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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

IEEE JOURNAL ON EMERGING AND SELECTED TOPICS IN CIRCUITS AND SYSTEMS 1

Caputo-Based Fractional Derivative inFractional Fourier Transform Domain

Kulbir Singh, Rajiv Saxena, and Sanjay Kumar

AbstractThis paper proposes a novel closed-form analyticalexpression of the fractional derivative of a signal in the Fouriertransform (FT) and the fractional Fourier transform (FrFT) do-

main by utilizing thefundamental principles of the fractional order

calculus. The generalization of the differentiation property in theFT and the FrFT domain to the fractional orders has been pre-sented based on the Caputos definition of the fractional differin-tegral, thereby achieving the flexibility of different rotation angles

in the timefrequency plane with varying fractional order param-eter. The closed-form analytical expression is derived in terms of

the well-known higher transcendental function known as confluenthypergeometric function. The design examples are demonstrated

to show the comparative analysis between the established and theproposed method for causal signals corrupted with high-frequency

chirp noise and it is shown that the fractional order differentiating

filter based on Caputos definition is presenting good performancethan the established results. An application example of a low-pass

finite impulse response fractional order differentiating filter in the

FrFT domain based on the definition of Caputo fractional differin-tegral method has also beeninvestigated taking into account ampli-

tude-modulated signal corrupted with high-frequency chirp noise.

Index TermsCaputo fractional derivative, fractional Fourier

transform (FrFT), fractional order calculus (FOC), fractionalorder derivative, Kummer confluent hypergeometric function.

I. INTRODUCTION

T HE CONCEPT of derivatives is traditionally associatedto an integer; given a function, one can differentiate itone, two, three times, and so on. One can have an interest to

investigate the possibility of derivatives for a real or noninteger

number of times of a function. So the concept of extending clas-

sical integer order calculus to noninteger order is by no means

new. The earliest systematic studies seem to have been made

in the beginning and middle of the nineteenth century by Liou-

ville, Riemann, and Holmgren [1]. This wonderful tool of math-

ematics called fractional order calculus (FOC), which is in fact

a generalization of the classical Newtonian calculus, aims to de-

scribe and discover the nature and the natural phenomena [2].In the research area of FOC, the integer order of the deriva-

tive of the function is general-

ized to the fractional order , where is a real number

Manuscript received February 15,2013; revised May03, 2013; accepted June10, 2013. This paper was recommended by Guest Editor G. Chen.

K. Singh and S. Kumar are with the Department of Electronics and Commu-nication Engineering, Thapar University, Patiala 147004, Punjab, India (e-mail:[email protected]; [email protected]).

R. Saxena is with the Department of Electronics and Communication Engi-neering, Jaypee University of Engineering and Technology, Raghogarh, Guna473226, Madhya Pradesh, India (e-mail: [email protected]).

Digital Object Identifier 10.1109/JETCAS.2013.2272837

[1]. In recent years, the concepts of fractional order operators

have been investigated extensively in science and engineering

applications [1][6] including the design of fractional order dig-

ital differentiators that have received a great attention to the re-

search community. Also, there has been a surge of research in

signal processing following the birth of the fractional Fourier

transform (FrFT) [5][7].

As the main theme of the proposed work revolves around the

fractional order differentiation, it can be emphasized that on two

occasions including [8] and [9], the differentiation property was

independently extended to the class of FT and FrFT, respec-tively, but not extended to the noninteger orders.

In this paper, the fractional derivative of a given signal in FT

and FrFT domain for different fractional orders is proposed, by

utilizing the inherent approach of the Caputo-based fractional

operators of FOC. The concept behind the study is that it in-

volves two different varying parameters: the fractional order pa-

rameter and the fractional Fourier transform parameter .

In a companion paper[10], we formulate a novel and unique

closed-form analytical expression of the fractional order differ-

entiation in FrFT domain by incorporating the RiemannLiou-

ville (RL) fractional order operator. The focus in [10] is on es-

tablishing the performance of the designed fractional order dif-ferentiating filterin FrFT domain with that of time-domain and

frequency-domain filtering. In the present work, a similar study

has been carried out for Caputo fractional derivative. The use of

this definition of the fractional derivative is justified since it has

good physical properties [11].

The focus here is on the usage of the Caputo-based defini-

tion for the FIR system and for a restrict class of signals and

noises: causal signals with high-frequency chirp noise. The Ca-

puto derivative has to be preferred because its initial conditions

have a nice physical meaning [11], [12]. But still there has been

a lot of debate about the usage and the practicality between

RL and Caputo-based fractional derivative definitions in the re-

search community, mainly concerned with different types of ini-

tial conditions when they are used to formulate the differential

equations [13], [14].

The motivation behind this study is provided by the work of

Tseng et al. [8], McBride et al. [9] and Kumaret al. [10]. The

main differences between the proposed study and the work of

Tseng et al. [8] are Tseng et al. used the Cauchy integral for-

mula and generalized it to define the fractional derivative of a

function in FT domain, whereas in the proposed study, the Ca-

puto-based definition for the general fractional differintegral is

used and proposed to derive the fractional derivative of a func-

tion in both FT and FrFT domain. Similarly, McBride et al. [9]

derive the differentiation property in the FrFT domain to integer

2156-3357/$31.00 2013 IEEE
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2 IEEE JOURNAL ON EMERGING AND SELECTED TOPICS IN CIRCUITS AND SYSTEMS

order only, whereas the proposed method generalize it to estab-

lish the differentiation property in the FrFT domain to nonin-

teger orders.

Thus, one could get the advantage of two degrees-of-freedom

(DOF) for achieving the goal, as compared to [8] where only

one DOF exists, thereby achieving the advantage of generating

the random fractal process for different rotation angles in the TF

plane with varying . The disadvantage of the proposed method

as compared to [8] is that it does not make use of the optimiza-

tion techniques in designing.

The proposed method has an asset over [9] is that it can be

applied to solve many engineering problems, like the modeling

of ordinary fractional order differential/partial differential equa-

tions or the design of robust control algorithms. The fractional

order differentiation in the FrFT domain can prove to be more

helpful than the usual ones. But the proposed method suffers

from the disadvantage is that it is valid only for varying be-

tween 0 and 1 i.e., .

The purpose of this paper is to design the fractional orderfilter

in the FrFT domain using the FOC approach based on Caputosfractional order derivative definition. An attempt has been made

to establish a closed-form analytical expression of the fractional

order differentiation in the FrFT domain. The results with the

proposed method lead to good performance as compared to [10].

Thus, the proposed filtering method offers good design flexi-

bility than the method in [10]. Therefore, the outcome of this

study is to establish a closed-form analytical expression for the

fractional order derivative in the FT and the FrFT domain based

on Caputos definition, is novel and unique.

The remainder of the paper proceeds as follows. Section II

presents the brief introduction of the confluent hypergeometric

function. In Section III, the mathematical foundation of FOCand its related properties are briefly introduced with some of

the essential mathematical background of the fractional Fourier

transform in Section IV. Section V presents the proposed

method of computing the fractional derivative in the FT and

the FrFT domain. A new fractional order differentiating filter

model in the FrFT domain has been described in Section VI

with the design examples investigated in Section VII. Finally,

some concluding remarks are drawn in Section VIII.

II. CONFLUENT HYPERGEOMETRIC FUNCTION

The confl

uent hypergeometric function (CHF) of thefi

rstkind is a degenerate form of the hypergeometric

function which arises as a solution of the con-

fluent hypergeometric differential equation. In this study, the

Kummer confluent hypergeometric function (KCHF) is consid-

ered for the derivation of the proposed analytical expression.

The KCHF is defined by the absolutely convergent infinite

power series [15], [16] as

(1)

where, the Pochhammer symbol

denotes the rising factorial and . The detailed

description of KCHF with properties can be found in [15], [16].

The KCHF is commonly available in numerical software pack-

ages, for example, Mathematica and Maple. The series in (1) is a

solution of the Kummer confluent hypergeometric equation that

is a second order ordinary differential equation, having regular

and irregular singularities at 0 and at , respectively

(2)

CHF includes, as special cases, the commonly used Bessel

functions, Hermite functions, Laguerre functions, etc. The

CHFs play an exceptional role in many branches of physics,

mathematics and also in electromagnetic theory.

III. FRACTIONAL ORDER CALCULUS

FOC, an extension of noninteger order derivative and inte-

gral, has received great attention in the last few decades, be-

cause of its ability to model systems more accurately than in-

teger orders [1]. Notwithstanding it represents more accurately

some natural behavior related to different areas of engineering

including signal processing [17], [18], electronics [19], [20],

robotics [21], [22], control theory [23], and bioengineering [24]

to name a few.

The theory of fractional order derivative [11] was developed

mainly in the nineteenth century. FOC is a generalization of

integration and differentiation to a fractional or noninteger order

fundamental operator , where and are the lower/upper

bounds of integration and the order of the operation

(3)

where is the real part of . Moreover, the fractional order

can be a complex number [25]. In this paper, the main focus

is on the case where the fractional order is a real number, i.e.,

.

The most frequently used equivalent definitions for the gen-

eral fractional differintegral are the RL, the GrnwaldLetnikov

(GL), and the Caputo definitions [1].

A. RiemannLiouville Definition

The RL definition for a function is given as

(4)

for where is the well-known Eulers

gamma function.

B. GrnwaldLetnikov Definition

The GL definition is the most popular definition for the gen-

eral fractional derivatives and integrals because of its discrete

nature. According to GL, the fractional derivative of a function

is given as

(5)
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SINGH et al.: CAPUTO-BASED FRACTIONAL DERIVATIVE IN FRACTIONAL FOURIER TRANSFORM DOMAIN 3

where is the sampling period, is the well-known Eulers

gamma function and means the integer part of

.

C. Caputo Definition

The Caputo definition for a function can be given as

(6)

where , is an integer and is a real

number. This definition is very famous and is used in many

studies and reports to define the fractional order derivatives and

integrals.

IV. FRACTIONAL FOURIER TRANSFORM

FT is undoubtedly one of the most valuable and frequentlyused tools in signal processing and analysis [26]. Little need be

said of the importance and ubiquity of the ordinary FT in many

areas of science and engineering. The FrFT has been found to

have several applications in the areas of optics and signal pro-

cessing [7], [9], [10], [27][31]. It also leads to the generaliza-

tion of notion of space (or time) and frequency domains, which

are central concepts of signal processing.

The FrFT is a generalization of the conventional FT, which

is richer in theory, flexible in application, and implementation

cost is at par with FT. With the advent of FrFT and the related

concept, it is seen that the properties and applications of thecon-

ventional FT are special cases of those of the FrFT. However, inevery area where FT and frequency domain concepts are used,

there exists the potential for generalization and implementation

by using FrFT.

The FrFT with a fractional Fourier order parameter corre-

sponds to the th fractional power of the Fourier transform op-

erator, . The th-order FrFT of is defined as

(7)

where , the transformation kernel

(8)

with [7] where indicates the rotation angle of the

transformed signalfor the FrFTand denotes the continuous

FrFT (CFrFT) operator.

The FrFT with corresponds to the conventional FT,

and the one with corresponds to the identity operator.

Also, two successive FrFTs with angles and are equivalent

to a single FrFT with an angle . Hence, the properties of

the conventional FT can be obtained by substituting

in the properties of FrFT.

V. COMPUTATION OF FRACTIONAL DERIVATIVE IN FOURIER

TRANSFORM AND FRACTIONAL FOURIER TRANSFORM DOMAIN

In this section, a novel closed-form analytical expression of

the fractional derivative of the signal in FrFT domain has been

derived. This has been obtained by utilizing the inherent ap-

proach of the FOC. The FOC approach in this paper is con-

fined to the Caputo definition for the general fractional differin-tegral [11].

Let be the Caputo fractional derivative operator of order

on the real axis, defined by

(9)

Here, is the well-known Eulers gamma function, and

. The operator represents the convolution

operation between the two signals of interest, here

and , respectively.

Taking the FT of the fractional derivative of (9) results in the

following expression:

(10)

Now, from the convolution property of the FT [26], (10) reduces

to

(11)

Therefore, the FT of the fractional derivative of (9) results in

the following expression:

(12)

where .

Thus, the FT of the Caputo fractional derivative of order of

a signal is times the FT of the signal of interest, where

.

Now, we will consider the FrFT of the Caputo fractional

derivative as follows.

Taking the FrFT of the Caputo fractional derivative of (9)

results in the following expression:

(13)

From the convolution property of the FrFT [28] and [31], the

above expression reduces to

(14)

According to the differentiation property of the FrFT [27]

(15)

where is the FrFT of the signal .
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Fig. 1. Variation of the relative error (in percentages) withthe number of terms, of the following CHF functions: (a)

, (b) ,

(c) , (d)

, (e), by letting .

Therefore, from (14) and (15), the following expression re-

sults:

(16)

(17)

Now, for solving the integral in (17) results in [32, Eq.

(A.1.55)]

(18)

The function on the right-hand side of (18), which is

known as the KCHF of thefi

rst kind is an infi

nite power series.For computing the KCHF using the computing machine, the

series must be truncated to some finite number of terms. So,

if the series truncation is used, there must exist a computation

error. The methodology for determining the truncation error of

an infinite power series is given in [16]. The variation of the

relative error (in percentages) after truncating an infinite power

series for different CHF functions (for different s and s) is

shown in Fig. 1 and it is clearly shown that the truncation error

decreases to zero pointwise, as the number of terms increases.

The functions and denotes the error function and

the imaginary error function, respectively, related by the relation

[16].

Now by letting, , , ,

and , (18) becomes

Fig. 2. Fractional order differentiating filter in fractional Fourier domain.

Fig. 3. Comparison of the fractional Fourier domain filtering between themethod in [10] (RL definition) and the proposed method (Caputo definition).

(19)

Thus, it can be seen that the integral representation (19) is a

generalized closed-form expression in terms of , and hence the

closed-form expression for the integral representation (19) can

be obtained by considering different values of the parameter ,

respectively. The example is provided below, by considering thedegenerate cases for and , respectively.

For example, by letting , (19) becomes

(20)

. Thus, (20) is a closed-form expression

for the integral representation (19) for the case .

Similarly, by letting , (19) becomes
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Fig. 4. Fractional orderfiltering results: (a), (b) amplitude-modulated corrupted signal, (real and imaginary parts, respectively) in timedomain; (c), (d) time-domain fil tered signal, (real and imaginary parts respectively).

(21)

Simplifying further, (21) becomes

(22)

Thus, (22) is a closed-form expression for the integral represen-

tation (21) for the case .

Now, from (17) and (19), the following expression results:

(23)

Thus, (23) gives the Caputo-based fractional derivative of the

input signal for varying fractional ordersfrom0 to 1 and for

different rotation angles in the time-frequency (TF) plane of

the FrFT.

Thus, (12) and (23) represents the FT and the FrFT represen-

tation of the Caputo fractional derivative of the input signal

with varying from 0 to 1 and for different s in the TF plane

of the FrFT.

Further, the FrFT approaches the conventional FT for the ro-

tation angle and the integer-order derivative in the

conventional FT can be obtained by substituting the parameters

and in deriving the FrFT representation of the

Caputo fractional derivative of the input signal as follows:

Substituting in (17) gives

(24)

where .

Solving (24)

(25)

Now substituting in (25) gives

(26)

Thus, (26) represents the conventional FT of the integer-order

derivative of the input signal using the FrFT representation

of the Caputo fractional derivative of the input signal .
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Fig. 4. (Continued.) Fractional orderfiltering results: (e), (f) frequency-domain filtered signal, (real and imaginary parts respectively); (g), (h)fractional order FrFT-domain filtered signal, (real and imaginary parts, respectively); (i) RMSE with respect to fractional order parameter, .

VI. FRACTIONAL ORDER DIFFERENTIATING FILTER MODEL

IN FRACTIONAL FOURIER DOMAIN

The proposed filtering scheme in the th FrFT domain is

shown in Fig. 2. In this configuration, first the th domain of

the FrFT of the input is obtained, and then the differentiation

in the th domain is carried out. Thereby, the fractional order

impulse response filter is applied in this domain. The

weighted convolution theorem for the FrFT of [28] is used in

the proposed filtering scheme. Finally, the resulting waveform

is transformed with order in order to obtain the output

signal in the time domain.

VII. DESIGN EXAMPLES

In this section, the design examples are used to demonstrate

the effectiveness of the proposed filtering method. The pro-

posed model describing the fractional order differentiation in

the fractional Fourier domain (FrFD) has been simulated on the

platform of Wolfram Mathematica software (version 8.0) on a
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system having configuration Pentium 4, Intel CPU 1.8 GHz pro-

cessor having 1 GB RAM.

Example 1: In this section, the design example of [10] is

used to evaluate the fractional orderfiltering in FrFD based on

Caputos fractional order operator proposed in this paper. The

proposed model (Fig. 2) is used to simulate the fractional order

differentiating filter in the FrFD. The filtering is performed to

establish a comparative analysis between the design example of

[10] and the proposed method.

Thecriterion used for the optimalfilteringis root mean square

error (RMSE) between the original signal and thefiltered signal.

Comparing the RMSE of the design example of [10] with that

of the proposed method with the optimal parameters

and , it is clear that the fractional order differentiating

filter designed by the proposed method has a smaller error, com-

puted as 0.175813 than that of[10] whose corresponding error

is computed as 0.241609 as shown in Fig. 3, respectively. Ob-

viously, the smaller the error is, the better the fractional order

differentiating filtering operation.

Example 2: In this example, the performance of thefractional order differentiating filter based on Caputos defi-

nition is studied from the point of view of modulation used

in communication engineering. Here the design example

presented is for filtering the amplitude-modulated signal cor-

rupted with high-frequency chirp noise and investigated its

performance from the point of view of time-domain, fre-

quency-domain, and FrFT domain filtering schemes based on

Caputos definition. The amplitude-modulated signal used here

is , which is

made to corrupt with high-frequency chirp noise , same

as in [10], to obtain the input signal to the filter. The input

signal is applied to the proposed model (Fig. 2).

The filtering is performed to compare the performance oftime-domain ( , ), frequency-domain ( .

), and FrFD filtering ( , ). The

various signals are, as shown in Fig. 4(a)(h). The criterion

used for the optimal filtering is RMSE between the original

signal and the filtered signal.

It can be shown by simulation that the FrFD filtered signal

matches maximally with the original signal, as compared

with the time-domain and frequency-domain filtered signals

as shown in Fig. 4. Finally, the RMSE between the original

and the filtered signals is observed for different values of ,

which varies from 0 to 1. This confirms that the FrFT domain

filtering produces minimum RMSE for optimum and ascompared with time-domain and frequency-domain filtering for

the amplitude-modulated signal corrupted with high-frequency

chirp noise.

VIII. CONCLUSION

A novel and unique closed-form analytical expression for the

fractional derivative in the FT and the FrFT domain is proposed

based on Caputos fractional order operator. This work is the

generalization of the differentiation property to the fractional

(noninteger) orders in the FT and theFrFT domains. It motivates

for the variation of two parametersfractional order parameter

and fractional Fourier transform parameter .

The fractional order differentiation derived in this paper is

a more generalized definition, since it achieves the flexibility

of different rotation angles in the TF plane of FrFT with

varying . The design examples have been demonstrated to

show the effectiveness of the proposed method. However, only

1-D fractional order differentiating filtering in the FrFT domain

is studied. Thus, it is interesting to extend the proposed method

to design 2-D fractional order differentiating filter in the FrFT

domain in the future. Thus, the freedom of utilizing varying

order of derivative (fractional derivative) in the entire TF plane

of the FrFT domain can be enjoyed for different potential signal

and image processing applications.

ACKNOWLEDGMENT

The authors would like to thank the Editorial Board and

acknowledge the suggestions made by learned reviewers in

shaping this article to its present form.

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Kulbir Singh was born in Batala, Punjab, India, in1975. He received the B.Tech. degree from PunjabTechnical University, Jalandhar, Punjab, India, in1997, and the M.Tech. degree and the Ph.D. degreefrom the Department of Electronics and Commu-

nication Engineering, Thapar University, Patiala,Punjab, India, in 2000 and 2006, respectively.

He is currently an Associate Professor in the De-partment of Electronics and Communication Engi-neering, ThaparUniversity,Patiala, Punjab, India.Hehas more than 13 years of teaching and research ex-

perience. His research interests include fractional Fourier transform, signal pro-cessing, digital filter design, linear canonical transforms and its applications,digital image processing, digital communication, FPGA-based system design.He has published numerous research papers in reputed peer-reviewed inter-national journals, domestic and international conferences. He is supervising 8Ph.D. degree students.

Dr. Singh is the recipient of the Best Paper Award from the Institutionof Electronics and Telecommunication Engineers (IETE), New Delhi, India, in2007, for his pioneering work in the fractional Fourier transform and its appli-cations in signal and image processing. He is a Member of Institution of Elec-tronics and Telecommunication Engineers, India

Rajiv Saxena received the B.E. degree in electronicsand telecommunication engineering from JabalpurUniversity, Jabalpur, India, and the M.E. degreein electronics and telecommunication engineeringfrom Jiwaji University, Madhya Pradesh, India,and the Ph.D. degree in electronics and computerengineering from Indian Institute of Technology,Roorkee, India.

He served with Reliance, GRASIM, CIMMCO,and Orimpex Industries and also had industrialexperience of automation in process industry. He

started his career as Lecturer in Electronics Engineering Department of MadhavInstitute of Technology and Science (MITS), Gwalior, India, were he becameProfessor in April 1997 and also worked as Head of the Department andChairman, Board of Studies. He moved, on lien from MITS, to Thapar Instituteof Engineering and Technology (TIET) (Now Thapar University), Patiala,India, as Professor in the Department of Electronics and CommunicationEngineering from June 2000 to June 2002. He was the founder Head of theDepartment of Electronics and Communication Engineering at TIET, Patiala(Now Thapar University, Patiala). He was also Principal at Rustam Ji Instituteof Technology, BSF Academy, Tekanpur, (again on lien from MITS Gwalior)for a period of two years from January 2004 to January 2006. He also servedas guest faculty at leading institutions including ABV-IIITM, Gwalior, IITTM,Gwalior, and in M.Tech. Program of U. P. Technical University, Lucknow. Hehas delivered various invited talks/keynote addresses at various platforms of

repute. He served as Coordinator of Project IMPACT for a period of two years(project was a joint venture of World Bank and Ministry of Communicationand IT, Government of India). He executed two major research projects fundedby AICTE besides the project under FIST program of DST. He has supervised13 Ph.D. degree candidates in the area of digital signal processing, digitalimage processing and application of DSP tools in electronic systems. Hehas published about 70 research articles in refereed journals of national andinternational repute.

Prof. Saxena received the Best Paper Award of IETE in 2008. He is a Fellowof the Institution of Electronics and Telecommunication Engineers (IETE) andsenior member of the Computer Society of India (CSI) and the Indian Societyfor Technical Education (ISTE).

Sanjay Kumar was born in Dehradun, Uttarakhand,

India, in 1981. He received the B.E. degree inelectronics and communication engineering fromMeerut University, Meerut, India, in 2003, and theM.Tech. degree in VLSI design from Thapar Univer-sity, Patiala, Punjab, India, in 2009. He is currentlyworking toward the Ph.D. degree in the Departmentof Electronics and Communication Engineering,Thapar University, Patiala, Punjab, India.

He has worked as scientist/engineer with IndianSpace Research Organization (ISRO), Bangalore,

India, working toward Indias Moon Mission CHANDRAYAANI beforejoining Thapar University as an Assistant Professor in the Department ofElectronics and Communication Engineering. His current research interestsinclude fractional order transforms, signal processing, implementation andapplication of fractional order circuits, and radar signal processing. He has pub-lished research papers in peer-reviewed international journals and internationalconferences.

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