Noninvertible Nonlinearities Research Articles

An Algebraic Approach to Efficient Identification of a Class of Wiener Systems

This paper considers the problem of identifying the linear portion of a Wiener system, for the case of a known, but non-invertible non-linearity. It is well known that this scenario, common in many practical applications, leads to NP-hard problems in the number of experiments. Thus, existing techniques scale poorly and are typically limited to relatively few points. We show that this difficulty can be circumvented by considering an algebraic motivated approach. Specifically, we show that it is equivalent to identification of a switched linear system generated from the data. In turn, we can solve this problem by recasting it as the problem of finding the vanishing ideal of an arrangement of subspaces, a task that reduces to finding the null space of an embedded data matrix constructed from observed data.

Non-Iterative Identification of IIR Wiener Systems with Non-Invertible Nonlinearities

In this paper, a non-iterative method for identifying a single input single output Wiener model consisting of an IIR digital filter in cascade with a potentially non-invertible static non-linearity is developed. Initially, the linear and nonlinear elements are expanded onto bases consisting of IIR Laguerre filters and polynomials, respectively. Extensions to other bases are also considered. The parameters of the linear subsystem and the coefficients of the non-linearity are estimated by using an over-parameterized linear regression generated using a multi-index based approach. The initial over-paramerized solution is projected onto the class of Wiener models using an approach based on singular value decomposition. Finally, simulation examples and results discussions are provided.

A semiparametric Bayesian approach to Wiener system identification

We consider a semiparametric, i.e. a mixed parametric/nonparametric, model of a Wiener system. We use a state-space model for the linear dynamical system and a nonparametric Gaussian process (GP) model for the static nonlinearity. The GP model is a flexible model that can describe different types of nonlinearities while avoiding making strong assumptions such as monotonicity. We derive an inferential method based on recent advances in Monte Carlo statistical methods, known as Particle Markov Chain Monte Carlo (PMCMC). The idea underlying PMCMC is to use a particle filter (PF) to generate a sample state trajectory in a Markov chain Monte Carlo sampler. We use a recently proposed PMCMC sampler, denoted particle Gibbs with backward simulation, which has been shown to be efficient even when we use very few particles in the PF. The resulting method is used in a simulation study to identify two different Wiener systems with non-invertible nonlinearities.

Recursive Identification of Systems with Noninvertible Output Nonlinearities

The paper deals with the recursive identification of dynamic systems having noninvertible output characteristics, which can be represented by the Wiener model. A special form of the model is considered where the linear dynamic block is given by its transfer function and the nonlinear static block is characterized by such a description of the piecewise-linear characteristic, which is appropriate for noninvertible nonlinearities. The proposed algorithm is a direct application of the known recursive least squares method extended with the estimation of internal variables and enables the on-line estimation of both the linear block parameters and the parameters of some noninvertible nonlinearities and their changes. The feasibility of the proposed method is illustrated on examples of time-varying systems.

Pseudo-random number generator based on asymptotic deterministic randomness

A novel approach to generate the pseudorandom-bit sequence from the asymptotic deterministic randomness system is proposed in this Letter. We study the characteristic of multi-value correspondence of the asymptotic deterministic randomness constructed by the piecewise linear map and the noninvertible nonlinearity transform, and then give the discretized systems in the finite digitized state space. The statistic characteristics of the asymptotic deterministic randomness are investigated numerically, such as stationary probability density function and random-like behavior. Furthermore, we analyze the dynamics of the symbolic sequence. Both theoretical and experimental results show that the symbolic sequence of the asymptotic deterministic randomness possesses very good cryptographic properties, which improve the security of chaos based PRBGs and increase the resistance against entropy attacks and symbolic dynamics attacks.

The asymptotic deterministic randomness

Abstract In this Letter, we focus on the deterministic randomness theory. Based on Lissajous map, which is constructed by the skewed parabola map and the non-invertible nonlinearity transform, we present conditions for generating asymptotic deterministic randomness. We rectify several popular statements, such as the function x n = sin 2 ( θ T z n ) cannot generate deterministic randomness, and the corresponding Lyapunov exponent is ln z , etc. In other words, we prove that such function can generate deterministic randomness only when the value of parameter z belongs to some relative prime fraction number which is larger than one. We further prove that the well-known autonomous system that has been stated to generate deterministic randomness can only act as an approximative asymptotic realizable model and any realizable models for deterministic randomness will degenerate to some special high dimensional chaotic system. Furthermore, we analyze the underlying dynamics such as the fixed point, bifurcation process, Lyapunov exponent spectrum, and symbolic dynamics, etc. in detail.

A recipe for an unpredictable random number generator

In this work we present a model for computing the random processes in digital computers which solves the problem of periodic sequences and hidden errors produced by correlations. We show that systems with noninvertible non-linearities can produce unpredictable sequences of independent random numbers. We illustrate our results with some numerical calculations related to random walks simulations.

Chaotic and stochastic phenomena in systems with non-invertible non-linearities

We show that systems with non-invertible non-linearities can produce sequences of deterministically independent values. We present autonomous dynamical systems that exhibit random behavior in such a way that all variables (taken separately) are unpredictable. We study a new mechanism for chaos in which a static system without inertial or dynamical elements can produce complexity. Using some results of exactly solvable chaotic systems we investigate the influence of different chaotic signals on the phenomenon of stochastic resonance. We report the results of real experiments concerning all these phenomena.

From exactly solvable chaotic maps to stochastic dynamics

For a class of nonlinear chaotic maps, the exact solution can be written as X n = P( θk n ), where P( t) is a periodic function, θ is a real parameter and k is an integer number. A generalization of these functions: X n = P( θz n ), where z is a real parameter, can be proved to produce truly random sequences. Using different functions P( t) we can obtain different distributions for the random sequences. Similar results can be obtained with functions of type X n = h[ f( n)], where f( n) is a chaotic function and h( t) is a noninvertible function. We show that a dynamical system consisting of a chaotic map coupled to a map with a noninvertible nonlinearity can generate random dynamics. We present physical systems with this kind of behavior. We report the results of real experiments with nonlinear circuits and Josephson junctions. We show that these dynamical systems can produce a type of complexity that cannot be observed in common chaotic systems. We discuss applications of these phenomena in dynamics-based computation.