data: 01 + 0/00 1 00 1/11 0/11 2 0/01 0 1 10 01 3 1/10 0/10 + 0 11 1/01 4 5

  21. Step 17: 1 Input data: 1 + 0 2 1 0 1 3 + 0 4 5

  22. Step 18: 1 Input data: 1 + 0/00 0 00 1/11 0/11 2 0/01 1 0 10 01 3 1/00 1/10 0/10 + 0 11 1/01 4 5

  23. Step 19: Trellis Diagram 1 00. . . . . . . . . . 10. . . . . . . . . . 01. . . . . . . . . . 11. . . . . . . . . . 0/00 0/00 2 1/11 1/11 1/11 0/11 0/11 0/01 1/00 1/00 3 1/10 1/10 0/10 0/10 1/01 1/01 t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 4 5

  24. Step 20: 1 00. . . . . . . . . . 10. . . . . . . . . . 01. . . . . . . . . . 11. . . . . . . . . . Input data: 010011101 0/00 2 1/11 1/11 0/11 0/01 1/00 3 1/10 0/10 1/01 t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 4 5

  25. Master Layout 2 1 Part 1 – Encoder Part 2 – Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 4 Trellis Diagram The output sequence is 11, 10, 10, 11, 11, 01, 00, 01 5

  26. Step 1: 1 2 3 4 5

  27. Step 2: 1 r0 =11 00 . . 10 . . 01 . . 11 . . 2 2 0 3 t=0 t=1 4 5

  28. Step 3: 1 r1 =10 00 . . . 10 . . . 01 . . . 11 . . . 3 2 3 2 3 0 t=0 t=1 t=2 4 5

  29. Step 4: 1 r2 =00 00 . . . . 10 . . . . 01 . . . . 11 . . . . 3 3 2 4 5 3 2 2 4 3 1 4 0 1 t=0 t=1 t=2 t=3 4 5

  30. Step 5: 1 r2 =00 00 . . . . 10 . . . . 01 . . . . 11 . . . . 3 2 2 3 1 1 t=0 t=1 t=2 t=3 4 5

  31. Step 6: 1 r3 =10 00 . . . . . 10 . . . . . 01 . . . . . 11 . . . . . 3 4 2 2 4 2 2 4 3 1 1 2 1 3 t=0 t=1 t=2 t=3 t=4 4 5

  32. Step 7: 1 r3 =10 00 . . . . . 10 . . . . . 01 . . . . . 11 . . . . . 2 2 2 3 1 2 t=0 t=1 t=2 t=3 t=4 4 5

  33. Step 8: 1 r4 =11 00 . . . . . . 10 . . . . . . 01 . . . . . . 11 . . . . . . 4 2 2 1 2 3 2 3 3 1 3 2 3 3 t=0 t=1 t=2 t=3 t=4 t=5 4 5

  34. Step 9: 1 r4 =11 00 . . . . . . 10 . . . . . . 01 . . . . . . 11 . . . . . . 2 1 2 3 3 3 t=0 t=1 t=2 t=3 t=4 t=5 4 5

  35. Step 10: 1 r5 =01 00 . . . . . . . 10 . . . . . . . 01 . . . . . . . 11 . . . . . . . 2 2 1 4 2 2 4 2 3 3 5 4 3 3 t=0 t=1 t=2 t=3 t=4 t=5 t=6 4 5

  36. Step 11: 1 r5 =01 00 . . . . . . . 10 . . . . . . . 01 . . . . . . . 11 . . . . . . . 2 2 2 2 3 3 t=0 t=1 t=2 t=3 t=4 t=5 t=6 4 5

  37. Step 12: 1 r6 =00 00 . . . . . . . . 10 . . . . . . . . 01 . . . . . . . . 11 . . . . . . . . 2 2 2 4 4 2 2 2 3 3 4 3 3 4 t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 4 5

  38. Step 13: 1 r6 =00 00 . . . . . . . . 10 . . . . . . . . 01 . . . . . . . . 11 . . . . . . . . 2 2 2 3 3 3 t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 4 5

  39. Step 14: 1 r7 =01 00 . . . . . . . . . 10 . . . . . . . . . 01 . . . . . . . . . 11 . . . . . . . . . 2 3 2 3 3 2 4 3 2 3 5 3 4 3 t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 4 5

  40. Step 15: 1 r7 =01 00 . . . . . . . . . 10 . . . . . . . . . 01 . . . . . . . . . 11 . . . . . . . . . 2 3 3 2 3 3 t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 4 5

  41. Step 16: 1 r7 =01 00 . . . . . . . . . 10 . . . . . . . . . 01 . . . . . . . . . 11 . . . . . . . . . 1/11 1/11 2 3 0/11 3 1/00 0/01 0/01 1/10 2 3 0/10 3 t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 4 5

  42. Electrical Engineering Slide 1 Slide 3 Slide 44,45 Slide 47 Slide 46 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ Encoder: Interactivity: + Try it yourself 00 • Place an input box to enter the input data • Place a dropdown box with options 0 and 1 10 01 11 + State Diagram Encoder 42 Credits

  43. Electrical Engineering Slide 1 Slide 3 Slide 44,45 Slide 47 Slide 46 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ Decoder: Interactivity: Try it yourself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Place an input box to enter the 16-bit output data sequence • Place a dropdown box with options 00, 10, 01, 11 Trellis Diagram 43 Credits

  44. Questionnaire 1 1. An encoder with n registers will have ________ states. Answers: a)2n b) 2n +1 ‏ • For the given encoder, what will be the output sequence? Input data: 01011100 Answers: a)00 11 01 00 10 01 10 11 b) 00 10 11 10 01 10 00 11 c) 00 10 11 01 01 10 00 11 d)‏ 00 11 01 00 01 01 10 11 2 + 3 4 + 5

  45. Questionnaire 1 3.The received sequence is 00 10 11 11 10 01. Among the four choices given below, which is the sequence nearest to the received sequence in terms of Hamming distance? Answers: a) 00 10 01 10 10 01 b) 00 11 01 10 00 01 c) 00 10 11 00 01 10 d) 00 10 11 10 00 11 The answers are given in red 2 3 4 5

  46. Links for further reading Reference websites: http://en.wikipedia.org/wiki/Convolutional_code Books: Error correction coding – Todd K. Moon, John wiley & sons,INC Research papers:

  47. Summary • A convolutional encoder is a finite state machine. An encoder with n binary cells will have 2n states. • The most commonly known graphical representation of a code is the trellis representation. A code trellis diagram is simply an edge labeled directed graph in which every path represents a code sequence. • This representation has resulted in a wide range of applications of convolutional codes for error control in digital communications. • Viterbi algorithmfor decoding a bitstream that has been encoded using forward error correctionbased on a convolutional code.

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