Invariant Measures of Tunable Chaotic Sources: Robustness Analysis and Efficient Estimation

Abstract

In this paper, a theoretical approach for studying the robustness of the chaotic statistics of piecewise affine maps with respect to parameter perturbations is discussed. The approach is oriented toward the study of the effects that the nonidealities derived from the circuit implementation of these chaotic systems have on their dynamics. The ergodic behavior of these systems is discussed in detail, adopting the approach developed by Boyarsky and Gora, with particular reference to the family of sawtooth maps, and the robustness of their invariant measures is studied. Although this paper is particularly focused on this specific family of maps, the proposed approach can be generalized to other piecewise affine maps considered in the literature for information and communications technology applications. Moreover, in this paper, an efficient method for estimating the unique invariant density for stochastically stable piecewise affine maps is proposed. The method is an alternative to Monte Carlo methods and to other methods based on the discretization of the Frobenius-Perron operator

Histogram test of ADCs with chaotic samples

“…2, as far as the parameter B gets close to 2, the stationary pdf φ * for the Sawtooth map better approximates an uniform pdf. This result has a theoretical explanation in the robustness property that this system has with respect to parameter perturbations [4].…”

“…In such case, the pdf φ * is not uniform, and only in few trivial cases it can be calculated analytically. Nevertheless, it can be always estimated with accurate numerical computations [4], [10]. As it can be observed in the first row of Fig.…”

“…IV). The parameter B and α can be set, adopting the same strategy discussed in [7], [8], and we assume for the parameter space 0 ≤ |α| < 0.17 and √ 2 < B ≤ 2 1+|α| , such that the Sawtooth map is an exact transformation over the attractor Λ = [−1, +1) [4]. If B > 2 1+|α| the dynamics diverge, and the output of any circuit implementing (1) saturates to a parasitic fixed point [9].…”

“…where B Λ is the Borel σ-algebra of subsets of Λ [4]. By denoting with D BV (Λ) the set of pdfs of bounded variation over Λ, we recall that exact systems like (1) are statistically stable [5], [6], i.e., they admit an unique stationary pdf φ * ∈ D BV (Λ) such that if φ 0 = φ * then the stochastic process {z m } is ergodic; otherwise, if φ 0 = φ * , regardless of φ 0 ∈ D BV (Λ) (i.e., regardless of the uncertainty about the initial condition z 0 ), the dynamics of the Sawtooth map (1) approaches an ergodic process with a convergence rate not slower than exponential (less than one hundred iterations suffices in practical cases) [4].…”

“…In the last decades, the Ergodic Theory developed for the analysis of dynamical systems has shown that chaotic systems can be used for the design of stochastic processes with predefined statistical characteristics [4]- [6]. In this paper we discuss the possible use of chaotic signals for testing ADCs, with particular reference to the CDT.…”