Fractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even odd positive integer orders. For each one, suitable integral formulations obtained. The computations of involved integrals lead new generalizations Cauchy derivative. To compute this integral, a special two‐straight‐line path was used. With referred well‐known Riesz potential operators their inverses that emerge as true fractional derivatives, but can be...

A brief introduction to the fractional continuous-time linear systems is presented. It will be done without needing a deep study of derivatives. We show that computation impulse and step responses very similar classic. The main difference lies in substitution exponential by Mittag-Leffler function. present also formulae defining

In this paper, we introduce a unified fractional derivative, defined by two parameters (order and asymmetry). From this, all the interesting derivatives can be obtained. We study one-sided show that most known are particular cases. consider also some myths of Fractional Calculus false derivatives. The results expected to contribute limit appearance differ from existing ones just because they on distinct domains, prevent ambiguous use concept derivative.

The actual state of interplay between Fractional Calculus, Signal Processing, and Applied Sciences is discussed in this paper. A framework for compatible integer fractional derivatives/integrals signals systems context described. It shown how suitable formulations are really extensions the order definitions currently used Processing. particular case linear considered problem initial conditions tackled.

This paper addresses the present day problem of multiple proposals for operators under umbrella “fractional derivatives”. Several papers demonstrated that various those “novel” definitions are incorrect. Here classical system theory is applied to develop a unified framework clarify this important topic in Fractional Calculus.

The mathematical description of the charging process time-varying capacitors is reviewed and a new formulation proposed. For it, suitable fractional derivatives are described. case that follow Curie–von Schweidler law considered. Through substitutions, similar scheme for inductors obtained. Formulae voltage/current input/output presented. Backward coherence with classic results established generalised to variable order case. concept tempered fractor introduced related Davidson–Cole model.

In the paper, class of continuous-time linear systems is enlarged with inclusion fractional systems. These are described by differential equations. It shown how to compute impulse, step, and frequency responses from transfer function. The theory supported definitions derivative integral, generalisations usual. An introduction fractal signals as outputs differintegrators presented. define a stationary fractal.

Fractional central differences and derivatives are studied in this article. These generalisations to real orders of the ordinary positive (even odd) integer order derivatives, also coincide with well known Riesz potentials. The coherence these definitions is by applying functions Fourier transformable functions. Some properties presented particular cases studied.

Heart Rate Variability (HRV) evaluates the autonomic nervous system regulation and can be used as a monitoring tool in conditions such cardiovascular diseases, neuropathies sleep staging. It extracted from electrocardiogram (ECG) photoplethysmogram (PPG) signals. Typically, HRV is obtained ECG processing. Being PPG sensor widely clinical setups for physiological parameters blood oxygenation ventilatory rate, question arises regarding adequacy extraction. There not consensus being able to...

This paper presents the continuous-time fractional linear systems and their main properties.Two particular classes of models are introduced: autoregresive-moving average type tempered system.For both classes, computations impulse response, transfer function, frequency response discussed.It is shown that such can have integer components.From component we deduce stability.The order always stable.The initial-condition problem analysed it verified depends on structure a correct definition...

A general framework for fractional signal processing is described and used to derive several interesting formulations. This scheme based on the Liouville approach that gave rise classic Riemann-Liouville Liouville-Caputo derivatives, here dismissed. Liouville's idea consisted of fractionalizing transfer function basic definition derivative. Various coherent formulations are introduced from suitable derivative definitions corresponding ARMA-type linear systems obtained. In particular, Euler...

In the paper, class of discrete linear systems is enlarged with inclusion discrete-time fractional systems. These are described by difference equations and frequency responses. It shown how to compute impulse response transfer function. Fractal signals introduced as output special systems: differaccumulators, that can be considered having poles or zeros. The concept differaccumulation discussed, generalising notions fractal 1/f noise, introducing two kinds differaccumulated stochastic...