Memoryless nonlinear response: A simple mechanism for the 1/f noise

Discovering the mechanism underlying the ubiquity of $"1/f^{\alpha}"$ noise has been a long--standing problem. The wide range of systems in which the fluctuations show the implied long--time correlations suggests the existence of some simple and general mechanism that is independent of the details of any specific system. We argue here that a {\it memoryless nonlinear response} suffices to explain the observed non--trivial values of $\alpha$: a random input noisy signal $S(t)$ with a power spectrum varying as $1/f^{\alpha'}$, when fed to an element with such a response function $R$ gives an output $R(S(t))$ that can have a power spectrum $1/f^{\alpha}$ with $\alpha < \alpha'$. As an illustrative example, we show that an input Brownian noise ($\alpha'=2$) acting on a device with a sigmoidal response function R(S)= \sgn(S)|S|^x, with $x<1$, produces an output with $\alpha = 3/2 +x$, for $0 \leq x \leq 1/2$. Our discussion is easily extended to more general types of input noise as well as more general response functions.Comment: 5 pages, 5 figure

Nonlinear system modeling based on constrained Volterra series estimates

A simple nonlinear system modeling algorithm designed to work with limited \emph{a priori }knowledge and short data records, is examined. It creates an empirical Volterra series-based model of a system using an $l_{q}$-constrained least squares algorithm with $q\geq 1$. If the system $m\left( \cdot \right)$ is a continuous and bounded map with a finite memory no longer than some known $\tau$, then (for a $D$ parameter model and for a number of measurements $N$) the difference between the resulting model of the system and the best possible theoretical one is guaranteed to be of order $\sqrt{N^{-1}\ln D}$, even for $D\geq N$. The performance of models obtained for $q=1,1.5$ and $2$ is tested on the Wiener-Hammerstein benchmark system. The results suggest that the models obtained for $q>1$ are better suited to characterize the nature of the system, while the sparse solutions obtained for $q=1$ yield smaller error values in terms of input-output behavior

Signal representation for compression and noise reduction through frame-based wavelets

Published versio

Phase-locked Loop Dynamics in the Presence of Noise by Fokker-planck Techniques

Phase error behavior of phase-locked loop tracking system in presence of gaussian noise determined by fokker-planck equatio