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Iterative Decoding Algorithms

  • Branka Vucetic
Chapter

Abstract

In this tutorial paper we present iterative algorithms which can be used for decoding of concatenated codes. The decoding operation is based on either a maximum a posteriori (MAP) algorithm or a Viterbi algorithm generating a weighted soft estimate of the input sequence. The iterative algorithm performs the information exchange between the two component decoders. The performance gain of the MAP algorithm over the Viterbi algorithm at low SNR leads to a slight performance advantage.

The MAP algorithm is computationally much more complex than the Viterbi algorithm. The operations in the MAP algorithm are multiplications and exponentiations while in the Viterbi algorithm they are simple add, compare and select operations.

As an example these algorithms are applied to decoding of turbo codes and their performance is compared on a Gaussian channel

Keywords

Turbo Code Convolutional Code Minimum Path Viterbi Algorithm Extrinsic Information 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    G. D. Forney, JR. “The Viterbi Algorithm”, Proceeding of IEEE, Vol. 61, No. 3, March 1973.Google Scholar
  2. [2]
    Yamamoto H. and Itoh K., “Viterbi Decoding for Convolutional Codes with Repeat Request”, IEEE on Inform. Theory, vol. IT-26, pp. 540–547, Sept. 1980.Google Scholar
  3. [3]
    J. Hagenauer, P. Hoeher, “A Viterbi Algorithm with Soft-Decision Outputs and its Applications,” Conf. Rec. GLOBECOM’89, Dallas, Texas, Vol. 3, pp. 47.1.1–47. 1. 7, Nov. 1989.Google Scholar
  4. [4]
    Y. Li, B. Vucetic and Y. Sato, “Optimum Soft Output Detection for Channels with Intersymbol Interference”, IEEE Trans. Inform. Theory, Vol-41, No-3, May 1995, p. 704–713.Google Scholar
  5. [5]
    Yunxin Li and Branka Vucetic, “A Low-complexity Soft-output TDMA Receiver”, TELFOR ‘95, Belgrade, 3–8 Dec. 1995, Yugoslavia.Google Scholar
  6. [6]
    Yunxin Li and Branka Vucetic, “A Genmeralized MLSE Algorithm”, INNSP ‘95, China, December 10–13, 1995.Google Scholar
  7. [7]
    T. Hashimoto “A list-Type Reduced-Constraint Generalization of the Viterbi Algorithm,” IEEE Trans. Inform. Theory, Vol. IT-33, No. 6, pp. 866–876, Nov. 1987.Google Scholar
  8. [8]
    B. Vucetic, “Bandwidth Efficient Concatenated Coding Schemes on Fading Channels” IEEE Trans. Commun., Jan. 1993.Google Scholar
  9. [9]
    T. Schaub and J.W. Modestino “An Erasure Declaring Viterbi Decoder and its Application to Concatenated Coding Systems,” ICC’86, IEEE Cat. No. CH23143/86, pp. 1612–1616, 1986Google Scholar
  10. [10]
    Branka Vucetic, Elvio Leonardo and Lin Zhang,”Soft Output Multistage Decoding of Multilevel Block Codes”, IEEE ITW’95 Proceedings, Ry-dzyna, Poland, June 15–19 1995.Google Scholar
  11. [11]
    N. Seshadri and C-E.W. Sundberg, “Generalized Viterbi Algorithms for Error Detection with Convolutional Codes,” Conf. Rec. GLOBECOM’89, Dallas, Texas, Vol$13, pp. 43.3.1–43. 3. 5, Nov. 1989.Google Scholar
  12. [12]
    C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon Limit Error-Correcting Coding and Decoding Turbo Codes (1), Proc. ICC’93, Geneva, Switzerland, pp. 1064–1070, May 1993.Google Scholar
  13. [13]
    S. Hirasawa et al, Modified Product Codes, IEEE Trans. Inform. Theory, Vol. IT-30, March 1984, pp. 299–306.Google Scholar
  14. [14]
    G.D. Forney, Concatenated Codes, MIT, Cambridge, MA, USA, 1966.Google Scholar
  15. [15]
    P. Elias, Error-free Coding, IEEE Trans. Inform. Theory, Vol. IT-4, pp. 29–37, Sept. 1954.Google Scholar
  16. [16]
    H. Imai and S. Hirakawa, A New Multilevel Coding Method Using Error-Correcting Codes”, IEEE Trans. Inform. Theory, Vol. IT-23, May 1975.Google Scholar
  17. [17]
    J. Hagenauer and P. Robertson, Iterative (Turbo) Decoding of Systematic Convolutional Codes with the MAP and SOVA Algorithms”, Proc. ITG Conf. Frankfurt, Germany, Oct. 1994.Google Scholar
  18. [18]
    S. Le Goff et al, Turbo Codes and High Spectral Efficiency Modulation, Proc. Globecom’94, San Francisco, California, USA, pp. 645–649, Dec. 1994.Google Scholar
  19. [19]
    P. Robertson, Illuminating the Structure of Code and Decoder of Parallel Concatenated Recursive Systematic (Turbo) Codes, Proc. Globecom’94, San Francisco, California, USA, pp. 1298–1303, Dec. 1994.Google Scholar
  20. [20]
    A.J. Viterbi and J.K. Omura, Principles of Digital Communications and Coding, New York McGraw Hill, 1979.Google Scholar
  21. [21]
    Claude Berrou and Alain Glavieux, Turbo-codes: General Principles and Applications, Audio and Video Digital Radio Broadcasing System and Techniques 1994, pp.215–226,Google Scholar
  22. [22]
    P. Jung, Novel Low Complexity Decoder for Turbo-Codes, Electronics Letters, 1995, Jan., Vol. 31, No. 2, pp. 86–87,Google Scholar
  23. [23]
    A.S. Barbulescu and S.S. Pietrobon, Terminating the Trellis of Turbo-Codes in the Same State, Electronics Letters, 1995, Jan., Vol. 31, No. 1, pp. 22–23CrossRefGoogle Scholar
  24. [24]
    P. Jung and M. Nabhan Performance Evaluation of Turbo Codes for Short Frame Transmission Systems, Electronics Letters, 1994, Jan., Vol. 30, No. 2, pp. 111–113CrossRefGoogle Scholar
  25. [25]
    Joachim Hagenauer and Lutz Papke, Iterative Decoding of Binary Block and Convolutional Codes, IEEE Trans. Inform. Theory, 1996, March, Vol. 42, No. 2, pp. 429–445.zbMATHGoogle Scholar
  26. [26]
    S. Benedetto and G. Montorsi, Average Performance of Parallel Concatenated Block Codes, Electronics Letters, 1995, Feb., Vol. 31, No. 3, pp. 156–158.CrossRefGoogle Scholar
  27. [27]
    S. Benedetto and G. Montorsi, Performance Evaluation fo Turbo Codes, Electronics Letters, 1995, Feb., Vol. 31, No. 3, pp. 163–165.Google Scholar
  28. [28]
    S. Benedetto and G. Montorsi, Design of Parallel Concatenated Convolutional Codes, IEEE Trans. Commun. 1996, May, Vol. 44, No. 5, pp. 591–600.zbMATHCrossRefGoogle Scholar
  29. [29]
    S. Benedetto, G. Montorsi, D. Divsalar and F. Pollara, Serial Concatenation of Interleaved Codes: Performance Analysis, Design, and Iterative Decoding, TDA Progress Report 42–126, Aug. 1996.Google Scholar
  30. [30]
    S. Benedetto and G. Montorsi, Unveiling Turbo Codes: Some Results on Parallel Concatenated Coding Schemes, IEEE Trans. Inform. Theory, Vol. 42, No. 2, March 1996, pp. 409–428.zbMATHCrossRefGoogle Scholar
  31. [31]
    Claude Berrou, Patrick Adde, Ettiboua Angui and Stephane Faudeil, A Low Complexity Soft-Output Viterbi Decoder Architecture, ICC93, 1993, pp. 737–740.Google Scholar
  32. [32]
    J. Lodge, R.Young, P. Hoeher, J. Hagenauer, Separable MAP “Filters” for the Decoding of Product and Concatenated Codes, ICC93, 1993, pp. 1740–1745.Google Scholar
  33. [33]
    Robert J. McEliece, Eugene R. Rodemich, Jung-Fu Cheng, The Turbo Decision Algorithm, The 33rd Alerton Conference on Communications, Computing and Control, October 1995, pp. 1–11.Google Scholar
  34. [34]
    K.Fazel, L.Papke, Combined Multilevel Turbo-code With 8PSK Modulation, GLOBECOM95. 1995, pp. 649–653.Google Scholar
  35. [35]
    Stephane Le Goff, Alain Glavieux and Claude Berrou, Turbo-Codes and High Spectral Efficiency Modulation, GLOBECOM94, 1994, pp. 645–649.Google Scholar
  36. [36]
    Lin Zhang, Weimin Zhang, Jeff T. Ball and Martin C. Gill, MILCOM96, An Extremely Robust Tutbo Coded HF Modem, 1996.Google Scholar
  37. [37]
    W.J. Blackert and S.G. Wilson, Turbo Trellis Coded Modulation, Proc. CISS’96, 1996, Princeton, NJ, USA.Google Scholar
  38. [38]
    W.J. Blackert, E.K. Hall and S.G. Wilson, An Upper Bound on Turbo Code Free Distance, ICC96, 1996, pp. 957–961.Google Scholar
  39. [39]
    Patrick Robertson and Thomas Worz, A Novel Bandwidth Efficient Coding Scheme Employing Turbo Codes, ICC96, 1996, pp. 962–967.Google Scholar
  40. [40]
    Ramesh Pyndiah, Annie Picart and Alain Glavieux, Performance of Block Coded 16-QAM and 64-QAM Modulations, GLOBECOM95, 1995, pp. 1039–1043Google Scholar
  41. [41]
    Joachim Hagenauer and Peter Hoeher, A Viterbi Algorithm with Soft-Decision Outputs and its Applications, GLOBECOM89, 1989, pp. 1680–1686.Google Scholar
  42. [42]
    U. C. G. Fiebig and P. Robertson, Soft Decision Decoding in Fast Frequency Hopping Systems with Convolutional Codes and Turbo Codes, ICC96, 1996, pp. 1064–1070.Google Scholar
  43. [43]
    Stefan Kaiser and Lutz Papke, Optimal Detection when Combining OFDM-CDMA with Convolutional and Turbo Channel Coding, ICC96, 1996, pp. 343–348.Google Scholar
  44. [44]
    Gerard Battail, Claude Berrou and Alain Glavieux, Pseudo-Random Recusive Convolutional Coding for Near-Capacity Performance, GLOBECOM93, 1993.Google Scholar
  45. [45]
    D. Divsalar, S. Dolinar, R. J. Mceliece and F. Pollara, Transfer Function Bounds on the Performance of Turbo Codes, JPL TDA Progress Report 42–122, Aug. 1995.Google Scholar
  46. [46]
    D. Divsalar and F. Pollara, Turbo Codes for Deep-Space Communications, JPL TDA Progress Report 42–120, Feb. 1995Google Scholar
  47. [47]
    D. Divsalar and F. Pollara, Mutiple Turbo Codes for Deep-Space Communications, JPL TDA Progress Report 42–121, May. 1995Google Scholar
  48. [48]
    K. Abend and B.D. Fritchman, “Statistical Detection for Communication Channels with Intersymbol Interference,” Proc. IEEE, Vol. 58, No. 5, pp. 779–785, May 1970.CrossRefGoogle Scholar
  49. [49]
    J. Raviv, “Decision Making in Markov Chains Applied to the Problem of Pattern Recognition,” IEEE Trans. Inform. Theory, Vol. IT-13, pp. 536–551, Oct. 1967Google Scholar
  50. [50]
    L.R. Bahl, J. Cocke, F. Jelinek and J. Raviv, “Optimal Decoding of Linear Codés for Minimizing Symbol Error Rate,” IEEE Trans. Inform. Theory, Vol. IT-20, pp. 284–287, Mar. 1974.Google Scholar
  51. [51]
    R.W. Chang and J.C. Hancock “On Receiver Structures for Channels Having Memory,” IEEE Trans. Inform. Theory, Vol. IT-12, pp. 463 - 468, Oct. 1966.Google Scholar
  52. [52]
    P.R. Chevillat and E. Eleftheriou, “Decoding of Trellis-Encoded Signals in the Presence of Intersymbol Interference and Noise,” IEEE Trans. Commun., Vol. 37, No. 7, pp. 669–676, Jul. 1989.CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 1997

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  • Branka Vucetic

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