Studies on fractional order differentiators and integrators: A survey

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Signal Processing

Signal Processing 91 (2011) 386–426

0165-16

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/sigpro

Studies on fractional order differentiators and integrators: A survey

B.T. Krishna

Department of ECE, GITAM University, Visakhapatnam, India

a r t i c l e i n f o

Article history:

Received 29 November 2009

Received in revised form

24 June 2010

Accepted 24 June 2010Available online 7 July 2010

Keywords:

Fractional order

Mittag–Leffler function

s to z transform

Digital differentiator

Digital integrator

Discretization

QRS complex

Edge detection

84/$ - see front matter & 2010 Elsevier B.V. A

016/j.sigpro.2010.06.022

ail address: [email protected]

a b s t r a c t

Studies on analysis, design and applications of analog and digital differentiators and

integrators of fractional order is the main objective of this paper. Time and frequency

domain analysis, different ways of realization of fractance device is presented. Active

and passive realization of fractance device of order 12 using continued fraction expansion

is carried out. Later, time and frequency domain analysis of fractance based circuits is

considered. The variations of rise time, peak time, settling time, time constant, percent

overshoot with respect to fractional order a is presented.

Digital differentiators and integrators of fractional order can be obtained by using

direct and indirect discretization techniques. The s to z transforms used for this purpose

are revisited. In this paper by using indirect discretization technique fractional order

differentiators and integrators of order 12 and 1

4 are designed. These digital differentiators

and integrators are implemented in real time using TMS320C6713 DSP processor and

tested using National instruments education laboratory virtual instrumentation system

(NIELVIS). The designed fractional order differentiators have been used for the detection

of QRS sequences as well as the occurrence of Sino Atrial Rhythms in an ECG signal and

also for the detection of edges in an image. The obtained results are in comparison with

the conventional techniques.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

This paper deals with the analysis, design andapplications of analog and digital fractional order differ-entiators and integrators. Fractional order differentiatorsand integrators are examples of fractional order systems.Fractional order systems are described by fractional orderdifferential equations [52,54]. A method for modeling andsimulation of fractional systems using state-space repre-sentation is presented in [33]. Fractional order differen-tiators and integrators are used to compute the fractionalorder time derivative and integral of the given signal[1–4]. Geometrical and physical interpretations of frac-tional order integrators and differentiators is discussed in[6,8,9,13,37]. In [35] Machado presented a probabilisticinterpretation of fractional order derivative based on

ll rights reserved.

Grunwald–Letnikov definition. In analog domain such anoperation can be called as fractance device. The expressionfor impedance function of a fractance device is given by,ZðsÞ ¼ k0=sa where k0 is a constant and a is a fractionalorder. Depending upon the values of a the behavior of theelement changes from Inductor to Capacitor [24,44].Terms fractance, fractional order differ integral, fractionalorder capacitor can be used synonymously.

In order to solve the fractional order differentialequations, which characterize such systems, a combina-tion of fractional calculus and Laplace transform techni-ques can be used. The solution of fractional orderdifferential equations contain Mittag–Leffler functions[7,10,12,36,43,56]. In this paper expressions for time-domain response of a fractance device of order a, 1

2 fordifferent excitations are derived.

With the advantages of fractance device in variousfields its realization has gained importance. The fractancedevice can be realized by using fractal structure. Nakagawa

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B.T. Krishna / Signal Processing 91 (2011) 386–426 387

and Sorimachi proposed a tree type circuit using resistorsand capacitors [24]. Oldham has proposed a chain typecircuit for the realization of fractance device [1]. Recently anet grid type circuit was proposed by Pu [38,42]. But thefractal based realization suffer from the problem ofoccupying high space and high cost.

The crucial point in the realization of fractance deviceis finding a rational approximation of its impedancefunction. There are so many procedures that can be usedto calculate the rational function approximation offractance device. Oustaloup method, Newton’s method,Matsudas method, etc. were some of them[17,22,32,47,58]. It has been proposed that using con-tinued fraction expansion fractance device can be rea-lized. The rational approximation thus obtained issynthesized as a ladder network. The results comparewell with the previous techniques [58].

The fractance can be used in circuits along with thethree passive elements resistor, inductor and capacitoreither as series or as a shunt element [18,21,26,34,60,62].As part of the paper time and frequency domain analysisof inverted-L type fractance based circuits has beenperformed. The effect of fractional order of the circuit onfrequency response is also studied. It can be observed thatthe performance of the higher integer order circuit couldbe obtained from the circuit with lesser fractional order.This also reduces the cost and space. Later, the expres-sions for peak overshoot, rise time, time constant, settlingtime, etc. were obtained for the fractional order circuitthat is considered [48].

The second part of this article concentrates on thedesign, application and real time implementations offractional order digital differentiators and integrators. Thedesign of digital differentiators and integrators involvesthe discretization of the fractional-order operator, sa

[77,78]. Direct discretization and indirect discretizationwere the commonly used discretization techniques. A lotof literature is available for direct discretization techni-que. In this paper indirect discretization technique isfollowed. An s to z transform has to be used to perform thediscretization [61–63]. As the s to z transform maps theleft hand plane of s-domain into the unit circle in z

domain it has to preserve the stability properties. Some ofthe common s to z transforms are Bilinear and Backwardtransform. Every integration rule can produce a new s to z

transform. Al-Alaoui has proposed a method for thecalculation of different s to z transforms from theintegration rules [64–72]. The first order s to z transform,called Al-Alaoui transform has shown to be much moreefficient than the previous transforms. Fractional orderdifferentiators and integrators can also be designed byusing least squares method [80,85], power series expan-sion [87], adaptive technique [81], etc.

The differentiators and integrators obtained usingdirect and indirect discretizations are compared. Theproposed approach is tested for differentiators andintegrators of order 1

4 and 12. The results obtained compare

favorably with the ideal characteristics. Fractional orderdigital differentiators and integrators designed are im-plemented in real time and the practical behavior iscompared with the theoretical behavior. The digital

fractional order differentiators and integrators are im-plemented in real time using TMS320C6713 DSP kit andtested using NIELVIS. For the real time implementationCascaded Direct Form-II structure is chosen. The theore-tical and practical results compare well within thereasonable limits. The error can be reduced by increasingthe gain.

QRS detection is an important topic in the area ofBiomedical Engineering. The electrocardiogram (ECG) is agraphical representation of the electrical activity of theheart and is obtained by connecting specially designedelectrodes to the surface of the body. Variety of methodsuse digital differentiators for the QRS detection [94–96].The template matching technique using digital differen-tiator is one of the traditional technique used by theresearch community [92]. In this paper, fractional orderdigital differentiator has been replaced with the tradi-tional differentiator. The results are comparable with theprevious techniques. Edge detection refers to the identi-fication of changes in brightness of an image. Applyingfractional order differentiators to detect edges of an imageis also performed in this paper. The performance of thefractional order differentiators is comparable to theconventional differentiators.

The paper is organized as follows. Section 2 deals withthe basic definition, time-domain response calculations offractance device. Realization of fractance device usingdifferent approximations is also presented in this section.Fractance based circuits, their time and frequency domainresponse calculations, time-domain parameter calcula-tions are presented in Section 3. An s to z transform is tobe used for the discretization of continuous time systems.Different types of s to z transforms (digital differentiators)and their comparisons are presented in Section 4. Section5 deals with the indirect discretization technique used forthe design of fractional order digital differentiators andintegrators. Design and real-time implementation of thedigital differentiators of fractional order are also discussedin this section. Some of the applications of fractional orderdifferentiators such as detection of QRS sequences in anECG signal, edge detection are discussed in Section 6.Finally, Section 7 deals with results and conclusions.

2. Fractance device

Of late, many researchers are paying attention to thefractance device. The origin of this device is from theworking principle of well known passive element capa-citor [20,23]. According to Curie’s Law when the initialstored energy is zero, in a capacitor and if DC Voltage V

has been applied, the current flowing through the devicewill be

iðtÞ ¼V

htafor t40 ð1Þ

where h and V are real. Taking Laplace transform ofEq. (1),

IðsÞ ¼Gð1�aÞV

hs1�a ð2Þ

B.T. Krishna / Signal Processing 91 (2011) 386–426388

When the applied voltage signal is DC,

ZðsÞ ¼hs�a

Gð1�aÞ ¼k0

sað3Þ

where k0 ¼ h=Gð1�aÞ, GðxÞ is gamma function and a isfractional order and takes the values as �1, 0, 1 forcapacitance, resistance and inductance, respectively. Ata¼�2, it represents the well-known frequency-depen-dent negative resistor (FDNR). In general a symbol F isused to denote fractance device. For an ideal fractancedevice the phase angle is constant independent of thefrequency range of operation but depends only on thevalue of fractional order, a. Fractance device is also calledas constant phase angle device, constant phase element (CPE)or simply fractor [24]. The magnitude and phase responsesof the fractance device for various values of a are shown inFigs. 1 and 2, respectively.

Fig. 1. Magnitude response of fract

Fig. 2. Phase response of fractan

2.1. Time-domain response calculations of fractance device

In order to estimate the time domain response offractance device Laplace transform based approach isselected. From Eq. (3), response I(s) can be written as

IðsÞ ¼sa

k0VðsÞ ð4Þ

The output current is expressed in terms of fractionaloperators viz. Mittag–Leffler functions ðEa,bðxÞ ¼

P1k ¼ 0 xk=

GðakþbÞ aZ0, bZ0Þ, gamma functions, etc. and thensimplified. The time-domain expressions of fractancedevice of order a and 1

2 for different excitations werepresented in Table 1. Figs. 3–8 illustrate the time-domainbehavior of fractance device of order 1

2 ðZðsÞ ¼ k0=ffiffisp¼ffiffiffiffiffiffiffiffiffiffi

R=Csp

Þ when R¼ 1 kO, C ¼ 1 nF and A=1 mV [10,31,57].

ance device for a¼ 7 12 ,7 1

4.

ce device for a¼ 7 12 ,7 1

4.

Fig. 3. Impulse response.

Fig. 4. Step response.

Table 1Time-domain response expressions for fractance device.

Excitation Fractance device of order aFractance device of order

1

2

dðtÞ t�a�1

k0Gð�aÞ�

t�3=2

2k0

ffiffiffiffipp

u(t) t�a

k0Gð1�aÞ1

k0

ffiffiffiffiffiptp

Asinot Aok0

� �t1�aE2,2�að�o2t2Þ

Aok0

t1=2E2,3=2ð�o2t2Þ ¼Ao1=2

k0

� �sin otþ

p4

� �þ

Ak0ffiffiffiffiffiptp

� � P1n ¼ 1

ð�1Þn1:3:5:7:��ð4n�3Þ

ð2otÞ2n�1

Acosot Aok0

� �t�aE2,1�að�o2t2Þ

Aok0

� �t�1=2E2,1=2ð�o2t2Þ ¼

Ao1=2

k0

� �cos otþ

p4

� �þ

Ak0ffiffiffiffiffiptp

� � P1n ¼ 1

ð�1Þnð1:3:5 . . . ð4n�1ÞÞ

ð2otÞ2n

Asinhot Aok0

� �t1�aE2,2�aðo2t2Þ

Aok0

� �t1=2E2,3=2ðo2t2Þ ¼

Ao1=2

k0

� �Dawð

ffiffiffiffiffiffiotpÞffiffiffiffi

pp þ

eoterf ðffiffiffiffiffiffiotpÞ

2

� �, where Dawð

ffiffiffiffiffiffiotpÞ is

Dawson’s Integral

Asinhotsinot 2Ao2t2�a

k0

� �E4,3�að�4o4t4Þ

2Ao2t3=2

k0

� �E4,5=2ð�4o4t4Þ

Acosotsinhot Aok0

� �½taþ1E4,2�að�4o4t4Þ�2o2t4�aE4,4�að�4o4t4Þ�

Aok0

� �½t3=2E4,3=2ð�4o4t4Þ�2o2t7=2E4,7=2ð�4o4t4Þ�


Fig. 5. Response to sinot.

Fig. 6. Response to cosot.

Fig. 7. Response to sinhot.


Fig. 8. Response to complicated function Asinho t sinot.

Table 2Rational approximations for sa .

No. of

terms

Rational approximation for a Rational approximation for a¼ 12

2 ð1�aÞþsð1þaÞð1þaÞþsð1�aÞ

3sþ1

sþ3

4 ða2þ3aþ2Þs2þð8�2a2Þsþða2�3aþ2Þ

ða2�3aþ2Þs2þð8�2a2Þsþða2þ3aþ2Þ

5s2þ10sþ1

s2þ10sþ5

6 ða3þ6a2þ11aþ6Þs3þð�3a3�6a2þ27aþ54Þs2þð3a3�6a2þ27aþ54Þsþð�a3þ6a2�11aþ6Þ

ð�a3þ6a2�11aþ6Þs3þð3a3�6a2þ27aþ54Þs2þð�3a3�6a2þ27aþ54Þsþða3þ6a2þ11aþ6Þ

7s3þ35s2þ21sþ1

s3þ21s2þ35sþ7

8

P0s4þP1s3þP2s2þP3sþP4

Q0s4þQ1s3þQ2s2þQ3sþQ4

P0 ¼Q4 ¼ a4þ10a3þ35a2þ50aþ24

P1 ¼Q3 ¼�4a4�10a3þ40a2þ320aþ384

P2 ¼Q2 ¼ 6a4�150a2þ864

P3 ¼Q1 ¼�4a4þ20a3þ40a2�320aþ384

P4 ¼Q0 ¼ a4�10a3þ35a2�50aþ24

9s4þ84s3þ126s2þ36sþ1

s4þ36s3þ126s2þ84sþ9

10

P0s5þP1s4þP2s3þP3s2þP4sþP5

Q0s5þQ1s4þQ2s3þQ3s2þQ4sþQ5

P0 ¼Q5 ¼�a5�15a4�85a3�225a2�274a�120

P1 ¼Q4 ¼ 5a5þ45a4þ5a3�1005a2�3250a�3000

P2 ¼Q3 ¼�10a5�30a4þ410a3þ1230a2�4000a�12 000

P3 ¼Q2 ¼ 10a5�30a4�410a3þ1230a2þ4000a�12 000

P4 ¼Q1 ¼�5a5þ45a4�5a3�1005a2þ3250a�3000

P5 ¼Q0 ¼ a5�15a4þ85a3�225a2þ274a�120

11s5þ165s4þ462s3þ330s2þ55sþ1

s5þ55s4þ330s3þ462s2þ165sþ11


2.2. Realization of fractance device

Network functions of fractance device are not easy toimplement for computational purposes [27–30,46,49,88,51]. So it becomes mandatory to find integer orderrational approximations. There are many different ways offinding such rational approximations.

2.2.1. The Oustaloup approximation

This method provides a continuous approximationbased on a recursive distribution of zeros andpoles at well chosen intervals. The starting pointis [47]

FðsÞ ¼ sd ð5Þ

where d is the fractional order. The structure of theapproximation FuðsÞ to F(s) is of the form

FuðsÞ ¼ CYN

k ¼ �N

1þs

ok

1þs

oku

ð6Þ

choosing the following set of synthesis formulae:

o0 ¼ a0:5ou ð7Þ

o0u¼ a�0:5ou ð8Þ

okþ1

ok¼okþ1u

oku¼ aZ41 ð9Þ


okþ1u

ok¼ Z40 ð10Þ

ok

oku¼ a40 ð11Þ

N¼logðoN=o0Þ

logðaZÞ41 ð12Þ

d¼logðaÞ

logðaZÞð13Þ

with ou being the unit gain frequency and the centralfrequency of a band of frequencies geometrically dis-tributed around it. That is, ou ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiohobp

,oh,ob are the high

Fig. 9. Comparison of magnitude responses of rati

Fig. 10. Comparison of phase responses of ration

and low transitional frequencies. Assuming oh ¼ 102,ob ¼ 10�2 the rational approximations obtained will be

1ffiffisp ¼

s5þ74:97s4þ768:5s3þ1218s2þ298:5sþ10

10s5þ298:5s4þ1218s3þ768:5s2þ74:97sþ1ð14Þ

ffiffisp¼

10s5þ298:5s4þ1218s3þ768:5s2þ74:97sþ1

s5þ74:97s4þ768:5s3þ1218s2þ298:5sþ10ð15Þ

2.2.2. The Carlson approximation

The method proposed by Carlson is derived from aregular Newton process. The starting point of the method

onal approximation functions with idealffiffisp

.

al approximation functions with idealffiffisp

.


states the following relationships [17]:

HðsÞ1=a�GðsÞ ¼ 0 ð16Þ

HðsÞ ¼ ðGðsÞÞa ð17Þ

Defining, a¼ 1=q, m¼ q=2 in each iteration, starting fromthe initial value H0(s)=1 an approximated rational func-tion is obtained in the form

HiðsÞ ¼Hi�1ðsÞðq�mÞH2

i�1ðsÞþðqþmÞGðsÞ

ðqþmÞH2i�1ðsÞþðq�mÞGðsÞ

ð18Þ

Carlson has obtained rational approximation of 1=ffiffisp

as

HðsÞ ¼s4þ36s3þ126s2þ84sþ9

9s4þ84s3þ126s2þ36sþ1ð19Þ

Fig. 11. Pole–zero

Fig. 12. Pole–zero

2.2.3. The Matsuda approximation

This method provides continuous approximation bycalculating gain at logarithmically spaced frequencies. Letthe frequencies chosen be o0,o1,o2, . . .oN . This methodrequires defining functions [22],

d0ðoÞ ¼ jFðjoÞj

d1ðoÞ ¼o�o0

d0ðoÞ�d0ðo0Þ

d2ðoÞ ¼o�o1

d1ðoÞ�d1ðo1Þ

^

dNðoÞ ¼o�oN�1

dN�1ðoÞ�dN�1ðoN�1Þð20Þ

plot offfiffisp

.

plot of 1ffiffisp .


Then (N+1)� (N+1) superior triangular matrix will beformed

D¼

d0ðo0Þ d0ðo1Þ d0ðo2Þ � � � d0ðoNÞ

d1ðo1Þ d1ðo2Þ � � � d1ðoNÞ

d2ðo2Þ � � � d2ðoNÞ

� � � � � � � � � � � � � � �

� � � � � � � � � � � � dNðoNÞ

26666664

37777775

ð21Þ

wherefrom a set of coefficients is defined as

ak ¼Dkk ¼

jFðjo0Þj if k¼ 0ok�ok�1

dk�1ðokÞ�dk�1ðok�1Þif k¼ 1,2, . . . ,N

8<: ð22Þ

The desired approximation is then given by the continuedfraction

FuðsÞ ¼ a0þs�o0

a1þs�o1

a2þs�o2

a3þ . . .

¼ a0þs�o0

a1þ

s�o1

a2þ� � � ð23Þ

Now the rational approximation obtained for 1=ffiffisp

is,

FuðsÞ ¼0:08549s4þ4:877s3þ20:84s2þ12:995sþ1

s4þ13s3þ20:84s2þ4:876sþ0:08551ð24Þ

Fig. 13. Passive realization of the fractance device.

Fig. 14. Active realization of the fractance device.

Fig. 15. Magnitude r

2.2.4. Continued fraction expansion (CFE) method

The procedure is outlined below [25,58].

1.

espo

By making use of continued fraction expansionformulae, obtain the rational approximation that bestfits sa in s-domain.

2.
Check for the convergence of the obtained rationalapproximation. Also, check whether the selectedrational approximation is stable or not. If the systemis stable and minimum phase then the rationalapproximation of 1=sa is simply the inverse of thetransfer function obtained in step 1.
3.
The circuit for fractance device is synthesized from therational approximation obtained using one-port net-work synthesis procedures. The active circuit isobtained by using an operational amplifier.
The above-mentioned procedure is implemented be-low.We have the continued fraction expansion forð1þxÞa as [5]

ð1þxÞa ¼1

1�

ax

1þ

ð1þaÞx2þ

ð1�aÞx3þ

ð2þaÞx2þ

ð2�aÞx5þ � � �

ð25Þ

The above continued fraction expansion converges in thefinite complex s-plane, along the negative real axis fromx¼�1 to � 1. Substituting x=s�1 and limiting numberof terms in Eq. (25), the rational approximations obtainedfor sa , s1=2 are presented in Table 2. In order to get therational approximation of 1=sa the expressions have to besimply inverted. Higher order rational approximations canbe obtained by increasing the number of terms in Eq. (25).

Figs. 9 and 10 compare the magnitude and phaseresponses of the rational approximations with the idealffiffi

sp

. It is observed that fifth order rational approximation isbest fit to ideal one compared to other rationalapproximations. So in the remaining part of the paperfifth order rational approximation is considered.

nse offfiffisp

.


So,

sa ¼P0s5þP1s4þP2s3þP3s2þP4sþP5

Q0s5þQ1s4þQ2s3þQ3s2þQ4sþQ5ð26Þ

The above rational approximation is stable if

P040

P140

P540

P1P2�P0P340

P1P2P3þP0P1P54P0P23þP2

1P4 ð27Þ

similar conditions are also possible if all P’s are replacedwith Q ’s. Considering a as 1

2, 14 the rational approximations

Fig. 16. Phase resp

Fig. 17. Error p

obtained will be

ffiffisp¼

11s5þ165s4þ462s3þ330s2þ55sþ1

s5þ55s4þ330s3þ462s2þ165sþ11ð28Þ

1ffiffisp ¼

s5þ55s4þ330s3þ462s2þ165sþ11

11s5þ165s4þ462s3þ330s2þ55sþ1ð29Þ

s1=4 ¼663s5þ12597s4þ41990s3þ35530s2þ7315sþ209

209s5þ7315s4þ35530s3þ41990s2þ12597sþ663ð30Þ

s�1=4 ¼209s5þ7315s4þ35530s3þ41990s2þ12597sþ663

663s5þ12597s4þ41990s3þ35530s2þ7315sþ209ð31Þ

In order to check for the stability of these rationalapproximations given in Eqs. (28) and (29), pole–zero plotare drawn and these are shown in Figs. 11 and 12.

onse offfiffisp

.

lot offfiffisp

.


respectively. From Figs. 11 and 12, it can be concludedthat the rational approximations were stable. It is alsoevident that pole and zeros interlace on negative real axis.So these rational approximation can be synthesized usingRC or RL elements [14]. The passive and active networkssynthesized for the fractance device of order 1

2 werepresented in Figs. 13 and 14, respectively. Figs. 15–20compare the magnitude response, phase response anderror plots for fractance device of order 1

2 and � 12 obtained

using Oustaloup method and the CFE method. From thefigures it can be observed that the magnitude responseusing the CFE method is closer to the ideal one comparedto Oustaloup method. The phase response is constant forlarger range of frequencies with the CFE methodcompared to Oustaloup method. The relative percentageerror is almost zero for large range of frequencies with theCFE method. The PSPICE simulations of the realized circuit

Fig. 18. Magnitude r

Fig. 19. Phase res

in Fig. 14 are shown in Figs. 21 and 22, respectively. Thesimulations have been carried out at frequency, f=10 and20 Hz.

3. Fractance based circuits

The six possible inverted-L type circuits using frac-tance device as series or as shunt element were shown inFig. 23 [21,32,59]. It has been observed that the transferfunction H(s) can be expressed in two ways as,

�

esp

pon

For R–F, L–F and C–F circuits, HðsÞ ¼ Z=ðZþsbÞ.
� For F–R, F–L and F–C circuits HðsÞ ¼ sb=ðZþsbÞ where
b¼a for R2F & F2R

aþ1 for L2F & F2L

a�1 for C2F & F2C

8><>:

onse of 1ffiffisp .

se of 1ffiffisp .

Fig. 20. Error plot of 1ffiffisp .

Fig. 21. Simulation results of fractance device of order � 12 at f=10 Hz.


and

Z¼

k0

Rfor R2F & F2R

k0

Lfor L2F & F2L

Ck0 for C2F & F2C

8>>>>><>>>>>:

3.1. Time-domain response of fractance based circuits

Time-domain response calculations for unit step andimpulse excitations were carried out for the circuitsshown in Fig. 23. The response for different excitationsis expressed are presented in Table 3. Table 3 also presentthe simplified expressions for impulse and step responsesby choosing a¼ 1

2 , R¼ 1O, L¼ 1 H, C ¼ 1 F,and k0=1.

The corresponding impulse and step responses areshown in Figs. 24 and 25, respectively. The impulse andstep responses of R-F and F-R circuits for different valuesof a, were shown in Figs. 26–29.

3.2. Frequency response of fractance based circuits

Substituting s¼ jo the magnitude and phase responseexpressions for R–F, L–F, C–F circuits are given by

jHðoÞj ¼ Zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2bþZ2þ2obcos

bp2

r ð32Þ

+HðoÞ ¼ �arctanobsin

bp2

Zþobcosbp2

ð33Þ

Fig. 22. Simulation results of fractance device of order � 12 at f=20 Hz.

Fig. 23. Inverted-L type fractance based circuits. (a) R–F circuit, (b) L–F circuit, (c) C–F circuit, (d) F–R circuit, (e) F–L circuit and (f) F–C circuit.


Similarly for F–R, F–L and F–C circuits,

jHðoÞj ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio�2bþZ2þ2Zo�bcos

bp2

r ð34Þ

+HðoÞ ¼�arctano�bZsin

bp2

1þZo�bcosbp2

ð35Þ

Frequency response of all the inverted-L type circuits isshown in Figs. 30–35. From the frequency response of R–Fcircuit it has been observed that it behaves as conventionalRC low pass circuit for a¼ 1. For a41, the slope hasstarted decreasing by �20adb=dec. Surprisingly, for thevalues of aZ1:5, response has exhibited peak, indicatingthat the circuit is behaving as a second order circuit. So,higher order circuit behavior can be obtained from acircuit with lesser fractional order. Similarly, the F–R

circuit acts as a single stage high pass circuit for a¼ 1. Fora41, the circuit behaves like a second order circuitrevealing the RLC nature of fractance device [21,26,34].

From Fig. 31, the frequency response of L–F circuit, itbehaves as a conventional LC circuit and provides zerodegrees phase shift at a¼ 1. For a41, the circuitexhibited high pass behavior and also exhibits low passaction for ao1. As per the F–L circuit is concerned, forvalues of 1oao2, because of resonant nature of thefractance device, circuit has behaved as second ordersystem [59]. For ao1 it behaved as low pass filter. Fora¼ 1 it provides zero phase shift. For smaller values of athe C–F circuit behaves as a single stage RC high passcircuit. For higher values of a it behaved as a single stageRC low pass filter. For an F–C circuit, at a¼ 1, the circuitbehaves as an attenuator. It behaves as RC high pass filterfor a41 and has shown low pass action for ao1.Recently Biswas et al. [45] have made an attempt towardsthe commercial realization of fractance device.

Ta

ble

3Im

pu

lse

an

dst

ep

resp

on

ses

of

inv

ert

ed

-Lfr

act

an

ceb

ase

dci

rcu

its.

Cir

cuit

Imp

uls

ere

spo

nse

Imp

uls

ere

spo

nse

fora¼

1 2S

tep

resp

on

seS

tep

resp

on

sefo

ra¼

1 2

R–

FZt

a�1Ea,að�Zt

aþ

1Þ

1 ffiffiffiffiffi ptp�

eter

fcðffiffi tpÞ

1�

Eaþ

1,1ð�Zt

aþ

1Þ

1�

eter

fcðffiffi tpÞ

L–F

ZtaEaþ

1,aþ

1ð�Zt

aÞ

1 3��

eter

fcðffiffi tpÞ

þ2

e�t=

2co

s

ffiffiffi 3p

2t�p 3

!

þ4ffiffi tp

ffiffiffiffi pp

Z t 0e�

tð1�

v2Þ=

2co

s

ffiffiffi 3p

tð1�

v2Þ

2

! d

v

2 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 5

1�

Ea,

1ð�Zt

aÞ

1�

1 3��et

erfcðffiffi tpÞ

þ2

e�t=

2co

s

ffiffiffi 3p

2t

!

þ4ffiffi tp

ffiffiffiffi pp

Z t 0eð�

tð1�

v2Þ=

2co

s

ffiffiffi 3p

tð1�

v2Þ

2þp 3

! d

v

2 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 5C

–F

Zta�

2Ea�

1,a�

1ð�Zt

a�1Þ

1�

1 ffiffiffiffiffi ptpþ

eter

fcðffiffi tpÞ

1�

Ea�

1,1ð�Zt

a�1Þ

eter

fcðffiffi tpÞ

F–R

1�Zt

aEa,að�Zt

aÞ

1�

1 ffiffiffiffiffi ptpþ

eter

fcðffiffi tpÞ

taEa,

1ð�Zt

aÞ

eter

fcðffiffi tpÞ

F–L

1�Zt

aEaþ

1,aþ

1ð�Zt

aþ

1Þ

1�

1 3��

eter

fcðffiffi tpÞ

þ2

e�t=

2co

s

ffiffiffi 3p

2t�p 3

!

þ4ffiffi tp

ffiffiffiffi pp

Z t 0eð�

tð1�

v2Þ=

2co

s

ffiffiffi 3p

tð1�

v2Þ

2

! d

v

2 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 5

taþ

1Eaþ

1,1ð�Zt

aþ

1Þ

1 3��et

erfcðffiffi tpÞ

þ2

e�t=

2co

s

ffiffiffi 3p

2t

!

þ4ffiffi tp

ffiffiffiffi pp

Z t 0eð�

tð1�

v2Þ=

2co

s

ffiffiffi 3p

tð1�

v2Þ

2þp 3

! d

v

2 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 5F–

C1�Zt

a�2Ea�

1,a�

1ð�Zt

a�1Þ

1 ffiffiffiffiffi ptp�

eter

fcðffiffi tpÞ

ta�

1Ea�

1,1ð�Zt

a�1Þ

1�

eter

fcðffiffi tpÞ


3.3. Calculation of time domain parameters

All the six fractance based circuits can be generalizedby the unity feedback system shown in Fig. 36. where R(s)is the input and C(s) is the output. The expression forHðsÞ ¼ CðsÞ=RðsÞ is same as the expression obtained for R–F,C–F and L–F circuits if a is positive and is same as theexpression obtained for F–R, F–L and F–C circuits if a isnegative. The time-domain response parameters such aspercent overshoot, peak time at which the overshootoccurs, rise time, settling time, etc. were given by Eqs.(36)–(43) [31,48].

�
The frequency domain response will have a resonantpeak Mr at the frequency or
Mr ¼1

sinðap=2Þð36Þ

or ¼ocjcosðap=2Þj1=a ð37Þ

where oc ¼ k1=a is the gain crossover frequency.
� The percent overshoot is given by the expression
Mp ¼ 0:8ða�1Þða�0:75Þ 1oao2 ð38Þ

Peak time TP at which the overshoot occurs is given by
� the approximate formula
Tp ¼1:106ða�0:255Þ2

ða�0:921Þoc1oao2 ð39Þ

Risetime Tr is given by
�
Tr ¼0:131ðaþ1:157Þ2

ða�0:724Þoc1oao2 ð40Þ

Settling time Ts is the time required for the response to
� settle within a small fraction of its steady state valueand to stay there. The approximate expressions for 2%and 5% criteria were given by the expressions
TSð2%Þ ¼4

cosðp�p=aÞoc1:39oao2 ð41Þ

TSð5%Þ ¼3

cosðp�p=aÞoc1:44oao2 ð42Þ

Time-constant TC, the time the response to rise to 63%
� of the final value is given by
TC ¼0:2ða�1Þ2þ1

oc1oao2 ð43Þ

Variations of Tr, Ts, TP, MP, TC are shown in Figs. 37–40. Itcan be observed that as the value of the fractional orderincreases the time-constant TC, and settling time, TS areincreased, whereas rise time Tr decreases.

4. IIR type digital differentiators

It is well-known that a digital differentiator can beused for the purpose of discretization [64–72]. Digitaldifferentiators are used to find the time-derivative of theincoming signal. A differentiator is defined as

GðjoÞ ¼ jo ð44Þ

Fig. 24. Impulse response of fractance based circuits.

Fig. 25. Step response of fractance based circuits.


where j¼ffiffiffiffiffiffiffi�1p

. In 1992, Al-Alaoui has proposed aprocedure for the design of IIR type digital differentiatorswhich are obtained by the inversion and magnitudestabilization of digital integrators [64].

Some of the commonly available digital integratorspresented in Table 4 are compared in Fig. 41.

The following approach is proposed by Al-Alaoui forthe design of IIR type digital differentiators.

1.
Design an integrator that has the same range andaccuracy as the desired differentiator.
2.
Invert the transfer function of the integrator proposedin step 1 and stabilize it.
3.
Compensate the change in magnitude.
The IIR type digital differentiators obtained by using theabove-mentioned procedure were summarized in Table 4.

The magnitude and phase responses of the digitaldifferentiators were compared in Figs. 42 and 43,respectively. It can be observed from Figs. 42 and 43 that,

�
Al-Alaoui first order differentiator approximates theideal differentiator till 0.78 of the full band. � The inverse Simpson differentiator has the poorest
high frequency response and has good low frequencyresponse up to 0.4 of the full band.

Fig. 26. Step response of R–F circuit for different values of a.

Fig. 27. Step response of F–R circuit for different values of a.


�
Differentiator from the tick integrator is linear till 0.5of the full band. � The third order digital differentiator can be used as a
wide-band digital differentiator.
� Differentiator from backward integrator, Al-Alaoui
second order differentiator has exhibited good lowfrequency response.
� Differentiator obtained from the inversion of the trape-
zoidal integrator has phase response closer to ideal one.
� The lower order of these digital differentiators makes
them suitable in real time applications like radars,sonar’s, bio-medical engineering, speech processing,global positioning system, etc.

Keeping in view of the resulting order of thetransfer function of fractional order digital differentia-tors and integrators, Al-Alaoui first order digitaldifferentiator, Bilinear transform were selected for thediscretization.

5. Fractional order digital differentiators and integrators

An ideal fractional order digital differentiator isdefined as [74–78]

HdðjoÞ ¼ ðjoÞa ð45Þ

Fig. 28. Impulse response of R–F circuit for different values of a.

Fig. 29. Impulse response of F–R circuit for different values of a.


where a is fractional order. Similarly an ideal fractionalorder integrator is defined as

HIðjoÞ ¼1

ðjoÞað46Þ

In general, there are two discretization methods,namely direct discretization and indirect discretization[50,77–79,82–84]. Chen and others proposed an IIR typefractional order digital differentiator based on directdiscretization method. The simplest and straight forwardmethod is the direct discretization which involve thedirect substitution of the s to z transform in sa. Thediscretization procedure involves producing a generatingpolynomial, oðz�1Þ. The rational approximation is gener-

ated using power series expansion (PSE), or continuedfraction expansion (CFE), or Muir recursion [77,78]. Thisrational approximation in digital domain should be astable minimum phase function. In indirect discretizationmethods, two steps are involved, i.e., frequency domainfitting the ideal magnitude response of sa in frequencydomain and then discretizing that approximation by usingappropriate s to z transform [90,91].

5.1. Indirect discretization

In indirect discretization, initially, rational approxima-tion for sa in s-domain is obtained by limiting its order

Fig. 30. Frequency response of R–F circuit.

Fig. 31. Frequency response of L–F circuit.

Fig. 32. Frequency response of C–F circuit.

Fig. 33. Frequency response of F–R circuit.

Fig. 34. Frequency response of F–L circuit.

Fig. 35. Frequency response of F–C circuit.



and then is to be digitized. The discretized transferfunction in z-domain is obtained by employing theexpressions for Bilinear and Al-Alaoui differentiators in

Fig. 36. Blockdiagram of the fractance based circuit.

Fig. 37. Variation of Tr and

Fig. 38. Variation of settling

Eq. (26) and is given by H(z) as

HðzÞ ¼A0z5þA1z4þA2z3þA3z2þA4zþA5

B0z5þB1z4þB2z3þB3z2þB4zþB5ð47Þ

where the digital filter co-efficients A0, A1 ?A5 and B0, B1

?B5 are given by the equations shown in Table 5.For, a¼ 1

2 , 14 and using the Bilinear and Al-Alaoui

transforms following transfer functions are obtained andare given by

HdT ðzÞ ¼8119z5�11721z4�1002z3þ6302z2�1469z�197

5741z5�2547z4�6126z3þ2474z2þ1073z�263ð48Þ

TP with respect to a.

time with respect to a.

HdAðzÞ ¼24 999 391z5�63 334 707z4þ55 367 574z3�18 716 230z2þ1 617 211z�99 529

23 384 789z5�45 881 457z4þ27 482 610z3�4 184 962z2�492 343zþ51 811ð49Þ

HIT ðzÞ ¼5741z5�2547z4�6126z3þ2474z2þ1073z�263

8119z5�11 721z4�1002z3þ6302z2�1469z�197ð50Þ

HIAðzÞ ¼23 384 789z5�45 881 457z4þ27 482 610z3�4 184 962z2�492 343zþ51 811

24 999 391z5�63 334 707z4þ55 367 574z3�18 716 230z2þ1 617 211z�99 529ð51Þ

HdT1ðzÞ ¼715 647z5�859 601z4�309 466z3þ551 374z2�63 381z�27 885

601 785z5�421 943z4�546 422z3þ356 002z2þ62 253z�30 459ð52Þ

HdA1ðzÞ ¼2 360 325 231z5�5 644 455 299z4þ4 553 637 686z3�1 341 809 894z2þ68 561 931zþ10 588 857

2 282 831 529z5�4 806 900 149z4þ3 217 087 162z3�651 189 386z2�27 613 587zþ7 509 615ð53Þ

HIT1ðzÞ ¼60 1785z5�42 1943z4�546 422z3þ356 002z2þ62 253z�30 459

715 647z5�859 601z4�309 466z3þ551 374z2�63 381z�27 885ð54Þ

HIA1ðzÞ ¼2 282 831 529z5�4 806 900 149z4þ3 217 087 162z3�651 189 386z2�27 613 587zþ7 509 615

2 360 325 231z5�5 644 455 299z4þ4 553 637 686z3�1 341 809 894z2þ68 561 931zþ10 588 857ð55Þ


where HdT(z), HIT(z) are the transfer functions of digitaldifferentiator and integrator obtained using Bilineartransform, and HdA(z), HIA(z) are the transfer functions ofdigital differentiator and integrator when Al-Alaouitransform is used with a¼ 1

2. Similarly, HdT 1(z), HIT 1(z)are the transfer functions of digital differentiator andintegrator obtained using Bilinear transform, and HdA 1(z),HIA 1(z) are the transfer functions of digital differentiatorand integrator when Al-Alaoui transform is used witha¼ 1

4.The magnitude and phase responses, pole–zero dia-

grams of integrators and differentiators evaluated withT=1 s are shown in Figs. 44–51. Figs. 44–47 depict themagnitude and phase responses of differentiators andintegrators of order 1

4 using Bilinear and Al-Alaouitransforms. Figs. 48–51 depict the magnitude and phase

Fig. 39. Variation of peak over

responses of differentiators and integrators of order 12 for

the same transforms. It can be inferred that Al-Alaouitransform improves the high frequency magnituderesponse compared to Bilinear transform, whereasBilinear transform provides better phase responsecompared to Al-Alaoui transform. Figs. 52 and 53 arethe pole–zero diagrams of differentiators and integratorsof order 1

2 and 14. One can observe from figures that the

poles and zeros are lying inside of the unit circle and areinterlacing on the segment of the real axis. So theproposed differentiators and integrators are stable andare of minimum phase. The proposed approach thusseems to be simple and accurate compared to directdiscretization. The percent relative error plots wereshown in Figs. 54 and 55. From the error plots it can beobserved that the differentiators and integrators obtained

shoot with respect to a.

Fig. 40. Variation of time constant with respect to a.

Table 4Transfer functions of IIR Type digital differentiators and integrators.

Digital integrator Digital differentiator

H1ðzÞ ¼zT

z�1G1ðzÞ ¼

z�1

Tz

H2ðzÞ ¼Tðzþ1Þ

2ðz�1ÞG2ðzÞ ¼

2ðz�1Þ

Tðzþ1Þ

HALðzÞ ¼Tðzþ7Þ

8ðz�1ÞGALðzÞ ¼

8ðz�1Þ

7T zþ1

7

� �

H3ðzÞ ¼Tðz2þ4zþ1Þ

3ðz2�1ÞG3ðzÞ ¼

0:8038ðz2�1Þ

Tðz2þ0:5358zþ0:0718Þ

H4ðzÞ ¼Tð0:3585z2þ1:2832zþ0:3584Þ

ðz2�1ÞG4ðzÞ ¼

0:852ðz2�1Þ

Tðz2þ0:611zþ0:0932Þ

HAL2ðzÞ ¼0:4Tðz2þ2:5zþ1Þ

ðz2�1ÞGAL2ðzÞ ¼

1:25ðz2�1Þ

Tðz2þzþ0:25Þ

H5ðzÞ ¼Tðzþ2:3658Þðz2þ1:1752zþ0:047Þ

2:7925z2ðz�1ÞG5ðzÞ ¼

1:1804z2ðz�1Þ

Tðz3þ0:1680z2�0:0607zþ0:0198Þ


by using Al-Alaoui transform has exhibited less errorcompared to that of Bilinear transform. The comparison ofthe magnitude and phase responses of digitaldifferentiators of order 0.5 obtained using Al-Alaouitransform with direct and indirect discretizationtechniques is shown in Figs. 56–59. It can be claimedthat theindirect discretization approaches are superior tothe direct discretization methods.

The Experimental setup for the real time implementationof the fractional order digital differentiators and integratorsof order 1

2 and 14 is as shown in Fig. 60. TMS320C6713 DSP

Processor and National Instruments Educational LaboratoryVirtual Instrumentation Suite (NIELVIS) have been used forthe purpose of implementation and measurements [73].A 2 V peak to peak sinusoidal signal is applied and thesampling frequency is chosen as 48 kHz. Figs. 61 and 62depict the time-domain response obtained when sinusoidalsignal is applied as in input. Figs. 63–68 are magnitude plots

of the differentiators and integrators for sampling frequency,fS=48 kHz. From Figs. 65 to 70 it can be observed that thedigital differentiators and integrators were approximatingthe theoretical one, upto 20 kHz with 48 kHz samplingfrequency, fS. A typical Bode diagram is shown in Fig. 69.There is an error of 10 db between theoretical and practicalresponses. However, this error can be reduced byintroducing gain.

6. Applications of fractional order digital differentiators

Fractional order differentiators and integrators aregaining importance in many fields. The application offractional calculus in the area of control systems, robotics,instrumentation is illustrated in [15,16,19,39,41,53,55,89].Debnath [11] has summarized the application of fractionalcalculus in various fields of science and engineering.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency

Mag

nitu

de

Magnitude Response

IdealG1 (z)

G2 (z)

G3 (z)

G4 (z)

GAL (z)

GAL2 (z)

G5 (z)

Fig. 42. Comparison of magnitude responses of IIR type digital differentiators.

Fig. 41. Comparison of magnitude responses of IIR type digital integrators.


In [40] Malti et al. have discussed about the application offractional order differentiation for system identification.

6.1. Detection of QRS signal using fractional order digital

differentiators

Electrocardiogram diagnosis require an accurate de-tection of QRS complex [93,99]. Various techniques used

for the detection of QRS complex using software issummarized in [97]. In [98] Ferdi et al. have usedfractional order differentiation for the detection anddelineation of R waves. Detection and delineation ofP and T waves using fractional order differentiators ispresent in [86]. Many of the QRS complex detectionalgorithms use digital differentiation followed by crosscorrelation techniques. The technique of correlating onesignal with another requires that the two signals be

0 0.2 0.4 0.6 0.8 1−100

−80

−60

−40

−20

0

20

40

60

80

100

Frequency

Pha

se, D

egre

es

Phase Response

IdealG1 (z)

G2 (z)

G3 (z)

G4 (z)

GAL (z)

GAL2 (z)

G5 (z)

Fig. 43. Comparison of phase responses of IIR type digital differentiators.

Table 5The equations for digital filter co-efficients.

Bilinear transform Al-Alaoui transform

A0 = 32P0 + 16P1T + 8P2T2 + 4P3T3 + 2P4T4 + P5T5 A0 = 32 768P0 + 28 672P1T + 25 088P2T2 + 21 952P3T3 + 19 208P4T4 + 16 807P5T5

A1 = � 160P0 � 48P1T � 8P2T2 + 4P3T3 + 6P4T4 + 5P5T5 A1 = � 163 840P0 � 110 592P1T � 68096P2T2� 34496P3T3

� 8232P4T4 + 12 005P5T5

A2 = � 320P0 �32P1T + 16P2T2 + 8P3T3� 4P4T4

� 10P5T5 A2 = 327 680P0 + 155 648P1 T + 54 272P2T2 +4480P3T3� 8624P4T4 + 3430P5T5

A3 = 320P0 + 32P1 T � 16P2T2� 8P3T3 + 4P4T4 + 10P5T5 A3 = � 327680P0 � 90 112P1T � 5120P2T2 + 6784P3T3

� 2128P4T4 + 490P5T5

A4 = 160P0 �48P1T + 8P2 T2 + 4P3T3� 6P4T4 + 5P5T5 A4 = 163 840P0 + 12 288P1T �5632P2T2 + 1216P3T3

� 216P4T4 + 35P5T5

A5 = � 32P0 �32P1T � 8P2T2 + 4P3T3� 2P4T4 + P5T5 A5 = � 32 768P0 + 4096P1T � 512P2T2 + 64P3T3

� 8P4T4 + P5T5

B0 = 32Q0 + 16Q1T + 8Q2T2 + 4Q3T3 + 2Q4T4 + Q5T5 B0 = 32 768Q0 + 28 672Q1 T + 25 088Q2T2 + 21 952Q3T3 + 19 208Q4T4 + 16 807Q5T5

B1 = � 160Q0 � 48Q1 T � 8Q2T2 + 4Q3T3 + 6Q4T4 + 5Q5T5 B1 = � 163 840Q0 � 110 592Q1T � 68 096Q2T2�34 496Q3T3

� 8232Q4T4 +

12 005Q5T5

B2 = 320Q0 + 32Q1 T � 16Q2T2� 8Q3T3 + 4Q4T4 + 10Q5T5 B2 = 327 680Q0 + 155 648Q1T + 54272Q2T2 +4480Q3T3

� 8624Q4T4 + 3430Q5T5

B3 = � 320Q0 + 32Q1T + 16Q2T2� 8Q3T3

� 4Q4T4 + 10Q5T5 B3 = � 327 680Q0 � 90 112Q1 T � 5120Q2T2 + 6784Q3T3�2128Q4T4 + 3430Q5 T5

B4 = 160Q0 �48Q1T + 8Q2T2 + 4Q3T3� 6Q4T4 + 5Q5 T5 B4 = 163 840Q0 + 12 288Q1T � 5632Q2T2 + 1216Q3T3

� 216Q4T4 + 35Q5T5

B5 = � 32Q0 + 16Q1T � 8Q2T2 + 4Q3T3� 2Q4T4 + Q5T5 B5 = � 32768Q0 + 4096Q1 T � 512Q2T2 + 646Q3T3

� 8Q4T4 + Q5T5

Fig. 44. Comparison of magnitude responses of fractional order digital integrators of order 14.


Fig. 47. Comparison of phase responses of fractional order digital differentiators of order 14.

Fig. 46. Comparison of magnitude responses of fractional order digital differentiators of order 14.

Fig. 45. Comparison of phase responses of fractional order digital integrators of order 14.


Fig. 49. Comparison of phase responses of fractional order digital differentiators of order 12.

Fig. 50. Comparison of magnitude responses of fractional order digital integrators of order 12.

Fig. 48. Comparison of magnitude responses of fractional order digital differentiators of order 12.


Fig. 51. Comparison of phase responses of fractional order digital integrators of order 12.

Fig. 52. Pole–zero diagram for fractional order digital differentiators and integrators of order 12.


aligned with one another. In this QRS detection techniquethe template of the signal that we are trying to matchstores a digitized form of the signal shape that we wish todetect. The alignment of the template with the incomingsignal to accomplish the task of correlation can be done intwo ways.

The first way of aligning the template and the incomingsignal is by using the fiducial points on each signal. Thesefiducial points have to be assigned to the signal by someexternal process. If the fiducial points on the template andthe signal are aligned, then the correlation can beperformed. Another implementation involves continuous

correlation between a segment of the incoming signal andthe template. The template can be thought of as a windowthat moves over the incoming signal one data point at atime. In this article correlation between template andsegment of the incoming signal have been used.

One of the objectives of this paper is analysis of ECGsignals to detect QRS sequences and occurrence of theSino-Atrial Rhythms. MIT-BIH has provided ‘‘n2000850,n2092910,n2321527’’ data of ECG signals (www.physionet.org). The block diagram used for the detection of QRS signalis shown in Fig. 70. The following is the procedure used forthe QRS detection.

www.physionet.org

www.physionet.org

www.physionet.org

www.physionet.org

www.physionet.org

Fig. 53. Pole–zero diagram for fractional order digital differentiators and integrators of order 14.

Fig. 54. Error plot for digital differentiators and integrators of order 12.


�
First step is to remove the DC offset. The instrumenta-tion used is a 1024-bit analog to digital converter andadds an offset of 1024 to it and we subtract it fromour signals. Wave detection block contains differen-tiator. � A manual template is to be selected, and all the other
sequences is to be correlated with this template.
� The correlation is done in time-domain. The correla-
tion is done by keeping the template steady and thedetected sequence five samples to the left and five

samples to the right. The sequence is then aligned tothe point of maximum correlation and is stored alongwith its new position marker at the same time thebeats are averaged to obtain an averaged beat tem-plate. Now using this averaged template the sameprocedure is to be repeated.
� From the saved and aligned QRS beats, we find the R–R
intervals, which give us the heart beat rates. Based onthese heart beat rates, we detect the sino-atrial rhythmin the subject as,

Fig. 55. Error plot for digital differentiators and integrators of order 14.

Fig. 56. Comparison of magnitude responses of digital differentiators using direct and indirect discretization for an order 12.


1. If heart rate is more than 100 beats per minute—

Sinus Tachycardia Rhythm.2. If heart rate is between 50 and 100 beats per

minute—Normal Sinus Rhythm.3. If heart rate is less than 50 beats per minute—Sinus

Bradycardia Rhythm.

The original signals n2000850, n2321529, n2092910

are shown in Figs. 71–73. Figs. 74, 77,78,81,82 representthe QRS signals detected using five point central

difference equation: dX (n) = x(n + 2) + 2x(n +1) �2x(n �1) � x(n � 2). Figs. 75,76, 79,80 represent the QRSsignals detected by using Al-Alaoui digital differentiator oforder 0.5.

6.2. Edge detection using fractional order digital

differentiators

Edge detection is an important task in image proces-sing which refers to the process of identifying andlocating sharp discontinuities in an image [106–108].

Fig. 57. Comparison of phase responses of digital differentiators using direct and indirect discretization for an order 12.

Fig. 58. Comparison of magnitude responses of digital integrators using direct and indirect discretization for an order 12.


There are many ways to perform edge detection such asgradient method and Laplacian method. The popularedge detection operators are Roberts, Sobel, Prewitt,Frei-Chen, and Laplacian, etc. Application of fractionaldifferentiation to detect edges of an image is discussed in[100]. But this article does not provide any visualinformation. Recently [105] Sparavigna et al. haveproved the efficiency of fractional order differentiationin increasing the visibility of feeble objects in anastronomical image. In a series of papers Pu et al.[101–104] have explained the use of fractional calculusin image processing. In this paper an attempt is madeto use fractional order digital differentiators for the

detection of edges. We have the transfer functionof the fractional order digital differentiator as given inEq. (47),

HðzÞ ¼A0z5þA1z4þA2z3þA3z2þA4zþA5

B0z5þB1z4þB2z3þB3z2þB4zþB5ð56Þ

The time-domain difference equation can be written as

y½n� ¼1

B0

A0x½n�þA1x½n�1�þA2x½n�2�

þA3x½n�3�þA4x½n�4�þA5x½n�5�

�B1y½n�1��B2y½n�2��B3y½n�3�

�B4y½n�4��B5y½n�5�

0BBBB@

1CCCCA ð57Þ

Fig. 60. Experimental setup.

Fig. 61. Time domain response of digital integrator of order 12 using

Al-Alaoui transform.


where A0, A1 y A5 and B0, B1 y B5 are given by theequations shown in Table 5. Consider an image f(x,y). Thegradient of the image can be written as

rf ðx,yÞ ¼df ðx,yÞ

dxuxþ

df ðx,yÞ

dyuy ¼ GxuxþGyuy ð58Þ

where ux, uy are the unit vectors in x and y directions. Theapproximated magnitude of the gradient is

G¼ jGxjþjGyj ð59Þ

Considering x[n]= f(x,y) and applying it to a digitaldifferentiator the outputs are calculated both in x and y

directions individually, and the gradient is calculatedusing Eq. (59). The output map is calculated as follows.The magnitude of the gradient G(x, y,c) is evaluated onthe function f(x,y) given by the image map b(x,y,c) foreach color tone c. For each color, the maximum valueGMax(c) on the image map is to be calculated. The outputmap is defined as follows:

bGðx,y,cÞ ¼ 255Gðx,y,cÞ

GMaxðcÞ

� �k

ð60Þ

where k is a parameter suitable to adjust the imagevisibility. The commonly used factor to compare theperformance of the edge detectors is, root-mean squareerror, ERMS which is given by

ERMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

MN

XM�1

x ¼ 0

XN�1

y ¼ 0

½foriginalðx,yÞ�fedgeðx,yÞ�2

vuut ð61Þ

where foriginal (x,y) is the original image of size M X N

and fedge (x,y) is the edge detected image. The resultsobtained by following the above procedure fordifferent values of k are shown in Figs. 83, 84 and 85,respectively. From the figures it is evident thatfor k = 0.5 the brightness of the image is enhanced[105]. The root mean square errors obtained by using

Fig. 59. Comparison of phase responses of digital integrators

various edge detection operators, integer andfractional order differentiators at k= 1 is present inTable 6.

using direct and indirect discretization for an order 12.

Fig. 63. Real time implementation of digital differ

Fig. 64. Real time implementation of digital inte

Fig. 62. Time domain response of digital integrator of order 14 using

Al-Alaoui transform.


7. Results and conclusions

In this paper, numerical calculations have beenperformed for the response characteristics of a typicalfractance device. The parameters used in the calculationare, R¼ 1 KO, C=1 nF and f=10 kHz. The response tocomplicated input functions for a fractance device canbe computed easily by employing fractional calculus.With fractional calculus approach the equations aresimple and are easily amenable for manipulation. A newmethod of realization of fractance device of order 1

2 usingcontinued fraction expansion is presented. From theresults, it can be observed that the magnitude and phaseresponses have shown considerable improvement overOustaloup method. The percent relative error is almostzero for broad range of frequencies using proposed

entiator of order 12 using Bilinear transform.

grator of order 12 using Bilinear transform.

Fig. 65. Real time implementation of digital differentiator of order 14 using Al-Alaoui transform.

Fig. 66. Real time implementation of digital integrator of order 14 using Al-Alaoui transform.

Fig. 67. Real time implementation of digital differentiator of order 14 using Bilinear transform.


Fig. 68. Real time implementation of digital integrator of order 14 using Bilinear transform.

Fig. 69. Typical Bode-diagram in NIELVIS.

Fig. 70. Block diagram of the program flow.


method and is almost comparable with that of Oustaloupmethod.

The time and frequency domain analysis of the circuitsinvolving fractance device either as series or as shuntelement is presented. The expressions for time and

frequency domain obtained for these circuits are simplewhen compared to previously existing methods. Theclosed form time domain expressions were derived byconsidering a fractance device of order 1

2. The response ofF–L and L–F circuits is oscillatory initially, for impulse andstep inputs for some time. From the frequency responsecurves it is observed that for a¼ 1:5, R–F and F–R circuitsexhibit the second order response characteristics. Thetime-domain response of R–F, C–F circuits is similar whilethe behavior of F–R, F–C is also similar but opposite tothat of R–F, C–F. That is responses of R–F and C–F islowpass and that of F–R and F–C is high pass. It is evidentfrom frequency response curves that by controlling thefractional order, a single fractance based circuit can beused in a variety of control applications. The expressionsfor peak over shoot, rise-time, peak time, etc. werecalculated and their variations with respect to fractionalorder a has been plotted.

Design of fractional order digital differentiators andintegrators using indirect discretization technique hasbeen presented. The rational approximation for thefractional order operator is calculated using continuedfraction expansion and is digitized using s to z transforms.The approximated transfer functions of digital differen-tiators and integrators of order 1

2 , 14 is obtained. The

magnitude response obtained by using Al-Alaoui trans-form is more closer to ideal one compared to Bilineartransform. But the phase response is better when Bilineartransform is used. The magnitude response of these digitaldifferentiators and integrators is almost similar to theideal one in the full Nyquist range when Al-Alaouitransform is used. But, the phase response is linear whichis not the desired feature. The magnitude response of thedesigned differentiators and integrators is closer to theideal one throughout the 90% of full band and phaseresponse is same as that of the ideal ones when Bilineartransform is used. As a whole, Al-Alaoui transform baseddiscretization is preferred when magnitude response isdesired. Bilinear transform based discretization is pre-ferred when both magnitude and phase responses are

Fig. 71. Original n2000850 signal and DC removed signal.




Fig. 74. Detected ECG signals with averaged template selection for n2000850—five point difference method.

Fig. 75. Detected ECG signals with manual template selection for 2092910—fractional differentiation.

Fig. 76. Detected ECG signals with averaged template selection for 2092910—fractional differentiation.


Fig. 77. Detected ECG signals with manual template selection for 2092910—five point differentiation.

Fig. 78. Detected ECG signals with averaged template selection for 2092910—five point differentiation.

Fig. 79. Detected ECG signals with manual template selection for n2321529—fractional differentiation.


Fig. 80. Detected ECG signals with averaged template selection for n2321529—fractional differentiation.

Fig. 81. Detected ECG signals with manual template selection for n2321529—five point differentiation.

Fig. 82. Detected ECG signals with averaged template selection for n2321529—Five point differentiation.


Fig. 83. Edge detection using fractional order digital differentiators of order 12.

Fig. 84. Edge detection using fractional order digital differentiators of order 14.


desired. From the pole–zero plots it can be easilyconcluded that the differentiators and integrators arestable and minimum phase. The proposed approach thusseems to be simple and accurate.

The digital differentiators and integrators of orders 12 , 1

4

obtained using indirect discretization technique has beenimplemented in real time using TMS320C6713 andNIELVIS. It has been observed that the digital differentia-tors and integrators were approximating the proposed

theoretical one, upto 4 kHz with 8 kHz sampling fre-quency, and upto 20 kHz with 48 kHz sampling frequency,fS and so on. The errors can be simply reduced by choosingproper gain.

Later, the designed digital differentiators of order 12 is

used in the detection of QRS complex and heart ratebeat estimation of the ECG signals, n2000850,n2092910,

n2321529 provided by the physionet.org. The wellknown template matching technique has been followed,

Table 6RMSE comparison.

Operator RMSE

Backward digital differentiator 0.0024167

Bilinear digital differentiator 0.0025429

Al-Alaoui first order digital differentiator 0.0024364

Simpson digital differentiator 0.0025993

Tick digital differentiator 0.0026045

Third order or Ngo digital differentiator 0.0021188

Gradient operator 0.0025219

Sobel operator 0.0027114

Bilinear 1/4 digital differentiator 0.0017098

Bilinear 1/2 digital differentiator 0.0017098

Al-Alaoui 1/2 digital differentiator 0.0017481

Al-Alaoui 1/4 digital differentiator 0.0017504

Fig. 85. Edge detection using digital differentiators at k=1.


by replacing the conventional five point digitaldifferentiator with the fractional order digital differentia-tor of order 1

2. The same process is also performedwith five point digital differentiator. The results obtainedwere comparable with the results obtained withthe conventional digital differentiator. The digital differ-entiators of order 1

2 and 14 has been used for the

detection of edges of an image, at different values ofbrightness constant. It has been observed from theresults that fractional order differentiators areproducing less error compared to conventional edgedetection operators and integer order digital differentia-tors.

Acknowledgements

The author wishes to express his gratitude to Prof. YangQuan Chen from Utah State University, USA, Prof. M.A.Al-Alaoui from American university of Beirut, Lebanon,Dr. Virginia Kiryakova, Editor, FCAA Journal and Dr. ManuelD. Ortigueira for their encouragement and suggestions. Theauthor would also like to thank the anonymous reviewersfor their useful comments. The author express the deep

sense of gratitude to the Board of Management, GITAMUniversity, Visakhapatnam for their encouragement.

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Studies on fractional order differentiators and integrators: A survey