Modelling and Simulation in Engineering

Volume 2018, Article ID 7280306, 10 pages

https://doi.org/10.1155/2018/7280306

## A Computational Fractional Signal Derivative Method

^{1}Escuela de Ingeniería C. Biomédica, Universidad de Valparaíso, Valparaíso, Chile^{2}Centro de Investigación y Desarrollo en Ingeniería en Salud, CINGS-UV, Universidad de Valparaíso, Valparaíso, Chile

Correspondence should be addressed to Carolina Saavedra; lc.vu@ardevaas.anilorac

Received 1 March 2018; Accepted 31 May 2018; Published 1 August 2018

Academic Editor: Jing-song Hong

Copyright © 2018 Matías Salinas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose an efficient computational method to obtain the fractional derivative of a digital signal. The proposal consists of a new interpretation of the Grünwald–Letnikov differintegral operator where we have introduced a finite Cauchy convolution with the Grünwald–Letnikov dynamic kernel. The method can be applied to any signal without knowing its analytical form. In the experiments, we have compared the proposed Grünwald–Letnikov computational fractional derivative method with the Riemman–Louville fractional derivative approach for two well-known functions. The simulations exhibit similar results for both methods; however, the Grünwald–Letnikov method outperforms the other approach in execution time. Finally, we show an application of how our proposal can be useful to find the fractional relationship between two well-known biomedical signals.

#### 1. Introduction

Signal processing has been a powerful tool for system analysis when the system’s analytical model is unknown. In order to analyse these kinds of systems, it is necessary to apply high-level transformation operations to understand the signal. Within this group of operations, fractional calculus has become very useful for feature extraction, systems control, and modelling speech and more complex applications such as finding the relationship between different biomedical signals [1–5].

The fractional calculus has emerged as a nonlocal theory described with operators of fractional nature [6]. Fractional calculus was born as a natural generalization of the traditional calculus (Leibniz 1695, Euler 1730, Fourier 1822, Abel 1823, among others [6–8]); however, until recently, this mathematical theory is playing an active role in disciplines such as physics and control theory [9].

In the last decade, several applications have emerged due to the fractional nature of the phenomena. For instance, in physics, fractional calculus has been applied to thermodynamics, materials, and waves [9, 10]. In a fractional optimal control, either the performance index or the differential equations governing the dynamics of the system contains a term with a fractional derivative [11]. Recently, Tapia et al. [1] and Salinas et al. [12] have applied fractional calculus to discover a fractional relationship between two different biomedical signal modalities.

*Fractional calculus* is a terminology that refers to the integration and differentiation of arbitrary order [6, 8]; in other words, the meaning of *k*-th derivative and *k*-th iterated integral are extended by considering a fractional parameter instead of the integer parameter.

On the other hand, the numerical computation of the derivatives of the signal has many uses in analytical signal processing. The first derivative can be interpreted as the slope of the tangent to the signal at each point, and the second derivative is a measure of the curvature of the signal (the rate of change of the slope). The derivatives have been used to smoothen the signals by filtering the noise, for peak detection, trace analysis, and feature extraction, among others [13–16].

The aim of this article is to propose an efficient computational method of the fractional calculus applications to digital signal processing. Thus, we introduce a numerical formulation of the fractional derivatives for continuous and discrete signals, and also, we introduce the concepts of *Cauchy finite convolution* and the *Grünwald–Letnikov (GL) dynamic kernel*. The fractional derivatives and integrals are obtained by computing these convolutions with the GL kernel and the signal of interest. Our proposal can be viewed as a dynamic filter applied to a causal realization of a stochastic process.

This article is organized as follows. In Section 2, we introduce the fundamentals concepts of fractional calculus. Afterwards, in Section 3, we formulate our proposal of the fractional derivative of a signal as a finite convolution. In Section 4, simulations results are obtained in order to numerically quantify the approximation error. Section 5 shows the mathematical relationship between two biosignals by applying our fractional derivative approach. Concluding remarks and further works are given in the last section.

#### 2. Basic Principles of Fractional Calculus

The fractional calculus can be obtained either from the generalization of the definition of the derivative or the definition of the integral. Depending on the type of approach selected, the resulting mathematical equation is different. If the extension is build from the derivative, then the Grünwald–Letkikov method is obtained. On the other hand, the Riemman–Louville (RL) method is achieved when the extension is made from the integral.

It is well known that the simple derivative of with respect to *x* is the function , defined as

The derivative of a function represents an infinitesimal change in the function with respect to one of its variables. For higher-order derivatives, let be the *k*-th derivative, and the *k*-th derivative is defined by taking the derivative of the previous function as follows:

If we apply the derivative recursively from the function , we obtain the following expression:where is the binomial coefficient indexed by a pair of integers, and , . If the notation corresponds to the *k*-th factorial, then the value of the coefficient is obtained as follows:

Equation (3) can be adjusted to define the *k*-th integral as an extension of the derivative, where a negative sign for the *k*-th derivative is used. As a result, the following expression for the integral is obtained:

In this case, the binomial coefficient for a negative value is computed as follows:

In Sections 2.1 and 2.2, we introduce the definitions of fractional derivatives and integrals given by both the Grünwald–Letnikov derivative approach and the Riemann–Liouville integrative approach.

##### 2.1. Grünwald–Letnikov Differintegral Operator

The Grünwald–Letnikov differintegral is a combined differentiation/integration operator. The Grünwald–Letnikov *α*-differintegral of function *f* is denoted by . The GL operator is based on a classical concept for which integer order derivatives can be represented as limits of finite differences (3).

The binomial coefficient of (4) can be generalized to real arguments by means of the Euler’s gamma function:where the following properties hold: , and if is an integer number, then . The previous equation converges for all . If we replace the argument with a real positive parameter in (4), then we obtain the following property:

Grünwald–Letnikov introduced (8) in (3), obtaining the following definition for the fractional *α*-th derivative:where and are the upper and lower limits of differentiation, respectively.

If we consider that the parameter can take negative values, then the fractional integrative is defined. For , we consider the for the GL integral definition to obtain the following expression:where the binomial coefficient is expanded as follows:

##### 2.2. Riemann–Liouville Differintegral Operator

On the other hand, Riemann and Liouville have extended traditional integral in order to obtain its fractional model. The notation for the is , and the being . Thus, the RL fractional integral is defined as

And, the fractional derivative of the RL is given bywhere is the evaluation point, *α* is the fractional order, and is the reference point of integration.

Several studies have shown important properties and conditions for the existence of the fractional models [17–19]. Although, the analysis of these properties is out of the scope of this article, a detailed discussion of Grünwald–Letnikov differential and integral operators can be found in the monograph of Samko et al. [20].

#### 3. Fractional Calculus for Signal Processing

In this section, we introduce our proposal of a computational method to obtain the *α* fractional derivative of the signal. Consider be a subset of . A set of stochastic or random variables indexed by is called a stochastic continuous-time process. When , the stochastic discrete-time process is obtained. The signals and are defined as a realization of a real-valued continuous or discrete random process, respectively. In this work, we will consider only causal signals, and the value of a signal equals to 0 when .

On a continuous state, a random variable can take any value between a time interval . The fractional derivative for a continuous signal can be expressed as follows:where is the spacing between points on a grid and directly related to the approximation error . Moreover, is the upper limit of differentiation, and in this case, is considered.

A continuous signal can be converted into a discrete signal to a sequence of samples by sampling the signal every seconds. The sampled signal is related to the continuous signal through the equation , where is called the sampling interval or sampling period. The sampling frequency is defined as the number of samples taken per second, thus . In order to capture all the information from the continuous signal, it is necessary to fulfill the Nyquist sampling theorem [21]. The Nyquist rate is the minimum sampling rate that satisfies the sampling theorem and corresponds to twice the maximum component frequency of the function being sampled.

The discrete signal , , can be expressed as a vector:where is the number of sample points. The fractional derivative of a discrete signal is given by

In order to optimize the definition above, it is possible to define the *α*-th Grünwald–Letnikov kernel function as follows:where the value of constant depends only on , as follows:

Let us define the *Grünwald–Letnikov (GL) dynamic kernel * of size *n* aswhere the coefficients , are given by (17).

The *n*-th *Cauchy finite discrete convolution* is defined as follows:

The Grünwald–Letnikov fractional derivative can be rewritten by computing the Cauchy finite discrete convolution, given by (20), between the signal and the *Grünwald–Letnikov (GL) dynamic kernel * (19) as follows:

In Figure 1, the values of the coefficients of the kernel , for and , are shown. Notice that, when increases the kernel rapidly converges to 0; however, simulation results have shown that even the small terms significantly contribute to the computation of the fractional derivative.