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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 795954, 7 pages
On Generalized Fractional Differentiator Signals
1Faculty of Computer Science and Information Technology, University Malaya, 50603 Kuala Lumpur, Malaysia
2Institute of Mathematical Sciences, University Malaya, 50603 Kuala Lumpur, Malaysia
Received 17 January 2013; Accepted 16 March 2013
Academic Editor: Jehad Alzabut
Copyright © 2013 Hamid A. Jalab and Rabha W. Ibrahim. 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.
By employing the generalized fractional differential operator, we introduce a system of fractional order derivative for a uniformly sampled polynomial signal. The calculation of the bring in signal depends on the additive combination of the weighted bring-in of cascaded digital differentiators. The weights are imposed in a closed formula containing the Stirling numbers of the first kind. The approach taken in this work is to consider that signal function in terms of Newton series. The convergence of the system to a fractional time differentiator is discussed.
Nowadays, fractional calculus (integral and differential operators) arises in signal processing and image possessing. The fractional calculation is able to enhance the quality of images, with interesting possibilities in edge detection and image restoration, to reveal faint objects in astronomical images and devoted to astronomical images analysis [1, 2]. Furthermore, fractional calculus is employed in image retrieval, design problems of variables and image denoising, digital fractional order for different filters [3–9]. In addition, the fractional calculus (differential operators) is used to reduce the error rate of handwritten signature verification system. All results based on the fractional calculus operators (differential and integral) show that this method is not only effective, but also good immunity. Therefore, the fractional calculus in the field of image processing and signal prosecuting that has broad application prospect.
The digital differentiator is a very helpful tool to compute and approximate the time derivatives of a given signal; such as, in radar and sonar applications, the velocity and acceleration are calculated from position measurements using differentiators. Digital fractional order differentiators are discrete-time digital systems fractional order differentiation. In view of signal processing, the generalization from integer to fractional orders that is an important concept for its possibility to enhance flexibility in designing digital differentiator has been well treated in the existing signal processing. There are comparatively published results respecting the fractional digital differentiators [10, 11].
In this work, by using the generalized Srivastava-Owa fractional differential operator  (involving two parameters , ), we introduce a system of generalized fractional order derivative for a uniformly sampled polynomial signal. The weights are obtained in a form containing the Pochhammer number. The convergence of the system to a fractional time differentiator is discussed. The output of the signal is determined by using the generalized hypergeometric function called the Fox-Wright function.
2. Design Technique
Ibrahim , has derived a formula for the generalized fractional integral. The -fold integral for and real , is defined by Employing the Cauchy formula for iterated integrals yields Repeating the previous step times, we have which implies the fractional operator type where and are real numbers, the function is analytic in simply connected region of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when . When , we arrive at the standard Srivastava-Owa fractional integral operator, which is used to define the Srivastava-Owa fractional derivatives.
Corresponding to the generalized fractional integrals (4), we define the generalized differential operator of order by where the function is analytic in simply connected region of the complex -plane containing the origin and the multiplicity of is removed by requiring to be real when .
Example 1. We find the generalized derivative of the function , . Let then we have
3. Fractional Digital Signal
In this section we will use the fractional differential operator (5) in order to compute the fractional signal. Assume the analytic signal , which can be represented as a Newton series around where is the Pochhammer symbol defined by such that denotes the Stirling numbers of the first kind , and is the backward difference operator defined by
Now we assume that ( is the sampling period), then we obtain where By truncating the Newton series expansion at the th term , we assume the polynomial signal such that for all the differences of order vanished.
The fractional differential digital signal is a discrete time system whose output is the uniformly sampled version of the th order derivative of . Therefore, we assume . Specifically, we write The input is supposed to be a polynomial of degree . In view of Example 1, we have Next we proceed to evaluate the fractional order of the rising factorial power term at . We expand using the binomial theorem, and we obtain where consequently; by using Example 1, we have as , , and ; thus we have where is the Fox-Wright function (the generalization of the hypergeometric function ) defined by where for all , for all , and for suitable values . Hence Substituting (20) into (14) we obtain (13)
4. Experimental Results
In this section, we propose to apply the formula (21) on the signal . We will assume . The Stirling numbers of the first kind take the values where is the delta function
Now for sufficient small value of , the Fox-Wright function reduces to the hypergeometric function such that satisfies
In virtue of (11), where and , we pose
For and , the 6th terms of become Furthermore, for and , we have Also, for and , we have
For given values of , fixed fractional number , , and different values of the second parameter , one can analyze the time-varying weights by plotting the time-varying impulse response of the system (Figures 1, 2, 3, and 4). For and any value of the system is a maximally linear differentiator (Figure 5). It follows from the relation (21) that the input/output characterizes the ideal digital differentiator, for fractional and integer values of with the help of the fractional value of , of the polynomial signal. This relation shows for integer case of that the weights are time-invariant. While the weights are varying time for fractional case.
Performance tests for the system proposed by this paper were implemented using MATLAB 2010a on Intel(R) Core i7 at 2.2 GHz, 4 GB DDR3 Memory, system type 64-bit, and Window 7.
The differential modeling of arbitrary order can be virtued as a signal processing method to develop numerical differential algorithms. There are various types of numerical fractional differential algorithms anticipated in the mathematics literature such as the Grünwald-Letnikov fractional differential operator and the Riemann-Liouville differential operator which based on one parameter . In this paper, we modified the Newton series by using two-parameters () fractional differential operator (generalized Srivastava-Owa operator). This approach implies zero error for the representation of the signal polynomial; thus it provides a means for the calculation of the fractional derivatives of . The system yields the arbitrary order derivative of the signal based on the current sample and past samples of the signal. The value of computed the truncation length in (14).
The authors would like to thank the reviewers for their comments on earlier versions of this paper. This research has been funded by university of Malaya, under UMRG 104-12ICT.
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