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**Nonlinear blind identification with three-dimensional tensor analysis.**
*(English)*
Zbl 1264.93252

Summary: This paper deals with the analysis of a third-order tensor composed of a fourth-order output cumulants used for blind identification of a second-order Volterra-Hammerstein series. It is demonstrated that this nonlinear identification problem can be converted in a multivariable system with multiequations having the form of \(Ax + By = c\). The system may be solved using several methods. Simulation results with the Iterative Alternating Least Squares (IALS) algorithm provide good performances for different signal-to-noise ratio (SNR) levels. Convergence issues using the reversibility analysis of matrices \(A\) and \(B\) are addressed. Comparison results with other existing algorithms are carried out to show the efficiency of the proposed algorithm.

### MSC:

93E12 | Identification in stochastic control theory |

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\textit{I. Cherif} et al., Math. Probl. Eng. 2012, Article ID 284815, 22 p. (2012; Zbl 1264.93252)

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### References:

[1] | M. Li, C. Cattani, and S. Y. Chen, “Viewing sea level by a one-dimensional random function with long memory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 2011. · doi:10.1155/2011/654284 |

[2] | S. C. Lim, C. H. Eab, K. H. Mak, M. Li, and S. Y. Chen, “Solving linear coupled fractional differential equations by direct operational method and some applications,” Mathematical Problems in Engineering, vol. 2012, Article ID 653939, 28 pages, 2012. · Zbl 1264.34009 · doi:10.1155/2012/653939 |

[3] | S. Y. Chen and Q. Guan, “Parametric shape representation by a deformable NURBS model for cardiac functional measurements,” IEEE Transactions on Biomedical Engineering, vol. 58, no. 3, pp. 480-487, 2011. · doi:10.1109/TBME.2010.2087331 |

[4] | S. Chen, H. Tong, and C. Cattani, “Markov models for image labeling,” Mathematical Problems in Engineering, vol. 2012, Article ID 814356, 18 pages, 2012. · Zbl 1264.68189 · doi:10.1155/2012/814356 |

[5] | E. G. Bakhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherence functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 436198, 2011. · Zbl 1248.37075 · doi:10.1155/2011/436198 |

[6] | L. Agarossi, S. Bellini, F. Bregoli, and P. Migliorati, “Simplified nonlinear receivers for the optical disc channel,” in Proceedings of the IEEE International Conference on Communications (ICC ’99), vol. 3, pp. 1945-1950, June 1999. |

[7] | K. Watanabe, Y. W. Chen, and K. Yamashita, “Blind nonlinear equalization using 4th -order cumulants,” in Proceedings of the Sice 2002, pp. 2128-2131, Osaka, Japan, August 2002. |

[8] | I. Cherif and F. Fnaiech, “Blind identification of a second order Volterra-Hammerstein series using cumulant cubic tensor analysis,” in Proceedings of the 34th Annual Conference of the IEEE Industrial Electronics Society (IECON ’08), pp. 1851-1856, Orlando, Fla, USA, November 2008. · doi:10.1109/IECON.2008.4758237 |

[9] | N. Kalouptsidis and P. Koukoulas, “Blind identification of volterra-hammerstein systems,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 2777-2787, 2005. · Zbl 1370.93089 |

[10] | G. Favier, A. Y. Kibangou, and T. Bouilloc, “Nonlinear system modeling and identification using Volterra-PARAFAC models,” International Journal of Adaptive Control and Signal Processing, vol. 26, pp. 30-53, 2012. · Zbl 1263.93227 |

[11] | T. Bouilloc and G. Favier, “Nonlinear channel modeling and identification using baseband VolterraParafac models,” Signal Processing, vol. 92, no. 6, pp. 1492-1498, 2012. · doi:10.1016/j.sigpro.2011.12.007 |

[12] | G. Favier, M. N. da Costa, A. L. F. de Almeida, and J. M. T. Romano, “Tensor space-time (TST) coding for MIMO wireless communication systems,” Signal Processing, vol. 92, pp. 1079-1092, 2012. |

[13] | T. Stathaki and A. Scohyers, “Constrained optimization approach to the blind estimation of Volterra kernels,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’97), vol. 3, pp. 2373-2376, April 1997. |

[14] | C. E. R. Fernandes, G. Favier, and J. C. M. Mota, “Tensor-based blind channel identification,” in Proceedings of the IEEE International Conference on Communications (ICC ’07), pp. 2728-2732, June 2007. · doi:10.1109/ICC.2007.453 |

[15] | P. Koukoulas and N. Kalouptsidis, “Blind identification of second order Hammerstein series,” Signal Processing, vol. 83, no. 1, pp. 213-234, 2003. · Zbl 1011.94510 · doi:10.1016/S0165-1684(02)00393-6 |

[16] | G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, Md, USA, 3rd edition, 1996. · Zbl 0865.65009 |

[17] | Y. Hua, A. B. Gershman, and Q. Cheng, High Resolution and Robust Signal Processing, Marcell Dekker, New York, NY, USA, 2004. |

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