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- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 638648, 6 pages
Local Fractional -Transforms with Applications to Signals on Cantor Sets
1School of Electrical Engineering, Southeast University, Nanjing 210096, China
2Department of Mathematics, University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy
3School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710048, China
4Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China
5Department of Electronic and Information Technology, Jiangmen Polytechnic, Jiangmen 529090, China
Received 28 January 2014; Accepted 3 February 2014; Published 10 March 2014
Academic Editor: Ming Li
Copyright © 2014 Kai Liu 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.
The -transform has played an important role in signal processing. In this paper the -transform has been generalized by the coupling of both the -transform and the local fractional complex calculus. In the literature the local fractional -transform is applied to analyze signals, in the following it will be used to analyze signals on Cantor sets. Some examples are also given to show the efficiency and accuracy for handling the signals on Cantor sets.
Integral transforms [1, 2], such as Fourier, Laplace, Mellin, Hilbert, and Hankel transforms, play important roles in solving the mathematical problems arising in applied mathematics, mathematical physics, and engineering science. In recent years, fractional calculus [3–11] was developed and used to model also some anomalous behaviors of diffusion [12–21] and transport [22–27]. Fractional integral transforms are suitable generalizations of the classical ones and were recently proposed by some researchers. For example, the fractional Fourier transforms were considered in [28, 29]. In , the fractional Hilbert transform was presented. The fractional Mellin transform [31, 32] was proposed to be used in image encryption. The fractional wavelet transform was presented and some applications were investigated in [33–35]. In , the fractional Hankel transform was reported in order to research the charge-amplitude state representations.
The -transform method [1, 2, 37] was applied to handle the linear time-invariant discrete-time systems (LTI discrete-time systems) and difference equations in -domain. However, the fractional derivative and integrals (the fractional PDIs) were used to transfer the fractional LTI discrete-time systems to -domain . There appear signals defined on Cantor sets, which are the most striking properties of nondifferentiable functions. The classical -transform method and PDIs did not deal with them. In order to overcome them, local fractional calculus [39–43] may be applied to handle the function defined on Cantor sets shown in Figure 1. The local fractional integral transforms via local fractional calculus theory were proposed in [44–51]. For example, local fractional Fourier transforms reported in [40, 44] were used to find nondifferentiable solutions for local fractional ODEs and PDEs [45–47]. Laplace transforms via local fractional calculus  were generalized and reported in order to solve the local fractional ODEs and PDEs [48–50].
Fractal signal processing [51–59] is a hot topic for scientists and engineers. Very recently, the concept of the -transform method via local fractional calculus was considered only in . However, there is no report on signal processing by using the local fractional -transforms. The main aim of this paper is to investigate the properties of local fractional -transforms and to present some examples for processing signals defined on Cantor sets.
The paper is organized as follows. In Section 2, the concepts of local fractional complex derivatives and integrals are given. In Section 3, the notions and properties of local fractional -transform method are presented. In Section 4, some examples and applications of this method are shown. Finally, Section 5 is the conclusions.
2. Local Fractional Derivatives and Integrals of Complex Functions and Recent Results
In this section, we introduce the concepts of local fraction derivative and integrals of complex functions. Let us first give the local fractional continuity of complex functions.
Definition 2 (see [40, 60]). The local fractional derivative of complex function of order is defined as
If the limit of (4) exists for all in a region , then the complex function is said to be local fractional analytic in a region .
The properties of the local fractional derivatives of some complex functions are presented as follows : where
Definition 3 (see [40, 46–50, 60]). The local fractional integral of complex function of order along the closed contour is defined as The properties of the local fractional integrals of some complex functions are suggested as follows :
Theorem 4 (see ). If is local fractional analytic within and on a simple closed contour and is any point interior to , then we have
Proof. See .
Definition 5 (see [40, 60]). If is an isolated singular point of , then we have a local fractional Laurent series of at given by The coefficient of is called the local fractional residue of at and is frequently written as
Theorem 6 (see ). If is local fractional analytic within and on the boundary of a region except at a number of poles within , having a residue , then
Proof. See .
Theorem 7 (see ). If is local fractional analytic within and on the boundary of a region except at a number of poles within , having numbers of residues, then
Proof. See .
3. Local Fractional -Transforms and Their Properties
In this section, we give the local fractional -transforms and their properties.
Definition 8 (see ). Local fractional -transform of of order is defined as where the above formula is convergent.
For a given sequence, the set of values of for which its local fractional -transform converges is called the region of convergence (ROC), namely, The inverse formula of local fractional -transform of of order reads as follows (see ): where is a counterclockwise closed fractal path encircling the origin and entirely in the region of convergence.
Let within the region of convergence and let within the region of convergence .
Property 1 (linearity). We have within the region of convergence .
Proof. From (15) we have within the region of convergence .
Property 2 (time shifting). If the variable has a useful interpretation in terms of time delay, then we have
Proof. From (15), we have
Property 3 (frequency modulation). We have
Proof. From (15), we have
4. Some Illustrative Examples
In this section, we give some samples for nondifferentiable signals defined on Cantor sets.
Example 1. Let us consider the following signal in the form: Taking local fractional -transform, we have
Example 2. We now suggest the following signal in the form: Taking local fractional -transform, we obtain When with the imaginary unit [40, 44–50], we get Hence, from (28), we get with the real part graph in Figure 2 and imaginary part graph in Figure 3.
Example 3. There is the signal in the following form: Taking local fractional -transform, we have When , we get with the graph of shown in Figure 4.
Example 4. We have the following signal in the form: Local fractional -transform gives the following form: with the region of convergence .
Example 5. We consider the following signal in the form: Taking local fractional -transform, we arrive at the following form: with the region of convergence .
Example 6. We present the following signal in the form: Local fractional -transform gives the following form: with the region of convergence .
In this work, we investigated the local fractional -transforms based on the local fractional complex calculus and some properties are also obtained. Some illustrative examples were also given. The obtained results show the accuracy and efficiency of the presented method.
Conflict of Interests
The authors declare that they have no competing interests in this paper.
This work was supported by Science and Technology Commission Planning Project of Jiangsu Province (no. BE2013737).
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