Laguerre-Volterra Filters Optimization Based on Laguerre Spectra
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Laguerre-Volterra Filters Optimization Based on Laguerre Spectra Alain Y. Kibangou Laboratoire I3S, CNRS-UNSA, Les algorithmes-Bˆat. Euclide B, 2000 route des lucioles, BP 121, 06903 Sophia Antipolis Cedex, France Email: [email protected] Laboratoire d’Electronique et Instrumentation, FSSM-UCAM, BP 2390, 40000 Marrakesh, Morocco
´ Gerard Favier Laboratoire I3S, CNRS-UNSA, Les algorithmes-Bˆat. Euclide B, 2000 route des lucioles, BP 121, 06903 Sophia Antipolis Cedex, France Email: [email protected]
Moha M. Hassani Laboratoire d’Electronique et Instrumentation, FSSM-UCAM, BP 2390, 40000 Marrakesh, Morocco Email: [email protected] Received 29 July 2004; Revised 1 April 2005; Recommended for Publication by Markus Rupp New batch and adaptive methods are proposed to optimize the Volterra kernels expansions on a set of Laguerre functions. Each kernel is expanded on an independent Laguerre basis. The expansion coefficients, also called Fourier coefficients, are estimated in the MMSE sense or by applying the gradient technique. An analytical solution to Laguerre poles optimization is provided using the knowledge of the Fourier coefficients associated with an arbitrary Laguerre basis. The proposed methods allow optimization of both the Fourier coefficients and the Laguerre poles. Keywords and phrases: Volterra filters, Laguerre basis, basis selection, nonlinear system identification.
1.
INTRODUCTION
Truncated Volterra filters constitute a class of nonrecursive polynomial models. The main drawback of these models is their over-parameterization. During the last decade, in order to reduce the parametric complexity, that is, the number of parameters, three main approaches have been considered: (i) approximations by means of parallel-cascade structures composed of linear filters and memoryless nonlinearities [1]; (ii) algebraic decompositions of matrices or tensors formed with kernels coefficients [2, 3, 4]; (iii) expansions on discrete orthonormal bases of functions (OBFs) [5, 6, 7, 8]. The class of OBFs generally used for modeling purposes is that of rational orthonormal bases, such as Laguerre basis [9]. The Laguerre functions have the property of being completely characterized by a single parameter, the Laguerre pole. When expanding a Volterra kernel on a Laguerre basis, the
parsimony of the expansion is strongly linked to the choice of the Laguerre pole. Expansion of Volterra kernels on Laguerre basis was first suggested by Wiener in the 1960s [10]. To the best of our knowledge, Campello et al. [11] were the first to derive an analytical solution to the Laguerre pole optimization for Volterra models. They have generalized the work in [12] and have also shown that using independent bases to expand each kernel gives better results than the use of a single basis. However the obtained analytical solution requires the knowledge of the Volterra kernels. Consequently, a step of Volterra kernels estimation or reconstruction is necessary before computing the Laguerre pole. Note again that this method is applicable if and only if the
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