# On Generalized Fractional Differentiator Signals

Authors: Hamid A. Jalab; Rabha W. Ibrahim
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

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 ofN 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.


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## Body

## 1. Introduction

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 [12] (involving two parameters α, μ), 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 [13],  has  derived a formula for the generalized fractional integral. The n-fold integral  for n∈ℕ={1,2,…} and real μ, is defined by
(1)Izα,μf(z)=∫0z‍ζ1μdζ1∫0ζ1‍ζ2μdζ2⋯∫0ζn-1‍ζnμf(ζn)dζn.
Employing the Cauchy formula for iterated integrals yields
(2)∫0z‍ζ1μdζ1∫0ζ1‍ζμf(ζ)dζ=∫0z‍ζμf(ζ)dζ∫ζz‍ζ1μdζ1=1μ+1∫0z‍(zμ+1-ζμ+1)ζμf(ζ)dζ.
Repeating the previous step n-1 times, we have(3)∫0z‍ζ1μdζ1∫0ζ1‍ζ2μdζ2⋯∫0ζn-1‍ζnμf(ζn)dζn=(μ+1)1-n(n-1)!∫0z‍(zμ+1-ζμ+1)n-1ζμf(ζ)dζ,
which implies the fractional operator type
(4)Izα,μf(z)=(μ+1)1-αΓ(α)∫0z‍(zμ+1-ζμ+1)α-1ζμf(ζ)dζ,
where α and μ≠-1 are real numbers, the function f(z) is analytic in simply connected region of the complex z-plane ℂ containing the origin, and the multiplicity of (zμ+1-ζμ+1)-α is removed by requiring log(zμ+1-ζμ+1) to be real when (zμ+1-ζμ+1)>0. When μ=0, 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 α by
(5)Dzα,μf(z)≔(μ+1)αΓ(1-α)ddz∫0z‍ζμf(ζ)(zμ+1-ζμ+1)αdζ;0<α≤1,
where the function f(z) is analytic in simply connected region of the complex z-plane ℂ containing the origin and the multiplicity of (zμ+1-ζμ+1)-α is removed by requiring log(zμ+1-ζμ+1) to be real when (zμ+1-ζμ+1)>0.Example 1.
We find the generalized derivative of the functionf(z)=zν, ν∈ℝ. Let η:=(ζ/z)μ+1 then we have
(6)Dzα,μzν=(μ+1)αΓ(1-α)ddz∫0z‍ζμ+ν(zμ+1-ζμ+1)αdζ=(μ+1)α-1Γ(1-α)ddzz(1-α)(μ+1)+ν×∫01‍η(ν+μ+1)/(μ+1)-1(1-η)(1-α)-1dη=(μ+1)α-1Γ(ν/(μ+1)+1)Γ(ν/(μ+1)+1-α)z(1-α)(μ+1)+ν-1.


## 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 f(z), which can be represented as a Newton series around z0(7)f(z)=∑n=0∞‍((z-z0)/T)nn!∇Tnf(z0),
where (a)n is the Pochhammer symbol defined by
(8)(a)n=Γ(a+n)Γ(a)={1,n=0a(a+1)⋯(a+n-1),n={1,2,…}=∑k=0∞[nk]ak
such that [nk] denotes the Stirling numbers of the first kind S1(n,k), and ∇Tnf(z0) is the backward difference operator defined by
(9)∇T0f(z0)=f(z0)∇T1f(z0)=f(z0)-f(z0-T)⋮∇Tnf(z0)=∇T1[∇Tn-1f(z0)]∇Tnf(z0)=∇Tn-1f(z0)-∇Tn-1f(z0-T).Now we assume thatz0=mT (T is the sampling period), s(m)=f(mT) then we obtain
(10)f(z)=∑n=0∞‍((z-mT)/T)nn!∇1ns(m),
where
(11)∇10s(m)=s(m)∇11s(m)=s(m)-s(m-1)⋮∇1ns(m)=∇1n-1s(m)-∇1n-1s(m-1).
By truncating the Newton series expansion at the Nth term (n=N), we assume the polynomial signal
(12)p(z)=∑n=0N‍cnzn,(zisreal)
such that for all the differences of order n≥N+1 vanished.The fractional differential digital signal is a discrete time system whose outputy(m) is the uniformly sampled version of the αth order derivative of f(z). Therefore, we assume α>0. Specifically, we write
(13)y(m)=Dzα,μf(z)|z=mT.
The input s(m)=f(mT) is supposed to be a polynomial of degree N. In view of Example 1, we have
(14)Dzα,μf(z)=∑n=0N‍∇1ns(m)n!Dzα,μ(z-mTT)n.
Next we proceed to evaluate the fractional order of the rising factorial power term at mT=z. We expand (z-mT)k using the binomial theorem, and we obtain
(15)(z-mT)k=(k0)zk(mT)0-(k1)zk-1(mT)1+⋯+(kk)(-mT)k=∑n=0k‍(kn)zn(-mT)k-n,
where
(16)(kn)=k!n!(k-n)!
consequently; by using Example 1, we have
(17)Dzα,μ(z-mT)k=∑n=0k‍(kn)[Dzα,μzn](-mT)k-n=∑n=0k‍k!n!(k-n)!(-mT)k-n×(μ+1)α-1Γ(n/(μ+1)+1)Γ(n/(μ+1)+1-α)z(1-α)(μ+1)+n-1=(μ+1)α-1Γ(k+1)z(1-α)(μ+1)-1(-mT)k×∑n=0k‍1Γ(k-n+1)Γ(n/(μ+1)+1)Γ(n/(μ+1)+1-α)(z/mT)nn!
as k→∞, n<k, and limk→∞Γ(k+1)/Γ(k-n+1)=1; thus we have
(18)Dzα,μ(z-mT)k=(μ+1)α-1z(1-α)(μ+1)-1(-mT)k×1Φ1[(1,1μ+1);(1-α,1μ+1);zmT],
where qΦp is the Fox-Wright function (the generalization of the hypergeometric function qFp) defined by
(19)qΦp[(α1,A1),…,(αq,Aq);(β1,B1),…,(βp,Bp);z]=qΦp[(αj,Aj)1,q;(βj,Bj)1,p;z]≔∑n=0∞Γ(α1+nA1)⋯Γ(αq+nAq)Γ(β1+nB1)⋯Γ(βp+nBp)znn!=∑n=0∞∏j=1qΓ(αj+nAj)∏j=1pΓ(βj+nBj)znn!,
where Aj>0 for all j=1,…,q, Bj>0 for all j=1,…,p, and 1+∑j=1p‍Bj-∑j=1q‍Aj≥0 for suitable values |z|<1. Hence
(20)Dzα,μ(z-mT)k|z=mT=(-1)k(mT)k+(1-α)(μ+1)-1(μ+1)α-11Φ1[1].
Substituting (20) into (14) we obtain (13)
(21)Dzα,μf(z)=∑n=0N‍∇1ns(m)n!Dzα,μ(z-mTT)n=∑n=0N‍∇1ns(m)n!Dzα,μ(∑k=0∞‍[nk](z-mTT)k)=∑n=0N‍∇1ns(m)n!(∑k=0∞‍[nk]Dzα,μ(z-mTT)k)=∑n=0N‍∇1ns(m)n!(∑k=0∞‍[nk]1TkDzα,μ(z-mT)k)=∑n=0N‍∇1ns(m)n!(∑k=0∞‍[nk]1Tk[(-1)k(mT)k+(1-α)(μ+1)-1∑k=0∞‍[nk]1Tkcc×(μ+1)α-11Φ1[1]]∑k=0∞‍[nk])=T(1-α)(μ+1)-11Φ1[1](μ+1)α-1∑n=0N‍∇1ns(m)n!×(∑k=0n‍[nk][(-1)k(m)k+(1-α)(μ+1)-1])≔y(m).


## 4. Experimental Results

In this section, we propose to apply the formula (21) on the signal f(z)=z. We will assume N=5. The Stirling numbers of the first kind S1(n,k) take the values
(22)S1(n,0)=δn,0S1(n,1)=(-1)n-1(n-1)!⋮S1(n,n-1)=-(n2)binomialcoefficients,
where δn,0 is the delta function
(23)δn,0={0n≠01n=0.Now for sufficient small value ofμ, the Fox-Wright function Φ11[1] reduces to the hypergeometric function qFp(24)qΦp[(α1,1),…,(αq,1);(β1,1),…,(βp,1);z]=∏j=1qΓ(αj)∏i=1pΓ(βi)qFp(α1,…,αq,β1,…,βp;z)
such that qFp satisfies
(25)2F1(a,b;c;1)=Γ(c-a-b)Γ(c)Γ(c-a)Γ(c-b),b≤0.Employing the relations (24) and (25) yields
(26)Φ11[1]≃Γ(α1)Γ(β1)2F1(1,0;1-α;1)=1Γ(1-α)Γ(-α)Γ(1-α)Γ(-α)Γ(1-α)=1Γ(1-α),
hence for α=1/2⇒Γ(1/2)=1.772, we impose 1Φ1[1]≃0.564.In virtue of (11), where f(z)=f(mT)=s(m)=Tm and T=1/25, we pose
(27)∇10s(m)=s(m)=m5,∇11s(m)=s(m)-s(m-1)=m-m-15,∇12s(m)=m-2m-1+m-25,∇13s(m)=m-3m-1+3m-2-m-35,∇14s(m)=m-4m-1+6m-2-4m-3+m-45,∇15s(m)=(m-5m-1+10m-2cc-10m-3+5m-4-m-5)×(5)-1.Forμ=0 and α=0.5, the 6th terms of y(m) become
(28)y(m)=2.82{(∇15s(m)5!)∇10s(m)m=2.82cc-m(∇11s(m)1!+∇12s(m)2!=2.82cc-mccc+2∇13s(m)3!+6∇14s(m)4!+24∇15s(m)5!)=2.82cc+m3(∇12s(m)2!+3∇13s(m)3!+11∇14s(m)4!=2.82cc+m3ccc+50∇15s(m)5!)=2.82cc-m5(∇13s(m)3!+6∇14s(m)4!+35∇15s(m)5!)=2.82cc+m7(∇14s(m)4!+10∇15s(m)5!)=2.82cc-m9(∇15s(m)5!)+⋯}.
Furthermore, for α=0.5 and μ=0.25, we have
(29)y(m)≃1.475{∇10s(m)m3-m23(∇11s(m)1!+∇12s(m)2!+2∇13s(m)3!≃1.475cc∇10s(m)m3-m23ccc+6∇14s(m)4!+24∇15s(m)5!)≃1.475cc+m53(∇12s(m)2!+3∇13s(m)3!+11∇14s(m)4!≃1.475cc+m53ccc+50∇15s(m)5!)≃1.475cc-m83(∇13s(m)3!+6∇14s(m)4!+35∇15s(m)5!)≃1.475cc+m113(∇14s(m)4!+10∇15s(m)5!)≃1.475cc-m143(∇15s(m)5!)+⋯}.
Also, for α=0.5 and μ=0.5, we have
(30)y(m)≃1.5{∇10s(m)m4-m34(∇11s(m)1!+∇12s(m)2!+2∇13s(m)3!≃1.5cc∇10s(m)m4-m34ccc+6∇14s(m)4!+24∇15s(m)5!)≃1.5cc+m74(∇12s(m)2!+3∇13s(m)3!+11∇14s(m)4!≃1.5cc+m74ccc+50∇15s(m)5!)≃1.5cc-m114(∇13s(m)3!+6∇14s(m)4!+35∇15s(m)5!)≃1.5cc+m154(∇14s(m)4!+10∇15s(m)5!)≃1.5cc-m194(∇15s(m)5!)+⋯}.


## 5. Discussion

For given values ofN, fixed fractional number α=0.5, α=0.75, and different values of the second parameter μ=0,0.25,0.5, one can analyze the time-varying weights by plotting the time-varying impulse response of the system (Figures 1, 2, 3, and 4). For α=1 and any value of μ 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 α with the help of the fractional value of μ, of the polynomial signal. This relation shows for integer case of α that the weights are time-invariant. While the weights are varying time for fractional case.Figure 1
The outcome of the fractional signal whenα=0.5, T=1/25.Figure 2
The outcome of the fractional signal whenα=0.75, T=1/25.Figure 3
The outcome of the fractional signal whenα=0.5, T=1.Figure 4
The outcome of the fractional signal whenα=0.75, T=1.Figure 5
The outcome of the fractional signal whenα=1.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.


## 6. Conclusion

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 parameterα. In this paper, we modified the Newton series by using two-parameters (α,μ) 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 s(m). The system yields the arbitrary order derivative of the signal based on the current sample and N past samples of the signal. The value of N computed the truncation length in (14).



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