1. Wireless Communication Research Lab. CGU What is Convolution Code? 指導教授：黃文傑 博士 學生：吳濟廷 2004.04.21

2. Outline Introduction Encoding structure State, tree, and trellis diagrams Veterbi decoding algorithm Soft decision decoding Applications Summary

4. Code taxonomy Today

5. Introduction continuous stream of source bits continuous stream of encoded bits sequence of source bits is convolved to produce output ‘ symbols ’ each encoded bit depends on Current input bit Previous sequence of input bits

6. Encoding structure (2,1,2) convolution code with generator polynomial code rate = 1/2

7. General encoding structure General (n,1,m) convolution code encoder

8. Encoding example Encoded Sequence 10 10 10 11 01 11 101011 S 1 S 2 S 3 Input Sequence Input sequence : 101011 Register number : 3 Generator polynomial : Initial state : 0 0 0

9. Polynomial representation Message polynomial: Generator polynomial:

10. State representation and diagram simpler representation two transitions emanating from each state not possible to move to any arbitrary state Code rate=1/2, m=2

11. Using state diagram We could also get the same output sequence by using state diagram “X” signifies “don’t know“ 10 10 01 00 01 01 11 U= Different time slot Code rate=1/2, m=2

12. Tree diagram the state diagram completely characterizes the encoder cannot represent time history tree diagram Code rate=1/2, m=2

13. Trellis diagram Branches increase ( L: number of branch words ) Trellis diagram Code rate=1/2, m=2

14. Viterbi decoding algorithm Discovered and analyzed by Viterbi in 1967 Advantage Maximum likelihood decoding Not a function of the number of symbols Reduces the decoding complexity

15. Example of Viterbi decoding Label each branch with Hamming distance error Decoder trellis diagram (rate=1/2, m=2) Using the encoder state diagram

16. Path remerging Two paths are remerged to state 00 at time t5 cumulative hamming path metric

17. Viterbi decoding procedure(1/2) Survivors and metric comparisons

18. Viterbi decoding procedure(2/2) Survivors and metric comparisons

19. Soft decision decoding Transform 2-level values to m-level values Measured by Euclidean distance instead of Hamming distance 2~3 dB coding gain better than hard decision decoding

20. Soft decision diagrammatic explanation (a) hard decision (b) soft decision (c) soft code symbols (d) encoding trellis section (e) decoding trellis section

21. Applications GSM: length= 5, rate= 1/2, free distance= 7 IS-95 Uplink: length= 9, rate= 1/3, d= 18 Downlink: length= 9, rate= 1/2, d= 12 UMTS (WCDMA), CDMA2000: turbo code (further development of convolution code)

22. Summary Convolution codes outperform block codes for the same implementation complexity Soft decision decoding decreases the error probability Widely used in wireless communication systems nowadays

23. Thank you ~