Performance Analysis of Adaptive Volterra Filters in the Finite-Alphabet Input Case
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Performance Analysis of Adaptive Volterra Filters in the Finite-Alphabet Input Case Hichem Besbes Ecole Sup´erieure des Communications de Tunis (Sup’Com), Ariana 2083, Tunisia Email: [email protected]
´ Meriem Ja¨ıdane Ecole Nationale d’Ing´enieurs de Tunis (ENIT), Le Belvedere 1002, Tunisia Email: [email protected]
Jelel Ezzine Ecole Nationale d’Ing´enieurs de Tunis (ENIT), Le Belvedere 1002, Tunisia Email: [email protected] Received 15 September 2003; Revised 21 May 2004; Recommended for Publication by Fulvio Gini This paper deals with the analysis of adaptive Volterra filters, driven by the LMS algorithm, in the finite-alphabet inputs case. A tailored approach for the input context is presented and used to analyze the behavior of this nonlinear adaptive filter. Complete and rigorous mean square analysis is provided without any constraining independence assumption. Exact transient and steadystate performances expressed in terms of critical step size, rate of transient decrease, optimal step size, excess mean square error in stationary mode, and tracking nonstationarities are deduced. Keywords and phrases: adaptive Volterra filters, LMS algorithm, time-varying channels, finite-alphabet inputs, exact performance analysis.
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INTRODUCTION
Adaptive systems have been extensively designed and implemented in the area of digital communications. In particular, nonlinear adaptive filters, such as adaptive Volterra filters, have been used to model nonlinear channels encountered in satellite communications applications [1, 2]. The nonlinearity is essentially due to the high-power amplifier used in the transmission [3]. When dealing with land-mobile satellite systems, the channels are time varying and can be modeled by a general Mth-order Markovian model to describe these variations [4]. Hence, to take into account the effect of the amplifier’s nonlinearity and channel variations, one can model the equivalent baseband channel by a time-varying Volterra filter. In this paper, we analyze the behavior and parameters tracking capabilities of adaptive Volterra filters, driven by the generic LMS algorithm. In the literature, convergence analysis of adaptive Volterra filters is generally carried out for small adaptation step size [5]. In addition, a Gaussian input assumption is used in order to take advantage of the Price theorem results. However, from a practical viewpoint, to maximize the rate of convergence or to determine the critical step size, one needs
a theory that is valid for large adaptation step size range. To the best knowledge of the authors, no such exact theory exists for adaptive Volterra filters. It is important to note that the so-called independence assumption, well known of being a crude approximation for large step size range, is behind all available results [6]. The purpose of this paper is to provide an approach tailored for the finite-alphabet input case. This situation is frequently encountered in many digital transmission systems. In fact, we develop an exact convergence analys
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