An Overview of Fractional Order Signal Processing (FOSP) Techniques

Abstract

This paper presents a brief overview of some existing fractional order signal processing (FOSP) techniques where the developments in the mathematical communities are introduced; relationship between the fractional operator and long-range dependence is demonstrated, and fundamental properties of each technique and some of its applications are summarized. Specifically, we presented a tutorial on 1) fractional order linear systems; 2) autoregressive fractional integrated moving average (ARFIMA); 3) 1/ f α noise; 4) Hurst parameter estimation; 5) fractional order Fourier transformation (FrFT); 6) fractional order linear transforms (Hartley, Sine, Cosine); 7) fractal and 8) fractional order splines. Whenever possible, we indicate the connections between these FOSP techniques.

An improved Hurst parameter estimator based on fractional Fourier transform

“…As explained in [32][33][34], the FGN can be considered as the output of a fractional integrator. The system can be defined by the transfer function H (s) = s ν = 1…”

Fractional model based Kaiman filters for angular rate estimation in vestibular systems

“…The areas where FC was successfully applied, cover almost all the fields of modern science and engineering with lists of related references in [5], [10], [11]. A significant progress on FC related techniques has been achieved in the control area [1], [2], [4] as well as in image and signal processing [3], [17]. Extensions of the classical system theory for both fractional linear and nonlinear systems are provided in [12], [13].…”

Estimation, Analysis and Smoothing of Self-Similar Network Induced Delays in Feedback Control of Nuclear Reactors

“…Popular method to find out the Hurst parameter is the R/S analysis. Apart from that there are a number of methods like aggregated variance method, absolute value method, Periodogram method, variance of residuals method, local whittle method, wavelet based method, Higuchi method and differenced variance approach etc [12]- [16]. Recently, Chen et al [17] has reported a new Fractional Fourier Transform (FrFT) based Hurst parameter estimator which is more robust than the existing ones.…”

“…Estimation of LRD time series enhances the importance of analyzing the self similarity in the time series. The concept of LRD was firstly introduced by Mandelbrot & Van Ness [11] in terms of Fractional Brownian Motion (FBM) [12] and since then it has been addressed by many other contemporary researchers to analyze degree of selfsimilarity in a time series and leads to the concept of Hurst parameter. A second order time series There are different methods to find out the Hurst parameter of a fractal time series.…”

“…The performance of these smoothing filters are tested for a step-change in reactivity in a nuclear reactor model governed the point-kinetic equations [10] given by (12)- (13) with { } , , P C ρ being the reactor power, delayed neutron precursor concentration and reactivity respectively. The measured power signals by the SPND are considered to be transmitted through the LAN from the reactor house to the control room or RRS.…”