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A Robust Maximum Likelihood Algorithm for Blind Equalization of Communication Systems Impaired by Impulsive Noise

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Abstract

To improve the performance of the blind equalizer (BE) in impulsive noise environments, a robust maximum likelihood algorithm (RMLA) is proposed for the communication systems using quadrature amplitude modulation signals. A novel robust maximum likelihood cost function based on the constant modulus algorithm is constructed to effectively suppress the influence of impulsive noise and ensure the computational stability. Theoretical analysis is presented to illustrate the robustness and good computational stability of the proposed algorithm under the impulsive noise ambient. Moreover, it is proved that the weight vector of the proposed BE can converge stably by LaSalle invariance principle. Simulation results are provided to further confirm the robustness and stability of the proposed RMLA.

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References

  1. Y. Cai, R.C. de Lamare, M. Zhao, J. Zhong, Low-complexity variable forgetting factor mechanism for blind adaptive constrained constant modulus algorithms. IEEE Trans. Signal Process. 60(8), 3988–4002 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Chitre, J. Potter, O.S. Heng, Underwater acoustic channel characterization for medium-range shallow water communications, in Proceedings of MTTS/IEEE OCEANS’04 Conference, vol 1 (2004), pp. 9–12

  3. Z. Ding, Y. Li, Blind Equalization and Identification (Marcel Dekker Inc., New York, 2001)

    Book  Google Scholar 

  4. L.M. Garth, A dynamic convergence analysis of blind equalization algorithms. IEEE Trans. Commun. 49, 624–634 (2001)

    Article  MATH  Google Scholar 

  5. D. Godard, Self-recovering equalization and carrier tracking in two dimensional data communications systems. IEEE Trans. Commun. 28(11), 1867–1875 (1980)

    Article  Google Scholar 

  6. S. Haykin, Blind Deconvolution (Prentice Hall, Upper Saddle River, 1994)

    Google Scholar 

  7. G.E. Hinton, S.J. Nowlan, The bootstrap Widrow-Hoff rule as a cluster-formation algorithm. Neural Comput. 2(3), 355–362 (1990)

    Article  Google Scholar 

  8. J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. D.L. Jones, A normalized constant-modulus algorithm, in Conference record of the Twenty-Ninth Asilomar Conference on Signals, Systems and Computers, vol. 1 (1995), pp. 694–697

  10. M. Kawamoto, K. Kohno, Y. Inouye, Robust eigenvector algorithms for blind deconvolution of MIMO linear systems. Circuits Syst. Signal Process. 26(4), 473–494 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Kuo, T. Chen, C.C. Fung, C. Lee, Second-order statistics based prefilter-blind equalization for MIMO-OFDM, in 2008 14th Asia-Pacific Conference on Communications (2008), pp. 1–5

  12. J.P. LaSalle, The Stability of Dynamical Systems (SIAM, Philadelphia, 1976)

    Book  MATH  Google Scholar 

  13. S. Li, L.-M. Song, T.-S. Qiu, Steady-state and tracking analysis of fractional lower-order constant modulus algorithm. Circuits Syst. Signal Process. 30, 1275–1288 (2011)

    Article  MATH  Google Scholar 

  14. J. Li, D.Z. Feng, W.X. Zheng, Space-time semi-blind equalizer for dispersive QAM MIMO system based on modified Newton method. IEEE Trans. Wirel. Commun. 13(6), 3244–3256 (2014)

    Article  Google Scholar 

  15. J. Liang, D. Wang, L. Su, B. Chen, H. Chen, H. So, Robust MIMO radar target localization via nonconvex optimization. Signal Process. 122, 33–38 (2016)

    Article  Google Scholar 

  16. J. Lim, Mixture filtering approaches to blind equalization based on estimation of time-varying and multi-path channels. J. Commun. Netw. 18(1), 8–18 (2017)

    Google Scholar 

  17. W. Liu, P. Pokharel, J. Principe, Correntropy: properties and application in non-Gaussian signal processing. IEEE Trans. Signal Process. 55(11), 5286–5296 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Mahmood, M. Chitre, M.A. Armand, On single-carrier communication in additive white symmetric alpha-stable noise. IEEE Trans. Commun. 62(10), 3584–3599 (2014)

    Article  Google Scholar 

  19. A. Naveed, I.M. Qureshi, T.A. Cheema, M.A.S. Choudhry, Blind channel equalization using second-order statistics: a necessary and sufficient condition. Circuits Syst. Signal Process. 25(4), 511–523 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. C.L. Nikias, M. Shao, Signal Processing with Alpha-Stable Distribution and Application (Wiley, New York, 1995)

    Google Scholar 

  21. T. Qiu, H. Tang, Capture properties of the generalized CMA in alpha-stable noise environment, in The 7th International Conference Signal Processing, Beijing, China (2005), pp. 439–442

  22. M. Richharia, Satellite Communication Systems: Design Principles. Tech. Eng. (Macmillan, Basingstoke, 1999)

    Book  Google Scholar 

  23. M. Rupi, P. Tsakalides, E.D. Re, C.L. Nikias, Constant modulus blind equalization based on fractional lower-order statistics. Signal Process. 84(8), 881–894 (2004)

    Article  MATH  Google Scholar 

  24. Y. Sato, A method of self-recovering equalization for multilevel amplitude modulation systems. IEEE Trans. Commun. 23(6), 679–682 (1975)

    Article  Google Scholar 

  25. V. Savaux, F. Bader, J. Palicot, OFDM/OQAM blind equalization using CMA approach. IEEE Trans. Signal Process. 64(9), 2324–2333 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Scarano, A. Petroni, M. Biagi, R. Cusani, Second-order statistics driven LMS blind fractionally spaced channel equalization. IEEE Signal Process. Lett. 24(2), 161–165 (2017)

    Google Scholar 

  27. L. Sen, Q. Tian Shuang, Z. Dai Feng, Adaptive blind equalization for MIMO systems under α-stable noise environment. IEEE Commun. Lett. 13(8), 609–611 (2009)

    Article  Google Scholar 

  28. O. Shalvi, E. Weinstein, Super-exponential methods for blind equalization. IEEE Trans. Inf. Theory 39(2), 505–519 (1993)

    Article  MATH  Google Scholar 

  29. M. Shao, C.L. Nikias, Signal processing with fractional lower order moment stable processes and their applications. IEEE Proc. 81(7), 986–1010 (1993)

    Article  Google Scholar 

  30. L. Tong, G. Xu, T. Kailath, Blind identification and equalization based on second-order statistics: a time domain approach. IEEE Trans. Inf. Theory 40, 340–349 (1994)

    Article  Google Scholar 

  31. J. Treichler, B. Agee, A new approach to multipath correction of constant modulus signals. IEEE Trans. Acoust. Speech Signal Process. 31(2), 459–471 (1983)

    Article  Google Scholar 

  32. S. Unawong, S. Miyamoto, N. Morinaga, A novel receiver design for DS-CDMA systems under impulsive radio noise environments. IEICE Trans. Commun. E82-B, 936–943 (1999)

    Google Scholar 

  33. B. Wang, Y. Zhang, W. Wang, Robust group compressive sensing for DOA estimation with partially distorted observations. EURASIP J. Adv. Signal Process. 2016, 128 (2016)

    Article  Google Scholar 

  34. B. Wang, Y. Zhang, W. Wang, Robust DOA estimation in the presence of miscalibrated sensors. IEEE Signal Process. Lett. 24(7), 1073–1077 (2017)

    Article  Google Scholar 

  35. J. Yuan, J. Chao, T. Lin, Effect of channel noise on blind equalization and carrier phase recovery of CMA and MMA. IEEE Trans. Commun. 60(11), 3274–3285 (2012)

    Article  Google Scholar 

  36. B. Zhang, J. Yu, W. Kuo, Fast convergence on blind and semi-blind channel estimation for MIMO–OFDM systems. Circuits Syst. Signal Process. 34(6), 1993–2013 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grants 61801363, 61271299 and 61501348, the Natural Science Foundation of Shaanxi Province under Grant 2017JM6039, the Basic Scientific Research Foundation of Xidian University under Grants 8002/20101166309 and 8002/20103166309.

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Correspondence to Jin Li.

Appendix A

Appendix A

1.1 Proof of Proposition 1

To prove the function, \( J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \) is a Lyapunov function, \( J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \) must satisfy the three conditions mentioned in Definition 1. So we divide the proof into three parts to prove Proposition 1.

  1. 1.

    \( J_{\text{RMLA}} ({\mathbf{w}}) \) is constructed of basic functions, such as linear function \( y(n) = {\mathbf{w}}^{H} {\mathbf{x}}(n) \), absolute value function \( \left| {y(n)} \right| \), quadratic function \( (\left| {y(n)} \right| - R)^{2} \). Obviously, the function \( J_{\text{RMLA}} ({\mathbf{w}}) \) is continuous.

  2. 2.

    We first construct iteration direction \( {\mathbf{d}}_{n + 1} = {\mathbf{w}}_{n + 1} - {\mathbf{w}}_{n} = \left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \times R\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right)\left| {y_{n} \left( k \right)} \right|^{ - 1} y_{n}^{*} \left( k \right){\mathbf{x}}\left( k \right)}}{{Y_{n} \left( k \right) + C}}} - {\mathbf{w}}_{n} \) and prove its descent property. It is well known that \( \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} ){\mathbf{d}}_{n + 1} < 0 \) implies that \( {\mathbf{d}}_{n + 1} \) is a descent direction when \( \nabla J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \ne 0 \). This is completely proved by

    $$ \begin{aligned} & \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} ){\mathbf{d}}_{n + 1} \\ &\quad = \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} ) \times \left( {\left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \times R\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right)\left| {y_{n} \left( k \right)} \right|^{ - 1} y_{n}^{*} \left( k \right){\mathbf{x}}\left( k \right)}}{{Y_{n} \left( k \right) + C}}} - {\mathbf{w}}_{n} } \right) \\ &\quad = \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} )\left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \\ &\quad \times \left( {R\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right)\left| {y_{n} \left( k \right)} \right|^{ - 1} y_{n}^{*} \left( k \right){\mathbf{x}}\left( k \right)}}{{Y_{n} \left( k \right) + C}}} - \left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right){\mathbf{w}}_{n} } \right) \\ &\quad = - \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} )\left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \nabla J_{\text{RLMA}} ({\mathbf{w}}_{n} ) < 0 . \end{aligned} $$
    (A.1)

    Because \( \left( {\sum\nolimits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \) is positive definite.

    Hence, the following equation holds

    $$ J_{\text{RMLA}} ({\mathbf{w}}_{n} + \lambda {\mathbf{d}}_{n + 1} ) \le J_{\text{RMLA}} ({\mathbf{w}}_{n} ) $$
    (A.2)

    where parameter \( \lambda \) is positive step size that is small enough. When \( \lambda = 1 \), we have \( {\mathbf{w}}_{n + 1} = {\mathbf{w}}_{n} + {\mathbf{d}}_{n + 1} = \left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \times R\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right)\left| {y_{n} \left( k \right)} \right|^{ - 1} y_{n}^{*} \left( k \right){\mathbf{x}}\left( k \right)}}{{Y_{n} \left( k \right) + C}}} \) and \( J_{\text{MMA}} ({\mathbf{w}}_{k + 1} ) \le J_{\text{MMA}} ({\mathbf{w}}_{k} ) \).

  3. 3.

    For any finite constant \( \xi \) in the codomain of \( J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \), it is obvious that set \( \left\{ {{\mathbf{w}}_{n} \left| {J_{\text{RMLA}} ({\mathbf{w}}_{n} )} \right. < \xi } \right\} \) is bounded. Because \( J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \ge \xi \) holds when \( {\mathbf{w}}_{n} \) is unbounded.

    Through the analysis of above three points, we can know that \( J_{\text{MMA}} ({\mathbf{w}}_{k} ) \) is a Lyapunov function. Finally, we can give the conclusion that the discrete sequence \( {\mathbf{w}}_{k} \) converges to the invariance set \( \tilde{\varOmega } = \left\{ {{\mathbf{w}}_{k} \left| {J_{\text{MMA}} ({\mathbf{w}}_{k + 1} ) - J_{\text{MMA}} ({\mathbf{w}}_{k} ) = 0} \right.} \right\} \) according to LaSalle invariance principle. This completes the proof of Proposition 1.

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Li, J., Feng, DZ., Li, B. et al. A Robust Maximum Likelihood Algorithm for Blind Equalization of Communication Systems Impaired by Impulsive Noise. Circuits Syst Signal Process 38, 2387–2401 (2019). https://doi.org/10.1007/s00034-018-0966-6

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