Wireless Personal Communications

, Volume 108, Issue 4, pp 2077–2102 | Cite as

Performance Analysis of Four Dimensional Turbo Code (4D-TC) Using Moment Based Simplified Augmented State Diagram (MSASD) Approach: Extension to LTE System

  • Subhabrata BanerjeeEmail author
  • Sudipta Chattopadhyay


Thought provoking “Turbo Code” (TC) is found to be as one of the competent error control coding schemes not only for its amazing “Bit Error Rate” (BER) act but also for its proficiency in secured data transmission with least errors. However, “TC” is incapable of lessening the knocking down effect because of low “Minimum Hamming Distance (MHD)” \(d_{\hbox{min} }\). This problem can be resolved either by designing greater memory driven “TCs” or by conniving a coding system driven by higher number of encoding units. Moreover, the evaluation of “constricted and accurate upper bound” on “BER” can be considered as another interesting scope of research in this regard. Inspired by this research orientation, in this article, the Moment based Simplified Augmented State Diagram (MSASD) has been proposed to evaluate the “Transfer Function” for the assessment of the “upper bound” on the “BER” act of the “4D-TC” scheme. For this purpose, “BER” act of the two memories driven “4D-TC” model has been inspected using “simulation”, “Augmented State Diagram” (ASD) and “MSASD” techniques. Moreover, a qualified investigation has been done on the “BER” performances of the “4D-TC” for different code rates. Furthermore, the “BER” behavior of the “4D-TC” has been compared with “3D-TC” using three different approaches namely, “simulation”, “ASD” and “MSASD”. Finally, a comprehensive BER comparison among different forms of TCs has been performed in LTE system under different fading channel conditions.


Augmented state diagram BER Code rate Four dimensional turbo code Shannon-Happ method 



  1. 1.
    Berrou, C., Glavieux, A., & Thitimajshima, P. (1993). Near Shannon limit error-correcting coding and decoding: Turbo-codes. In: Proceedings of IEEE international conference on communications (ICC’93), Geneva, Switzerland (pp. 1064–1070).
  2. 2.
    Divsalar, D., Dolinar, S., McEliece, R. J., & Pollara, F. (1995). Transfer function bounds on the performance of turbo codes. In JPL, Cal. Tech., TDA Progr. Rep. 42-121.Google Scholar
  3. 3.
    Benedetto, S., & Montorsi, G. (1996). Unveiling turbo codes: Some results on parallel concatenated coding schemes. IEEE Trans. Inform. Theory, 42(2), 409–429. Scholar
  4. 4.
    Benedetto, S., Divsalar, D., Montorsi, G., & Pollara, F. (1998). Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding. IEEE Transactions on Information Theory, 44(3), 909–926. Scholar
  5. 5.
    Perez, L. C., Seghers, J., & Costello, D. J. (1996). A distance spectrum interpretation of turbo codes. IEEE Transactions on Information Theory, 42(6), 1698–1709. Scholar
  6. 6.
    Takeshita, O. Y., Collins, O. M., Massey, P. C., & Costello, D. J. (1999). A note on asymmetric turbo-codes. IEEE Communication Letter, 3(3), 69–71. Scholar
  7. 7.
    IEEE standard for local and metropolitan area networks. IEEE 802.16a, 2003.Google Scholar
  8. 8.
    Berrou, C., i Amat, A. G., Ould-Cheikh-Mouhamedou, Y., Douillard, C., & Saouter, Y. (2007). Adding a rate-1 third dimension to turbo codes. In Proceedings of IEEE Information Theory Workshop (ITW’07), Lake Taho, CA, (pp. 156–161).
  9. 9.
    Berrou, C., i Amat, A. G., Ould-Cheikh-Mouhamedou, Y., & Saouter, Y. (2009). Improving the distance properties of turbo codes using a third component code: 3D turbo codes. IEEE Transactions on Communication, 57(9), 2505–2509. Scholar
  10. 10.
    Rosnes, E., & i Amat, A. G. (2011). Performance analysis of 3-D turbo codes. IEEE Transactions on Information Theory, 57(6), 3707–3720. Scholar
  11. 11.
    Kbaier, D., Douillard, C., & Kerouedan, S. (2012). Analysis of three-dimensional turbo codes. Annales des Telecommunications, 67(5–6), 257–268. Scholar
  12. 12.
    Morero, D. A., & Hueda, M. R. (2013). Novel serial code concatenation strategies for error floor mitigation of low-density parity-check and turbo product codes. Canadian Journal of Electrical and Computer Engineering, 36(2), 52–59. Scholar
  13. 13.
    Breddermann, T., & Vary, P. (2014). Rate-compatible insertion convolutional turbo codes: analysis and application to LTE. IEEE Transactions on Wireless Communications, 13(3), 1356–1366. Scholar
  14. 14.
    Kayani, A., Aziz, K., & Khattak, S. (2014). Optimising the activation cycle of turbo decoders using three-dimensional extrinsic information transfer charts. IET Communication, 8(15), 2706–2712. Scholar
  15. 15.
    Moloudi, S., Lentmaier, M., & i Amat, A. G. (2015). Threshold saturation for spatially coupled turbo-like codes over the binary erasure channel. In Proceedings of IEEE information theory workshop - fall (ITW) (pp. 138–142).
  16. 16.
    Kraidy, G. M. (2016). On progressive edge-growth interleavers for turbo codes. IEEE Communications Letters, 20(2), 200–203. Scholar
  17. 17.
    Brejza, M. F., Li, L., Maunder, R. G., Al-Hashimi, B. M., Berrou, C., & Hanzo, L. (2016). 20 years of turbo coding and energy-aware design guidelines for energy-constrained wireless applications. IEEE Communication Surveys & Tutorials, 18(1), 8–28. Scholar
  18. 18.
    Tonnellier, T., Leroux, C., Le Gal, B., Gadat, B., Jego, C., & Van Wambeke, N. (2016). Lowering the error floor of turbo codes with CRC verification. IEEE Wireless Communications Letters, 5(4), 404–407. Scholar
  19. 19.
    Mukhtar, H., Al-Dweik, A., & Shami, A. (2016). Turbo product codes: Applications, challenges, and future directions. IEEE Communication Surveys & Tutorials, 18(4), 3052–3069. Scholar
  20. 20.
    Garzón-Bohórquez, R., Nour, C. A., & Douillard, C. (2016). Improving turbo codes for 5G with parity puncture-constrained interleavers. In Proceedings of 9th international symposium on turbo codes and iterative information processing (ISTC) (pp. 151–155).
  21. 21.
    Banerjee, S., & Chattopadhyay, S. (2016). Improved three dimensional turbo code using superposition modulation techniques: Extension to WiMAX system. In Proceedings of IEEE Uttar Pradesh Section international conference on electrical, computer and electronics engineering (UPCON-2016). Google Scholar
  22. 22.
    Trifina, L., & Tarniceriu, D. (2017). On the equivalence of cubic permutation polynomial and ARP interleavers for turbo codes. IEEE Transactions on Communications, 65(2), 473–485. Scholar
  23. 23.
    Chatzigeorgiou, I., Rodrigues, M. R. D., Wassell, I. J., & Carrasco, R. A. (2006). A novel technique for the evaluation of the transfer function of parallel concatenated convolutional codes. In Proceedings of 6th international ITG-conference on source and channel coding (pp. 1–6).Google Scholar
  24. 24.
    Chatzigeorgiou, I., Rodrigues, M. R. D., Wassell, I. J., & Carrasco, R. A. (2006). A novel technique to evaluate the transfer function of punctured turbo codes. In Proceedings of IEEE international conference on communications (pp. 1166–1171).
  25. 25.
    Chatzigeorgiou, I., Rodrigues, M. R. D., Wassell, I. J., & Carrasco, R. A. (2007). A union bound approximation for rapid performance evaluation of punctured turbo codes. In Proceedings of IEEE 41st annual conference on information sciences and systems (pp. 474–479).
  26. 26.
    Caldeira, L. G., de Souza, A. F. G., Pimentel, C., Fernandes, H. C. C. (2009). A new technique to evaluate an expurgated bound on punctured turbo trellis coded modulation. In Proceedings of SBMO/IEEE MTT-S international microwave and optoelectronics conference (IMOC) (pp. 578–581).
  27. 27.
    Chatzigeorgiou, I., Rodrigues, M. R. D., Wassell, I. J., & Carrasco, R. A. (2009). The augmented state diagram and its application to convolutional and turbo codes. IEEE Transactions on Communication, 57(7), 1948–1958. Scholar
  28. 28.
    Jaspar, X., & Vandendorpe, L. (2010). Performance bounds and distance spectra of variable length codes in turbo/concatenated systems. IEEE Transactions on Communication, 58(9), 2537–2548. Scholar
  29. 29.
    Xia, Shao, Weidang, Zhang, & Ping, Li. (2014). Improving the bit error rate performance by optimising energy allocation based on union bound. IET Communication, 8(18), 3366–3371. Scholar
  30. 30.
    Banerjee, S., & Chattopadhyay, S. (2016). A novel simplified augmented state diagram of three dimensional turbo code. In Proceedings of IEEE international conference on power electronics, intelligent control and energy systems (ICPEICES-2016) (pp. 3366–3371).
  31. 31.
    Banerjee, S., & Chattopadhyay, S. (2017). Evaluation of system performance by adding a fourth dimension to turbo code. International Journal of Communication System. Scholar
  32. 32.
    Mason, S. J. (1953). Feedback theory-some properties of signal flow graphs. Proceedings of IRE, 41(9), 1144–1156.CrossRefGoogle Scholar
  33. 33.
    Johnson, D. E., & Johnson, J. R. (1972). Graph theory, with engineering applications. New York: Ronald Press.zbMATHGoogle Scholar
  34. 34.
    Ren, Y. (2011). The methodology of flowgraph models. Ph.D. dissertation. London: Department of Statistics, London School of Economics and Political Science.Google Scholar
  35. 35.
    Gangadi, A. (2010). Object-oriented implementation of LTE turbo codes. Thesis for Master of Science, West Virginia University.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electronics and Communication EngineeringFuture Institute of Engineering and ManagementKolkataIndia
  2. 2.Department of Electronics and Telecommunication EngineeringJadavpur UniversityKolkataIndia

Personalised recommendations