Signal Modeling Using Piecewise

CHAOTIC GENERATORS Thomas Schimming, Marco Gotz, Wolfgang Schwarz Institut f ur Grundlagen der Elektrotechnik und Elektronik, Technische Universitat Dresden, Mommsenstr. 13, 01062 Dresden, Germany e-mail: [email protected] ABSTRACT In this paper the modeling of a signal by a chaotic generator with respect to a speci ed signal statistic will be considered. To accomplish that, the considered class of n-dimensional piecewise linear Markov generators will rst be analyzed analytically, yielding an algebraic expression for the statistical quantity in question. Based on this analytical result an optimal set of parameters minimizing the modeling error with respect to the considered statistical quantity will be calculated. 1 MOTIVATION Statistical signal modeling is an important method in modern digital signal processing. Speci cally, parametric signal models have been widely used for the estimation of the power density spectrum, but also for higher order spectra. Usually, linear (AR, MA, ARMA) models are preferred due to the relatively straight-forward estimation of their parameters. Linear models have the disadvantageous necessity to be driven by a noise source (due to their stability). In contrast, chaotic systems need no driving source, which makes the implementation of the corresponding signal generator in a digital signal processing context much easier. A number of methods has been proposed by several authors allowing the design of spectral properties of relatively narrow classes of chaotic systems (e.g. [1]). In this paper an approach is described, where a wider class of chaotic systems (n-dimensional piecewise linear Markov systems) is rst analyzed analytically (in an automated analysis scheme), yielding an algebraic expression for the signal statistic of interest (or, at least, the statistic can be calculated exactly numerically). Based on this, the system parameters can either be obtained directly or can be found by an optimization scheme. Even in the case of numerical calculation or (LMS) optimization, the performance is by far superior to the design of chaotic generators based on simulation. 2 ANALYSIS OF THE MARKOV SYSTEM In the following the steps of the analysis will be discussed. For more detail, see [2, 3, 4, 5]. They consist of 1. analysis of the map 2. functional representation of the FPO 3. representation of the FPO on a nite polynomial subspace 4. eigensystem decomposition of the FPO 5. exact calculation of the pdf and moments (e.g. acf) and will be discussed in the following. They have all been implemented in a Maple library which allows the analysis to run automatically [3]. 2.1 System Class For the proposed approach we consider Markov systems in discrete time, on the state space Z = [ 1; 1]n where in each iteration a piece wise linear Markov map is applied: