Abstract

A look into fractional calculus and its 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 also 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 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 frequency 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.

Highlights

  • Fractional calculus has been attracting the attention of scientists and engineers from long time ago, during this period the main applications involved the using of the so called fractional integral operators to obtain explicit solutions of regular models

  • We must remark that in the 80% of the papers that appear in the Scientifics literature, in the framework of the fractional calculus and their applications, the corresponding author use different fractional differential operators but at the end they contrast their model using a numerical approach based in a finite number of terms of the series that define the known Grünwald-Letnikov derivative [30]

  • We can conclude that a generalization of the linear systems of differential equation it is very useful to be used in modeling much process [28,137]

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Summary

Introduction

Fractional calculus has been attracting the attention of scientists and engineers from long time ago, during this period the main applications involved the using of the so called fractional integral operators to obtain explicit solutions of regular models. We must remark that in the 80% of the papers that appear in the Scientifics literature, in the framework of the fractional calculus and their applications, the corresponding author use different fractional differential operators but at the end they contrast their model using a numerical approach based in a finite number of terms of the series that define the known Grünwald-Letnikov derivative [30]. We always assume that they are either of exponential order or tempered distributions

Definitions
Existence
Causality
Derivative of a product
Inverse element
The exponential
The constant function
Integral representations
Riemann-Liouville and Caputo derivatives
The Fourier transform of the fractional derivative and the frequency response
Trans-finite circuits
Band-limited approximations
Transfer function and frequency response
From the Transfer Function to the Impulse Response
The Stability Problem
The Initial Conditions
Discrete-time implementations
Input-output numerical computations in general nonlinear systems
Adams-Bashforth-Moulton method
The variational iteration method
Homotopy-perturbation method
Some considerations
Some considerations concerning fractional order models
Fractional Dynamics Model
Fractional Impedance Model
The fractional Brownian motion
Findings
CONCLUSIONS
Full Text
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