Manuel Ortigueira | Universidade Nova de Lisboa (original) (raw)

Papers by Manuel Ortigueira

The formulations of Riemann-Liouville and Caputo derivatives in the complex plane are presented. ... more The formulations of Riemann-Liouville and Caputo derivatives in the complex plane are presented. Two versions corresponding to the whole or half plane. It is shown that they can be obtained from the Grünwald-Letnikov derivative.

Fractal and fractional, Jun 25, 2023

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Bulletin of The Polish Academy of Sciences-technical Sciences, Aug 1, 2018

A look into Fractional Calculus and their applications from the Signal Processing point of view i... more A look into Fractional Calculus and their applications from the Signal Processing point of view is done in this paper. A coherent approach to the fractional derivative is presented leading to notions that are, not only compatible with the classic, but constitute a true generalization. This means that the classic are recovered when the fractional domain is left. This happens in particular with the impulse response and transfer function. An interesting feature of the systems is in the causality that the fractional derivative imposes. The main properties of the derivatives and their representations are presented. A brief and general study of the fractional linear systems is done, by showing how to compute the impulse, step and frequecy responses, how to test the stability and how to insert the initial conditions. The practical realization problem is focussed and it is shown how to perform the input-ouput computations. Some Biomedical applications are described.

$Research paper thumbnail of On the properties of some operators under the perspective of fractional system theory$

Communications in Nonlinear Science and Numerical Simulation, Mar 1, 2020

Mathematics, Feb 5, 2019

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

IEEE Circuits and Systems Magazine, 2022

IFAC Proceedings Volumes, 2006

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

$Research paper thumbnail of New discrete-time fractional derivatives based on the bilinear transformation: Definitions and properties$

Journal of Advanced Research, Sep 1, 2020

This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

$Research paper thumbnail of Introduction to fractional linear systems. Part 1: Continuous-time case$

IEE proceedings, 2000

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

This document presents a sliding window algorithm for the calculation of the empirical mode decom... more This document presents a sliding window algorithm for the calculation of the empirical mode decomposition for long signals. The spline calculation of very long signals requires a long computation time. Our aim is to improve the calculation time of the empirical mode decomposition for Long signals. Some authors have used sliding windows for the whole decomposition. Our main contribution is to reduce the computation time calculating each intrinsic mode function on a sliding window basis. That ensures the obtained intrinsic mode function has no discontinuities on the junction regions between consecutive windows. Moreover, the sliding window size changes adaptively according to the number of extrema in the previous intrinsic mode function. The effectiveness of the proposed method increases with the length of the signal obtaining computation times of the order of 30 % of the time required to obtain the decomposition using only a window as in the classical manner. Those results are important to apply the empirical mode decomposition to long signals. Particularly, to biomedical signals like long-term ECG or long term EEG.

The Mittag-Leffler function (MLF) [5, 2] plays an important role in many applications of fraction... more The Mittag-Leffler function (MLF) [5, 2] plays an important role in many applications of fractional calculus (FC) [6, 4, 7]. The MLF establishes a connection between purely exponential and power law behaviors that characterize integer and fractional order phenomena, respectively. The numerical computation of the MLF poses problems both with accuracy and convergence [9, 8, 3, 1, 10]. In this paper we propose two distinct methods for efficiently computing the MLF. In the first approach we calculate the MLF series directly. In the second scheme we obtain we compute the MLF by means of the fast Fourier transform (FFT).

The signals used to resolve molecular spectra in nuclear magnetic resonance (NMR) spectroscopy an... more The signals used to resolve molecular spectra in nuclear magnetic resonance (NMR) spectroscopy and to generate images in magnetic resonance imaging (MRI) are based on statistical averaging in space and time of the dynamic behavior of the magnetic moments from billions of water molecules. Gaussian and Cauchy (Lorentzian) distributions arise naturally for molecular motion in simple liquids and solids; however, in complex mixtures, emulsions and biological tissues, so-called, 'anomalous' relaxation and diffusion are observed. In this paper we present the idea that such behavior reflects the presence of non-Gaussian stable distributions of fractional order and describe a new way to calculate the standard stable density function. For NMR relaxation this model is developed by generalizing the molecular correlation function from an exponential to a stretched exponential, while for NMR diffusion, Gaussian phase averaging is replaced by a non-Gaussian stable signal. Both situations view the molecular systems as exhibiting anomalous Brownian motion (typically, rotational for relaxation and translational for diffusion).

The analysis of long signals is relevant in many fields, as biomedical signal analysis. In this p... more The analysis of long signals is relevant in many fields, as biomedical signal analysis. In this paper, a revision of the Empirical Mode Decomposition (EMD), from the application point of view, is done. The increase in number of Intrinsic Mode Functions (IMF) and computational time in long signals are the main problems that have been faced in this work. A solution based on a sliding window is proposed. An adaptive process is used to calculate the size of the sliding Windows. As a result, the effectiveness of the proposed algorithm increases with the length of the signal. Two examples are introduced to illustrate both problems mentioned above.

Mathematics, Sep 23, 2022

Applied sciences, Oct 21, 2020

Digital systems require sample and hold (S&H) systems to perform the conversion from analog to di... more Digital systems require sample and hold (S&H) systems to perform the conversion from analog to digital and vice versa. Besides the standard zero and first order holds, we find in the literature other versions, namely the fractional and exponential order holds, involving parameters that can be tuned to produce a superior performance. This paper reviews the fundamental concepts associated with the S&H and proposes a new fractional version. The systems are modeled both in the time and Laplace domains. The new S&H stemming from fractional calculus generalizes these devices. The different S&H systems are compared in the frequency domain and their relationships visualized by means of hierarchical clustering and multidimensional scaling representations. The novel strategy allows a better understanding of the possibilities and limitations of S&H systems.

ICFDA'14 International Conference on Fractional Differentiation and Its Applications 2014, Jun 1, 2014

The Riemann-Liouville and Caputo derivatives are analysed in the context of the linear system the... more The Riemann-Liouville and Caputo derivatives are analysed in the context of the linear system theory. For it an analysis framework is presented. It is shown that those derivatives are unsuitable for studying the linear systems and in particular define transfer function.

Mathematics, May 23, 2022

Mathematics, Aug 10, 2020

The paper reviews the unilateral and bilateral, one-and two-dimensional Laplace transforms. The u... more The paper reviews the unilateral and bilateral, one-and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one-and two-dimensional cases with the bilateral Laplace transform. The case of fractional-order systems is also included. General two-dimensional linear systems are introduced and the corresponding transfer function is defined.