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  1. Towards ideal codes: looking for new turbo code schemes Ph.D student: D. Kbaier Ben Ismail Supervisor: C. Douillard Co-supervisor: S. Kerouédan

  2. What is a good code? • Ideal system • Limits to the correction capability of any code • Established by Shannon (1947-48) Good convergence High asymptotic gain Asymptotic gain Extract from  «Codes and Turbo Codes» Under the direction of Claude Berrou Dilemma: good convergence versus high Minimum Hamming Distance Ph.D defense Monday 26th September 2011 Ph.D defense Monday 26th September 2011

  3. Turbo codes: a breakthrough in digital communications How to combat the floor while keeping a good convergence? • Turbo codes (TCs): various communication standards • (-) High floors of errors • Lower error rates are required for real-time & demanding applications • Asymmetric turbo codes with different RSC encoders • Devising more sophisticated internal permutations • Component encoders with a large number of states • Different types of concatenation: serial, hybrid, multiple… • 3D TCs [1] • Irregular TCs [2] [1] C. Berrou, A. Graell i Amat, Y. Ould-Cheikh-Mouhamedou, C. Douillard, and Y. Saouter, “Adding a rate-1 third dimension to turbo codes,” in Proc. IEEE Inform. Theory Workshop, Lake Tahoe, CA, Sep. 2007, pp. 156–161. [2] B. Frey and D. MacKay. Irregular turbocodes. In Proc. 37th Allerton Conference on Communication, Control and Computing, Illinois, page 121, September 1999. Ph.D defense Monday 26th September 2011

  4. Outline • Introduction • 3-Dimensional turbo codes (3D TCs) • 3D coding scheme • Parameters: post-encoder, Π’ and λ • Improving the asymptotic performance • Improving the convergence threshold • Irregular turbo codes • Conclusion Ph.D defense Monday 26th September 2011

  5. 3D coding scheme: encoding structure X (1-λ) Y1 P U N C T U R I N G data RSC 1 Y1 λ Y1 Π Post Encoder W P/S Π’ λ Y2 RSC 2 Y2 (1-λ) Y2 Classical turbo encoder λ =1/4  {1000} Parameters: Permeability rate λ Post-encoder Permutation Π’ The added part is placed just behind the pre-existing turbo encoder C. Berrou, A. Graell i Amat, Y. Ould-Cheikh-Mouhamedou, C. Douillard, and Y. Saouter, “Adding a rate-1 third dimension to turbo codes,” in Proc. IEEE Inform. Theory Workshop, Lake Tahoe, CA, Sep. 2007, pp. 156–161 Ph.D defense Monday 26th September 2011 Ph.D defense Monday 26th September 2011

  6. Choice of the post-encoder Influences performance in the waterfall and error floor region Must be simple  low memory RSC codes The code is made tail biting  accumulator Must not exhibit too much error amplification Our contribution: EXIT analysis Ph.D defense Monday 26th September 2011 Ph.D defense Monday 26th September 2011

  7. Post encoders EXIT analysis k = 570 bits λ = 1/4 R = 1/3 Max-Log-MAP 10 iterations AWGN channel Ph.D defense Monday 26th September 2011

  8. Permutation Π’ • A "composite"input weight 4 square error pattern • Weight of the codeword: d=28 • Puncturing to R=1/2 d=16 • Role of the 3D part: • A few 1s of the redundancy part of the error pattern will be moved away to each other • Produce a significant of additional 1s • Increasing the total codeword weight Role? Importance of the spread i=Π’(j)=(P0j+i0) mod P P0=sqrt(2P) i0~P0/2 Regular permutation Ph.D defense Monday 26th September 2011 Ph.D defense Monday 26th September 2011

  9. Choice of the permeability rate λ Convergence loss / required dmin trade-off A large value of λ: (+) a higher dmin (-) convergence FER / BER R1, λ1 R1, λ2 > λ1 Eb/N0 (dB) Ph.D defense Monday 26th September 2011 Ph.D defense Monday 26th September 2011

  10. Performance of 3GPP2 based 3D TCs k = 3066 R= 1/3 k = 570 R = 4/5 dmin= 23 dmin= 4 dmin= 4 dmin= 38 dmin <= 43 All simulations use the MAP algorithm with 10 decoding iterations Ph.D defense Monday 26th September 2011

  11. Improving the asymptotic performance of 3D TCs: optimization method • All-zero iterative decoding algorithm [3] determine low weight codewords & estimate multiplicity • First terms : low multiplicity Regular pattern λ = 1/4 Systematic part Parity y Parity w 000000000000000000000000000000000000000000000000000000000000000000000000 [3] R. Garello and A. Casado, “The All-Zero Iterative Decoding Algorithm for Turbo Code Minimum Distance Computaion," IEEE International Conference on Communications, pp. 361–364, June 2004. 000000000001….00000000011000000000000000..100000000010000000001000000... 000001000000….00100000010000.0010000000000000001000010000000001……..0001 Low weight codeword Ph.D defense Monday 26th September 2011

  12. Improving the asymptotic performance of 3D TCs: optimization method • All-zero iterative decoding algorithm [3] determine low weight codewords & estimate multiplicity • First terms : low multiplicity • Pattern of post-encoding: not regular any more Non regular pattern Systematic part Parity y Parity w 000000000000000000000000000000000000000000000000000000000000000000000000 001000100010000100001000010001000100000100010010010000100010100100100100 000000000001….00000000011000000000000000..100000000010000000001000000... 000001000000….00100000010000.0010000000000000001000010000000001……..0001 Low weight codeword Ph.D defense Monday 26th September 2011

  13. Optimization results for k = 1146 data bits k = 1146 R = 2/3 λ = 1/4 Ones concentrated in the systematic part The new minimum distance of the optimized 3D TC is 33 (compared to 7) Address 5 Address 13 Address 1 Address 9 Ph.D defense Monday 26th September 2011 Ph.D defense Monday 26th September 2011

  14. Assessment: optimization method Yes! Optimization method applicable for any family of TCs Provided that the distance spectrum has low multiplicities at the beginning For the 3GPP2: Tail bits  singular points in the trellis Tail bits  cause the codewords to be truncated But the method “cannot” be applied with the WiMAX permutation (ARP) Periodic distribution of the bits High codewords multiplicity Tail biting termination  better distances Can we generalize? • A slight irregular post-encoding pattern • improvement in the distance properties • Optimistic results implement the optimization method especially for high coding rates Ph.D defense Monday 26th September 2011 Ph.D defense Monday 26th September 2011

  15. Outline • Introduction • 3-Dimensional turbo codes (3D TCs) • 3D coding scheme • Parameters: post-encoder, Π’ and λ • Improving the asymptotic performance • Improving the convergence threshold • Irregular turbo codes • Conclusion Ph.D defense Monday 26th September 2011

  16. Improving the convergence threshold of the 3D TC • Loss in the convergence threshold (dB) for 3GPP2 3D TCs over AWGN channel: R λ Rayleigh channel 0.19 • Reducing the convergence loss of 3D TCs: • Costello [4] Time Varying (TV) post-encoder • Specific Gray mapping for 3D TCs associated with high order constellations [4] D. Costello Jr. Free distance bounds for convolutional codes. IEEE Transactions on Information Theory, 20(3):356-365, May 1974. Ph.D defense Monday 26th September 2011

  17. Recursivity : polynomial 5 Input 4 7 W1(4) W1(4) W1(4) W1(4) time W2(7) W2(7) W2(7) W2(7) W2(7) Replacement period L Reducing the convergence loss of 3D TCs: time varying post encoder • 4-state post-encoder with time-varying parity construction (5, 4:7) • 4-state post-encoder with time-varying parity construction (5, 4:7) • (5,4:7) Distance = 2 • (5,4) Distance =3 and (5,7)  Distance =5 • Replace periodically some redundancies W1=4 by W2=7 Time varying trellis • BER out = 2* (BER in +ξ) Convergence/distance trade-off Ph.D defense Monday 26th September 2011

  18. General results for the time varying technique • Loss of convergence reduced by 10% to 50% of the value expressed in dB • The asymptotic performance is not degraded • For a fixed code memory, the choice of the post-encoder does not influence dmin of the 3D TC • Higher local minimum distance of the post-encoder = • Better level of the extrinsic information which the predecoder supplies to the two SISO decoders • The TV technique acts as a convergence accelerator of the 3D TC Ph.D defense Monday 26th September 2011

  19. Error rate performance example of time varying 3D TCs Max-Log-MAP 10 iterations k = 1146 bits Loss of convergence reduced by 35% from 0.23 dB to 0.15 dB Ph.D defense Monday 26th September 2011

  20. 3D TCs for high spectral efficiency transmissions • BICM approach • Among the bits forming a symbol in M-QAM or M-PSK modulations, the average probability of error is not the same for all the bits Three constellation mappings: Configuration 1: mapping uniformly distributed Configuration 2: systematic bits mapped to better protected places as a priority Configuration 3: systematic bits (then if possible) post- encoded parity bits protected as a priority Ph.D defense Monday 26th September 2011

  21. Example: 3D TCs associated with a 16-QAM modulator 4 bits of a 16-QAM symbol k = 2298 bits Gain: 0.22 Systematic bits & post-encoded parity bits mapped to better protected places 1867 16-QAM symbols R = 1/3 λ = 1/8 Gaussian channel 2298 x 2298 y1 2298 y2 574 w Ph.D defense Monday 26th September 2011

  22. Design rules • Configuration 1 loss of convergence still observed • Configuration 2 or 3 gain in the waterfall region • Configuration 3 must be used as far as possible • Otherwise, implement at least the configuration 2 • Significant gain: • Even for transmissions over Rayleigh fading channels • Increases with the coding rate R for the same λ Ph.D defense Monday 26th September 2011

  23. Properties of 3-Dimensional turbo codes • Increase in dmin • But: • Loss in the convergence threshold • Increase in complexity • Why? • The answer is in the decoding process Ph.D defense Monday 26th September 2011

  24. What about the 3D decoding complexity? Π y1 8-state SISO DEC1 Classical Turbo Decoder w 4-state SISO PRE-DEC S/P Π’-1 8-state SISO DEC2 y2 Π Π-1 P/S Π’ Extrinsic information about the post-encoded parity bits Ph.D defense Monday 26th September 2011

  25. Complexity figures High throughputs # Proc increases  additional complexity decreases k = 1530 bits λ= 1/8 R = 1/2 Ph.D defense Monday 26th September 2011

  26. Summary: 3D TCs (1/2) • Time varying post-encoder (5, 4:7) with a little irregularity • Irregularity in the Gray mapping for 3D TCs associated with high order modulations • Non regular post-encoding pattern to improve the asymptotic performance BER/FER Classical TC 3D TCs + high order modulations + specific Gray mapping 3D TC Time varying Optimization method Eb/N0 (dB) Ph.D defense Monday 26th September 2011

  27. Summary: 3D TCs (2/2) • Time varying post-encoder (5, 4:7) with a little irregularity • Irregularity in the Gray mapping for 3D TCs associated with high order modulations • Non regular post-encoding pattern to improve the asymptotic performance • Work on irregular LDPC codes  significant gain • Frey & MacKay introduced irregularity to TCs • Sawaya & Boutros  lower the floor of irregular TCs Irregularity • The next step of the study concerns the investigation of irregular TCs • Why? • Obtain an irregular TC which performs well in both the waterfall and the error floor regions Ph.D defense Monday 26th September 2011

  28. Outline • Introduction • 3-Dimensional turbo codes (3D TCs) • Irregular turbo codes • Basics of irregular TCs • Selecting the degree profile EXIT diagrams • Design of suitable permutations for irregular TCs Principle & simulation results • Irregular TCs with post-encoding • Conclusion Ph.D defense Monday 26th September 2011

  29. Self-concatenated turbo encoder • Equivalent encoding structure for a regular turbo encoder: • Merge two trellis encoders •  double size interleaver + 2-fold repetition • Interest: introduce an irregular structure Ph.D defense Monday 26th September 2011

  30. Information bits Repetition (d2) kf2 k infobits kfj Repetition (dj) Interleaver RSC kfmax Repetition (dmax) Irregular turbo encoder • Performance of an irregular TC strongly depends on the degree profile • Number of degrees and fractions: 2(dmax-1) • Only two equations to optimize all these parameters! Parity bits • Two non-zero fractions:d =2 and d >2 : • f2 + fmax=1 • 2 f2 + dmaxfmax = dAverage • Only three parameters Degree profile (2, 3,…, dmax) or (f2, f3,…, fmax) Ph.D defense Monday 26th September 2011

  31. What is a good irregular turbo code? Our approach = we separate the problems It depends on: Degree profile • Search for a good degree profile using a random interleaver • Optimize the interleaver RSC code Fixed Our contribution: analyzing the degree profile using hierarchical EXIT charts Ph.D defense Monday 26th September 2011

  32. Analyzing the degree profile using hierarchical EXIT charts Ph.D defense Monday 26th September 2011

  33. Performance example of irregular TCs k = 1146 bits Interleaver length: 3438 dav = 3 R = ¼ MAP 8 iterations Ph.D defense Monday 26th September 2011

  34. Outline • Introduction • 3-Dimensional turbo codes (3D TCs) • Irregular turbo codes • Basics of irregular TCs • Selecting the degree profile EXIT diagrams • Design of suitable permutations for irregular TCs Principle & simulation results • Irregular TCs with post-encoding • Conclusion Ph.D defense Monday 26th September 2011

  35. Proposed algorithm for the permutation design (1/2) • Reduce the correlation effect between the pilot groups while improving the distance properties of irregular TCs Information sequence: 0 1 0 1 0 0 1 0 1 1 0 ... Appropriate repetition 00000000 11 00 11 00 00 11111111 00 … 4 3 Copy 3 / Address =120 2 1 weight 1 Original Address =565 Copy 2 Address =273 weight 0 Interleaver size: 576 The Dijkstra’ s algorithm [5]: [5] E. Dijkstra. A note on two problems in connexion with graphs. Numerische mathematik, 1(1):269-271, 1959. Ph.D defense Monday 26th September 2011

  36. In the example: d = 8 weight = 0 Copy 7 Address =47 weight = 1 Copy 3 / Address =120 Address =1 Copy 6 Address =189 Original Address =565 Copy 2 Address =273 Copy 6 Address =500 Copy 4 Address =440 Copy 5 Address =356 Proposed algorithm for the permutation design (2/2) • Reduce the correlation effect between the pilot groups while improving the distance properties of irregular TCs Information sequence: 0 1 0 1 0 0 1 0 1 1 0 ... Appropriate repetition 00000000 11 00 11 00 00 11111111 00 … 4 3 Copy 3 / Address =120 2 1 weight 1 Original Address =565 Copy 2 Address =273 weight 0 Ph.D defense Monday 26th September 2011

  37. Error rate performance of irregular TCs with an optimized interleaver R = 1/4 Interleaver size: 144 Interleaver size: 576 Gain: 3.5 decades Gain: 2.5 decades All simulations use the MAP algorithm with 10 decoding iterations Ph.D defense Monday 26th September 2011

  38. Error rate performance of irregular TCs with an optimized interleaver R = 1/4 • Proposed algorithm: very fast for short block sizes • For medium sizes and large blocks: • Unacceptable computational time • Uncertainty about detecting all the possible cases • Drawback: Necessity to store all the interleaved addresses • Devising good interleavers for irregular TCs proves to be a difficult task CPU: Two quad core processors (Xéon) RAM: 8Go Gain: > 2 decades Interleaver size: 3438 All simulations use the MAP algorithm with 8 decoding iterations Ph.D defense Monday 26th September 2011

  39. Outline • Introduction • 3-Dimensional turbo codes (3D TCs) • Irregular turbo codes • Basics of irregular TCs • Selecting the degree profile EXIT diagrams • Design of suitable permutations for irregular TCs Principle & simulation results • Irregular TCs with post-encoding • Conclusion Ph.D defense Monday 26th September 2011

  40. Information bits 1-λ λ Non-uniform repetition Post-encoder Π RSC Π’ Parity bits Adding a post-encoder to irregular TCs • We propose an irregular TC inspired by our work about 3D TCs • Ensure large asymptotic gain at very low error rates • Even with non optimized internal permutation • Improve the distance properties of irregular TCs Ph.D defense Monday 26th September 2011

  41. Performance example of irregular TCs with post-encoding • All simulations use the MAP algorithm with 10 decoding iterations • Degree profile (f2,f8), dav = 3, R = 1/4 , λ = 1/8 and k = 4096 bits • 3GPP2 interleaver, interleaver size: 12282 Gain: 2.5 decades dmin= 33 dmin= 44 dmin= 50 Ph.D defense Monday 26th September 2011

  42. Summary: irregular TCs BER/FER Classical TC Irregular TC Irregular TC + Post-encoder Suitable permutations Eb/N0 (dB) Ph.D defense Monday 26th September 2011

  43. Conclusion Towards ideal codes? 3D TCs • Asymptotic performance: The 3D TC significantly improves performance in the error floor region • Convergence: We can implement methods which reduce significantly the loss of convergence Irregular TCs • Performance: Closer to capacity but very poor asymptotic performance • Improve the distance properties: Graph-based permutations (Dijkstra's algorithm + estimation of the minimum distance) Irregular TCs + post-encoder Ph.D defense Monday 26th September 2011

  44. Perspectives Towards ideal codes? • 3D TCs: • New structures • Diversity techniques: MIMO, rotated constellations… • Double binary • Hardware implementation complexity of 3D turbo decoder • Irregular TCs: • Post-encoding pattern • The design of suitable permutations for irregular TCs is an important future research work • Eliminate the interleavers producing low minimum distances early in the search process  Reduce the space of search  Promising algorithm even for large blocks Ph.D defense Monday 26th September 2011

  45. Thank you for your attention Contributions to the literature Conference papers: • KBAIER BEN ISMAIL Dhouha, DOUILLARD Catherine and KEROUÉDAN Sylvie, "Improving 3-dimensional turbo codes using 3GPP2 interleavers", ComNet'09: 1st International Conference on Communications and Networking, 03-06 November 2009, Hammamet, Tunisia, 2009. • KBAIER BEN ISMAIL Dhouha, DOUILLARD Catherine and KEROUÉDAN Sylvie, "Reducing the convergence loss of 3-dimensional turbo codes", 6th International Symposium on Turbo Codes & Iterative Information Processing, 06-10 September 2010, France, pp. 146-150. Journal papers: • KBAIER BEN ISMAIL Dhouha, DOUILLARD Catherine and KEROUÉDAN Sylvie, "Analysis of 3-dimensional turbo codes", Annals of Telecommunications, available online at http://www.springerlink.com/content/1r8785617q48n106/ • KBAIER BEN ISMAIL Dhouha, DOUILLARD Catherine and KEROUÉDAN Sylvie, "Design of suitable permutations for irregular turbo codes", Electronics Letters, June 2011, vol. 47, n° 13, pp. 748-749. • KBAIER BEN ISMAIL Dhouha, DOUILLARD Catherine and KEROUÉDAN Sylvie, "Improving irregular turbo codes", Electronics Letters, to appear. Submitted journal paper: • KBAIER BEN ISMAIL Dhouha, DOUILLARD Catherine and KEROUÉDAN Sylvie, "Improving 3GPP2 3-dimensional turbo codes and aspects of irregular turbo codes", submitted to EURASIP Journal on Wireless Communications and Networking. Ph.D defense Monday 26th September 2011