1. Improving the Performance of Turbo Codes by Repetition and Puncturing Youhan Kim March 4, 2005

2. Improving the Performance of Turbo Codes 2 Outline  Conventional Turbo Codes  Motivation  Proposed Turbo Coding Scheme  Codeword Distance Spectrum  Iterative Decoder Structure  Simulation Results  Conclusions

3. Improving the Performance of Turbo Codes 3 Turbo Codes  Parallel Concatenated Convolutional Code  Pseudo-random interleaver between RSC1 and RSC2 Reduce the chance of both RSC1 and RSC2 generating low weight parity sequences at the same time  Codeword distance spectrum thinning Turbo Code Encoder

4. Improving the Performance of Turbo Codes 4 Turbo Codes  Near channel capacity performance achieved when very large interleaver is used  Near ML performance can be achieved using iterative decoder as well

5. Improving the Performance of Turbo Codes 5 Interleaver Size  Conventional Turbo code  Interleaver gain Performance enhanced as interleaver size increases Interleaver size = Frame length  Require very large frame size for good performance  Limit on frame length due to Transmission delay Decoding delay Hardware complexity  Unsuitable for Short frames Applications requiring very low error rate Interleaver gain

6. Improving the Performance of Turbo Codes 6 Motivation  Why limit ourselves to the case of interleaver size = frame length?  Design turbo codes with interleaver size > frame length For short frames,  Large interleaver gain even with short transmission delay For large frames,  Achieve very low error rate Reuse existing encoder/decoder hardware available for conventional turbo codes

7. Improving the Performance of Turbo Codes 7 Repeat-Puncture Turbo Code (RPTC)  Repeat each bit L times prior to interleaving  Interleaver size = L x Frame length  Puncture RSC2 parity sequence to control code rate  Asymmetry between CC1 and CC2

8. Improving the Performance of Turbo Codes 8 Benefits of Repeating  Greater interleaver gain  Low weight parity sequences generated by weight 2 input sequences dominate the performance of turbo codes [Benedetto96, Divsalar96]

9. Improving the Performance of Turbo Codes 9 Benefits of Repeating  RPTC: RSC2 encodes sequence of weight 2L  In the case of weight 1 input sequence with L=2

10. Improving the Performance of Turbo Codes 10 Puncturing Pattern  Puncture RSC2 parity sequence to control code rate  Simple puncturing pattern  Transmit the first n bits out of every L bits E.g.) L=4, n=2  Easy to compute codeword distance spectrum

11. Improving the Performance of Turbo Codes 11 Performance Analysis  Transition matrix approach for Turbo codes with puncturing [Kousa02]  Transition matrix for RSC  Encoder state transition over 2 input bits

12. Improving the Performance of Turbo Codes 12 Performance Analysis (2)  Assume both RSC encoders start and end in the all-zero state  Component code 1  Encoder state transition of CC1 over entire frame  Conditional weight enumerating function

13. Improving the Performance of Turbo Codes 13 Performance Analysis (3)  Component code 2  Period transition matrix Only first n bits out of L bits are transmitted  Encoder state transition of CC2 over entire frame  Conditional weight enumerating function

14. Improving the Performance of Turbo Codes 14 Performance Analysis (4)  Assuming uniform interleaver,  Conditional weight enumerating function of RPTC  Union bound on FER

15. Improving the Performance of Turbo Codes 15 Codeword Distance Spectrum of RPTC  Uniform Interleaver  Code rate = 1/3 (n=1)  RSC polynomial: (1+D 2 )/(1+D+D 2 )

16. Improving the Performance of Turbo Codes 16 Codeword Distance Spectrum of RPTC  More than 10 fold decrease in A d for L=2

17. Improving the Performance of Turbo Codes 17 Union Bound  Uniform interleaver  Code rate = 1/3 (n=1)  N=128

18. Improving the Performance of Turbo Codes 18 Iterative Decoder: Factor Graph

19. Improving the Performance of Turbo Codes 19 Simulation Parameters  RSC polynomial: (1+D 2 )/(1+D+D 2 )  Code rate = 1/3 (n=1)  Max. 40 iterations  Non-fading channel

20. Improving the Performance of Turbo Codes 20 Performance with Uniform Intlv.  SNR gain at FER = 10 -3  N=256: 1.0 dB  N=1024: 1.5 dB  N=8192: 1.9 dB  SNR gain at BER = 10 -5  N=256: 0.7 dB  N=1024: 0.6 dB

21. Improving the Performance of Turbo Codes 21 Performance with Prime Intlv.  Note: Prime interleaver is optimized for conventional Turbo codes

22. Improving the Performance of Turbo Codes 22 Price for Improvement  Increase in encoder and decoder complexity  Memory  Computational requirement  Hardware complexity  Many systems support multiple frame lengths  Short frames may use larger interleaver structures already available at the transmitter/receiver  Memory requirement not a problem

23. Improving the Performance of Turbo Codes 23 Conclusions  Repeat-Puncture Turbo Code  Use simple repetition and puncturing Repeater: Interleaver size > Frame length Puncturer: No loss in code rate  Improved codeword distance spectrum  Iterative decoding Superior performance than conventional turbo codes for moderate to high SNRs  Suitable for Improving the performance of short frames in systems supporting multiple frame lengths Applications requiring very low error rate

24. Improving the Performance of Turbo Codes 24 Thank you!