Caputo-Based Fractional Derivative in Fractional Fourier ... - IEEE Xplore

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IEEE JOURNAL ON EMERGING AND SELECTED TOPICS IN CIRCUITS AND SYSTEMS, VOL. 3, NO. 3, SEPTEMBER 2013

Caputo-Based Fractional Derivative in Fractional Fourier Transform Domain Kulbir Singh, Rajiv Saxena, and Sanjay Kumar

Abstract—This paper proposes a novel closed-form analytical expression of the fractional derivative of a signal in the Fourier transform (FT) and the fractional Fourier transform (FrFT) domain by utilizing the fundamental principles of the fractional order calculus. The generalization of the differentiation property in the FT and the FrFT domain to the fractional orders has been presented based on the Caputo’s definition of the fractional differintegral, thereby achieving the flexibility of different rotation angles in the time—frequency plane with varying fractional order parameter. The closed-form analytical expression is derived in terms of the well-known higher transcendental function known as confluent hypergeometric function. The design examples are demonstrated to show the comparative analysis between the established and the proposed method for causal signals corrupted with high-frequency chirp noise and it is shown that the fractional order differentiating filter based on Caputo’s definition is presenting good performance than 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 differintegral method has also been investigated taking into account amplitude-modulated signal corrupted with high-frequency chirp noise. Index Terms—Caputo fractional derivative, fractional Fourier transform (FrFT), fractional order calculus (FOC), fractional order derivative, Kummer confluent hypergeometric function.

I. INTRODUCTION

T

HE CONCEPT of derivatives is traditionally associated to an integer; given a function, one can differentiate it one, 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 classical 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 Liouville, Riemann, and Holmgren [1]. This wonderful tool of mathematics called fractional order calculus (FOC), which is in fact a generalization of the classical Newtonian calculus, aims to describe and discover the nature and the natural phenomena [2]. In the research area of FOC, the integer order of the derivative of the function is generalized to the fractional order , where is a real number Manuscript received February 15, 2013; revised May 03, 2013; accepted June 10, 2013. Date of publication July 19, 2013; date of current version September 09, 2013. This paper was recommended by Guest Editor G. Chen. K. Singh and S. Kumar are with the Department of Electronics and Communication Engineering, Thapar University, Patiala 147004, Punjab, India (e-mail: [email protected]; [email protected]). R. Saxena is with the Department of Electronics and Communication Engineering, Jaypee University of Engineering and Technology, Raghogarh, Guna 473226, 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 digital differentiators that have received a great attention to the research 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, respectively, 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 involves two different varying parameters: the fractional order parameter 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 differentiation in FrFT domain by incorporating the Riemann–Liouville (RL) fractional order operator. The focus in [10] is on establishing the performance of the designed fractional order differentiating filter in 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 definition for the FIR system and for a restrict class of signals and noises: causal signals with high-frequency chirp noise. The Caputo 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 research community, mainly concerned with different types of initial 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 Kumar et 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 formula and generalized it to define the fractional derivative of a function in FT domain, whereas in the proposed study, the Caputo-based definition for the general fractional differintegral is used and proposed to derive the fractional derivative of a function in both FT and FrFT domain. Similarly, McBride et al. [9] derive the differentiation property in the FrFT domain to integer

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SINGH et al.: CAPUTO-BASED FRACTIONAL DERIVATIVE IN FRACTIONAL FOURIER TRANSFORM DOMAIN

order only, whereas the proposed method generalize it to establish the differentiation property in the FrFT domain to noninteger 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 optimization 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 equations 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 between 0 and 1 i.e., . The purpose of this paper is to design the fractional order filter in the FrFT domain using the FOC approach based on Caputo’s fractional 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 flexibility 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 Caputo’s 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 FOC and 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.

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The KCHF is commonly available in numerical software packages, 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 integral, has received great attention in the last few decades, because of its ability to model systems more accurately than integer 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)

is the real part of . Moreover, the fractional order where 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 general fractional differintegral are the RL, the Grünwald–Letnikov (GL), and the Caputo definitions [1]. A. Riemann–Liouville Definition The RL definition for a function

II. CONFLUENT HYPERGEOMETRIC FUNCTION The confluent hypergeometric function (CHF) of the first kind is a degenerate form of the hypergeometric function which arises as a solution of the confluent hypergeometric differential equation. In this study, the Kummer confluent hypergeometric function (KCHF) is considered for the derivation of the proposed analytical expression. The KCHF is defined by the absolutely convergent infinite power series [15], [16] as

is given as (4)

for gamma function.

where

is the well-known Euler’s

B. Grünwald–Letnikov Definition

(1)

The GL definition is the most popular definition for the general fractional derivatives and integrals because of its discrete nature. According to GL, the fractional derivative of a function is given as

where, the Pochhammer symbol denotes the rising factorial and . The detailed description of KCHF with properties can be found in [15], [16].

(5)

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where is the sampling period, is the well-known Euler’s gamma function and means the integer part of . C. Caputo Definition The Caputo definition for a function

can be given as

(6) , is an integer and is a real where 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 frequently used 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 processing [7], [9], [10], [27]–[31]. It also leads to the generalization 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 the conventional FT are special cases of those of the FrFT. However, in every 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 corresponds to the th fractional power of the Fourier transform opis defined as erator, . The th-order FrFT of (7)

Ŧ

where

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 approach of the FOC. The FOC approach in this paper is confined to the Caputo definition for the general fractional differintegral [11]. Let be the Caputo fractional derivative operator of order on the real axis, defined by (9) is the well-known Euler’s 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:

Here,

(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)

, the transformation kernel

From the convolution property of the FrFT [28] and [31], the above expression reduces to (8) with [7] where indicates the rotation angle of the transformed signal for the FrFT and denotes the continuous FrFT (CFrFT) operator. corresponds to the conventional FT, The FrFT with corresponds to the identity operator. and the one with Also, two successive FrFT’s 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.

(14) According to the differentiation property of the FrFT [27] (15) where

is the FrFT of the signal

.


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

Fig. 1. Variation of the relative error (in percentages) with of the following CHF functions: (a) the number of terms, , (b) , (c) , (d) , (e) , by letting .

Therefore, from (14) and (15), the following expression results:

(16)

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

Now by letting, and

,

, , (18) becomes

,

(17) Now, for solving the integral in (17) results in [32, Eq. (A.1.55)]

(18) on the right-hand side of (18), which is The function known as the KCHF of the first kind is an infinite 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].

(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 the degenerate cases for and , respectively. For example, by letting , (19) becomes

(20)

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Fig. 4. Fractional order filtering results: (a), (b) amplitude-modulated corrupted signal, (real (real and imaginary parts respectively). domain; (c), (d) time-domain filtered signal,

. Thus, (20) is a closed-form expression for the integral representation (19) for the case . Similarly, by letting , (19) becomes

(21) Simplifying further, (21) becomes

and imaginary

Thus, (23) gives the Caputo-based fractional derivative of the input signal for varying fractional orders from 0 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 representation 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 rotation 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

(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:

parts, respectively) in time

(24) where . Solving (24) (25) Now substituting

in (25) gives (26)

(23)

Thus, (26) represents the conventional FT of the integer-order using the FrFT representation derivative of the input signal of the Caputo fractional derivative of the input signal .


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

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 “ signal in the time domain.

” in order to obtain the output

VII. DESIGN EXAMPLES In this section, the design examples are used to demonstrate the effectiveness of the proposed filtering method. The proposed 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 processor having 1 GB RAM. Example 1: In this section, the design example of [10] is used to evaluate the fractional order filtering in FrFD based on Caputo’s 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. The criterion used for the optimal filtering is root mean square error (RMSE) between the original signal and the filtered 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, computed as 0.175813 than that of [10] whose corresponding error is computed as 0.241609 as shown in Fig. 3, respectively. Obviously, the smaller the error is, the better the fractional order differentiating filtering operation. Example 2: In this example, the performance of the fractional order differentiating filter based on Caputo’s definition 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 corrupted with high-frequency chirp noise and investigated its performance from the point of view of time-domain, frequency-domain, and FrFT domain filtering schemes based on Caputo’s 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 of time-domain ( , ), frequency-domain ( . ). The ), and FrFD filtering ( , 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 as compared 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 Caputo’s fractional order operator. This work is the generalization of the differentiation property to the fractional (noninteger) orders in the FT and the FrFT domains. It motivates for the variation of two parameters—fractional 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. REFERENCES [1] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. New York: Academic, 1974. [2] S. Das, Functional Fractional Calculus, 2nd ed. New York: Springer, 2011. [3] A. Le Mehauté, J. A. T. Machado, J. C. Trigeassou, and J. Sabatier, Fractional Differentiation and its Applications. Diedorf, Germany: Ubooks, 2005, vol. 1. [4] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers. New York: Springer, 2011. [5] Y. F. Luchko, H. Matrínez, and J. J. Trujillo, “Fractional Fourier transform and some of its applications,” Fract. Cal. Appl. Anal., vol. 11, no. 4, pp. 457–470, 2008. [6] G. Jumarie, “Fourier’s transform of fractional order via Mittag-Leffler function and modified Riemann-Liouville derivative,” J. Appl. Math. Inf., vol. 26, no. 5–6, pp. 1101–1121, 2008. [7] H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform With Applications in Optics and Signal Processing. New York: Wiley, 2001, vol. 39. [8] C. C. Tseng, S. C. Pei, and S. C. Hsia, “Computation of fractional derivatives using Fourier transform and digital FIR differentiator,” Signal Process., vol. 80, pp. 151–159, 2000. [9] A. C. McBride and F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math., vol. 39, no. 2, pp. 159–175, 1987. [10] S. Kumar, K. Singh, and R. Saxena, “Closed-form analytical expression of fractional order differentiation in fractional Fourier transform domain,” Circuits Syst. Signal Process., vol. 32, no. 4, pp. 1875–1889, 2013. [11] I. Podlubny, Fractional Differential Equations. San Diego, CA: Academic, 1999. [12] C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls. New York: Springer, 2010. [13] J. C. Trigeassou, N. Maamri, and A. Oustaloup, “Automatic initialization of the Caputo fractional derivative,” in Proc. IEEE Conf. Decision Control Eur. Control Conf., Orlando, FL, 2011, pp. 3362–3368. [14] H. Sheng, Y. Q. Chen, and T. S. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications. New York: Springer, 2012. [15] L. J. Slater, Confluent Hypergeometric Functions. Cambridge, U.K.: Cambridge Univ. Press, 1960. [16] M. Abramowitz and I. A. Shegun, “Confluent hypergeometric functions,” in Handbook of Mathematical Functions With Formulas, Graphs, Mathematical Tables. New York: Dover, 1972, ch. 13, pp. 503–515. [17] R. Panda and M. Dash, “Fractional generalized splines and signal processing,” Signal Process., vol. 86, no. 9, pp. 2340–2350, 2006. [18] Z. Z. Yang and J. L. Zhou, “An improved design for the IIR-type digital fractional order differential filter,” Proc. Int. Seminar Future Bio Med. Inf. Eng., pp. 473–476, 2008. [19] B. T. Krishna and K. V. V. S. Reddy, “Active and passive realization of fractance device of order 1/2,” Act. Passive Electron. Compon., vol. 2008. [20] Y. Pu, X. Yuan, K. Liao, J. Zhou, N. Zhang, X. Pu, and Y. Zeng, “A recursive two-circuits series analog fractance circuit for any order fractional calculus,” in Proc. SPIE Opt. Inf. Process., 2006, vol. 6027, pp. 509–519.


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Kulbir Singh was born in Batala, Punjab, India, in 1975. He received the B.Tech. degree from Punjab Technical University, Jalandhar, Punjab, India, in 1997, and the M.Tech. degree and the Ph.D. degree from the Department of Electronics and Communication Engineering, Thapar University, Patiala, Punjab, India, in 2000 and 2006, respectively. He is currently an Associate Professor in the Department of Electronics and Communication Engineering, Thapar University, Patiala, Punjab, India. He has more than 13 years of teaching and research experience. His research interests include fractional Fourier transform, signal processing, 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 international journals, domestic and international conferences. He is supervising 8 Ph.D. degree students. Dr. Singh is the recipient of the “Best Paper Award” from the Institution of Electronics and Telecommunication Engineers (IETE), New Delhi, India, in 2007, for his pioneering work in the fractional Fourier transform and its applications in signal and image processing. He is a Member of Institution of Electronics and Telecommunication Engineers, India

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Rajiv Saxena received the B.E. degree in electronics and telecommunication engineering from Jabalpur University, Jabalpur, India, and the M.E. degree in electronics and telecommunication engineering from Jiwaji University, Madhya Pradesh, India, and the Ph.D. degree in electronics and computer engineering from Indian Institute of Technology, Roorkee, India. He served with Reliance, GRASIM, CIMMCO, and Orimpex Industries and also had industrial experience of automation in process industry. He started his career as Lecturer in Electronics Engineering Department of Madhav Institute of Technology and Science (MITS), Gwalior, India, were he became Professor in April 1997 and also worked as Head of the Department and Chairman, Board of Studies. He moved, on lien from MITS, to Thapar Institute of Engineering and Technology (TIET) (Now Thapar University), Patiala, India, as Professor in the Department of Electronics and Communication Engineering from June 2000 to June 2002. He was the founder Head of the Department of Electronics and Communication Engineering at TIET, Patiala (Now Thapar University, Patiala). He was also Principal at Rustam Ji Institute of Technology, BSF Academy, Tekanpur, (again on lien from MITS Gwalior) for a period of two years from January 2004 to January 2006. He also served as guest faculty at leading institutions including ABV-IIITM, Gwalior, IITTM, Gwalior, and in M.Tech. Program of U. P. Technical University, Lucknow. He has 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 Communication and IT, Government of India). He executed two major research projects funded by AICTE besides the project under FIST program of DST. He has supervised 13 Ph.D. degree candidates in the area of digital signal processing, digital image processing and application of DSP tools in electronic systems. He has published about 70 research articles in refereed journals of national and international repute. Prof. Saxena received the Best Paper Award of IETE in 2008. He is a Fellow of the Institution of Electronics and Telecommunication Engineers (IETE) and senior member of the Computer Society of India (CSI) and the Indian Society for Technical Education (ISTE).

Sanjay Kumar was born in Dehradun, Uttarakhand, India, in 1981. He received the B.E. degree in electronics and communication engineering from Meerut University, Meerut, India, in 2003, and the M.Tech. degree in VLSI design from Thapar University, Patiala, Punjab, India, in 2009. He is currently working toward the Ph.D. degree in the Department of Electronics and Communication Engineering, Thapar University, Patiala, Punjab, India. He has worked as scientist/engineer with Indian Space Research Organization (ISRO), Bangalore, India, working toward India’s Moon Mission “CHANDRAYAAN—I” before joining Thapar University as an Assistant Professor in the Department of Electronics and Communication Engineering. His current research interests include fractional order transforms, signal processing, implementation and application of fractional order circuits, and radar signal processing. He has published research papers in peer-reviewed international journals and international conferences.