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Modèles de volterra à complexité réduite : estimation paramétrique et application à l'égalisation des canaux de communication

Abstract : A broad class of physical systems can be represented using the Volterra model. Particularly, it was shown that a truncated Volterra model could represent any non-linear system, time-invariant with fading memory. This model is thus particularly attractive for non-linear systems modeling and identification purpose. One of the main advantages of the Volterra model is its linearity-in-parameters, i.e. the kernel coefficients. This property allows the extension of some results established for linear model identification to this model. In addition, one can note that the Volterra model can be viewed as the natural extension of the impulse response concept of linear systems to non-linear systems. However, some limitations are to be circumvent: the number of parameters which can be very high and the ill conditioning of the matrix of the input moments used in the model estimation in the Minimal Mean Square Error (MMSE) sense. It should be noted that this ill conditioning is also the origin of the convergence slowness of the LMS (Least Mean Squares). This thesis mainly deals with these two issues. The proposed solutions are primarily based on the orthogonality concept. Firstly, orthogonality is considered according to the model structure by expanding the Volterra kernels on an orthogonal basis of rational functions. The parsimony of the expansion is strongly linked to the basis selection. In order to get parsimonious expansions, we developed new tools for Laguerre bases and Generalized Orthonormal bases (GOB) optimization. Secondly, orthogonality is considered in relation with the input signals. By exploiting the statistical properties of the input signal, multivariable orthogonal polynomials were built. The parameters of the Volterra model expanded on such bases are then estimated without any matrix inversion, which significantly simplifies the parametric estimation in the MMSE sense. The input signals orthogonalization was also considered via a Gram-Schmidt procedure. In an adaptive context, the speed convergence of LMS type algorithms is accelerated without an excessive additional computational cost. Some physical systems can be represented using a simplified Volterra model, with low parametric complexity, such as the Hammerstein and the Wiener models. That is the case of the Radio-over-fiber channel. We show particularly that a Wiener model and a Hammerstein model can respectively represent the corresponding uplink and downlink channels. In the SISO case, by using an input precoding, we suggest new algorithms for the joint estimation of the channel coefficients and the transmitted symbols in a semi-blind approach. For the uplink channel, a multi-sensors configuration is also considered. For such a configuration, thanks to a specific input precoding, we exploit the spatial diversity introduced by the sensors and the temporal diversity to obtain a tensorial representation of the received signal. By applying the tensorial decomposition technique known as PARAFAC, we carry out the joint estimation of the channel and the transmitted symbols in a blind way. Keywords: Modeling, Identification, Orthogonal Bases, Laguerre basis, Generalized orthonormal basis, orthogonal polynomials, pole optimization, parametric complexity reduction, equalization, Volterra model, Wiener model, Hammerstein model, PARAFAC decomposition.
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https://tel.archives-ouvertes.fr/tel-00190985
Contributor : Estelle Nivault <>
Submitted on : Friday, November 23, 2007 - 2:15:29 PM
Last modification on : Monday, October 12, 2020 - 10:30:28 AM
Long-term archiving on: : Monday, April 12, 2010 - 4:49:45 AM

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  • HAL Id : tel-00190985, version 1

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Alain Y. Kibangou. Modèles de volterra à complexité réduite : estimation paramétrique et application à l'égalisation des canaux de communication. Automatique / Robotique. Université Nice Sophia Antipolis, 2005. Français. ⟨tel-00190985⟩

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