Diagonal Kernel Point Estimation of th-Order Discrete Volterra-Wiener Systems
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Diagonal Kernel Point Estimation of nth-Order Discrete Volterra-Wiener Systems Massimiliano Pirani Dipartimento di Elettronica, Intelligenza artificiale e Telecomunicazioni, Universit`a Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy Email: [email protected]
Simone Orcioni Dipartimento di Elettronica, Intelligenza artificiale e Telecomunicazioni, Universit`a Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy Email: [email protected]
Claudio Turchetti Dipartimento di Elettronica, Intelligenza artificiale e Telecomunicazioni, Universit`a Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy Email: [email protected] Received 1 September 2003; Revised 18 February 2004 The estimation of diagonal elements of a Wiener model kernel is a well-known problem. The new operators and notations proposed here aim at the implementation of efficient and accurate nonparametric algorithms for the identification of diagonal points. The formulas presented here allow a direct implementation of Wiener kernel identification up to the nth order. Their efficiency is demonstrated by simulations conducted on discrete Volterra systems up to fifth order. Keywords and phrases: nonlinear system identification, Wiener kernels, Volterra filtering.
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INTRODUCTION
Among the identification techniques based on input-output correlations, the one proposed by Lee and Schetzen [1] is the most widely adopted due to its versatility, even if more recent techniques and up-to-date insights on these arguments can be found in [2] and more references in [3]. The application of the Lee-Schetzen technique on discrete nonlinear systems is straightforward and also gains some validity advantages versus the continuous time version, as stated rigorously in [4] and in [5]. In [6], the authors describe some characteristic behaviors of the Lee-Schetzen method for discrete systems and propose practical suggestions on its use. The estimation of diagonal elements of a Wiener model kernel is a well-known problem. Such problem can be found documented in [6, 7, 8]. It arises from the higher estimation error variance exhibited by the estimation process of the kernel points having at least two equal coordinates. In [6], some explanations for this phenomenon, which augments increasing the number of equal coordinates, are given. The original Lee-Schetzen identification technique was particularly subject to this kind of errors. Goussard et al., in [9], made a ma-
jor contribution to the solution of the diagonal point estimation problem, although their work contains explicit solutions and proofs only up to the third order. Koukoulas and Kalouptsidis, in [10], using the results on the calculation of cumulants due to the work of Leonov and Shiryaev [11], proposed a proof of the nth-order case valid also for inputs drawn from nonwhite Gaussian distributions. In the white Gaussian input case, the general formulas in [10] can be shown to reduce to Goussard’s method. Other formulas using cumulants to estimate Wiener kernels
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