1

Multifunction Fractional Inverse Filter Based onOTRA

Nariman A. Khalil1, Lobna A. Said2, Ahmed G. Radwan3,4, and Ahmed M. Soliman51FacultycofcEngineering, Nahda University (NUB), Egypt

2 NanoelectronicscIntegratedcSystems Centerc(NISC),cNilecUniversity,cGiza,cEgypt3EngineeringcMathematicscandcPhysicscDept,cFacultycof Engineering,cCairocUniversity,cGiza,c12613,cEgypt

4 SchoolcofcEngineeringcandcAppliedcSciences,cNilecUniversity, 12588, Giza,cEgypt5ElectronicscandcComm.cEng.cDept.,cCairo University, Egypt

Abstract—This paper proposes a generalized topology of afractional-order inverse filter (FOIF) using operational trans-resistance amplifiers (OTRA) block. Seven different configura-tions are extracted from the introduced topology employinggeneralized admittances. The generalized admittances increasethe flexibility to provide different types of FOIFs such as inversefractional high pass filter(IFHPF), inverse fractional low passfilter (IFLPF), inverse fractional bandpass filter (IFBPF), andinverse fractional notch filter (IFNF). Numerical and PSPICEsimulation results are presented for selected cases to approvethe theoretical findings. The fractional-order parameters increasedesign flexibility and controllability, which are validated experi-mentally.

Keywords—Inverse filter, Fractional-order circuits, OTRA, FOC.

I. INTRODUCTIONThe Operational Transresistance Amplifier (OTRA) is a very

important analog block in many electrical and electronicsapplications [1]. The OTRA block has three-terminals aspresented in Fig.1(a) as a symbol and its realization usingAD844 in Fig.1(b). The characteristic matrix of OTRA iswritten as follows:

[V+V−VO

]=

[0 0 00 0 0

Rm −Rm 0

][I+I−IO

]. (1)

Both input terminals are virtually grounded, that decreases theeffect of parasitics at both terminals. The input and outputterminals have low impedance, which is very useful to designcascaded circuits based on OTRA block. OTRA is employedin many analog applications such as filters [1] and oscillators[2].

Fractional Calculus (FC) is the generalization of integer-order integration and differentiation to the fractional domain.The fractional-order (FO) models improve system flexibilityand controllability through adding extra parameters whichare the fractional-order [3]. Due to the advantages of FC,several applications have been designed based on FC suchas inverse filter [4], oscillators [2], [5–7], and filters [8–10].The implementation of FO filters is achieved using differenttechniques such as using the integer-order approximation of theFO transfer function or using the fractional-order capacitors

(a) (b)

Fig. 1. OTRA (a) symbol diagram, and (b) realization using AD844

(FOC) instead of the conventional ones. In this paper, thefractional-order inverse filter will be designed using FOCproposed in [11].

Several applications employ the inverse filter to enhance thebehavior deterioration in the transmitted and received signals[12]. The inverse filter is a very useful block to reconstruct thesignal, that has the reciprocal frequency response of the systemthat causes distortion. Inverse filters are applied in manyapplication for example acoustic system [13], digital filters[14], and proportional integral derivatives (PID) [15]. Thecurrent mode inverse filter employing four-terminal floatingnullor (FTFN) was presented in [16] using the analysis ofnullor and dual transformation (RC: CR). Many active blockswere employed to realize the inverse filter like current feed-back operational amplifier (CFOA), Differential difference cur-rent conveyor (DDCC), Second-generation current conveyor(CCII), and current differencing buffered amplifier (CDBA)[17–19]. In [20], two configurations for multifunction FOIFwere proposed based on operational amplifier.

The main purpose of this paper is to present a generalizedFOIF topology based on OTRA block. It was proposed in [1]as a generalized universal filter. Seven possible general con-figurations are extracted from the introduced prototype. Eachconfiguration can be used to realize FOIFs such as IFHPF,IFLPF, IFNF, and IFBPF filters by selecting the appropriateadmittance. All possible realizations for selected configurationsare investigated. The numerical results of FOIF are presentedat different fractional-order and implemented on PSPICE. Thesimulations results are verified experimentally to approve thetheoretical findings.

This paper is organized as follows; the effect of the

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162

nu978-1-7281-3173-3/19/$31.00 ©2019 IEEE

2

0

40

80

120160

Mag

nitu

de (d

B)

10-2 100 102 1040

45

90

135

180

Phas

e (d

eg)

=0.2 =0.4 =0.6 =0.8 =1

f, Hz

Fig. 2. Numerical simulation of IFLPF at different α

020

40

6080

Mag

nitu

de (d

B)

10-2 100 102 104-90

-45

0

45

90

Phas

e (d

eg)

=0.2 =0.4 =0.6 =0.8 =1

f, Hz

Fig. 3. Numerical simulation of IFBPF for different values of α

fractional-order α on the different types of FOIF are presentedin section II. Different configurations from the proposed gen-eral topology are investigated where all possible cases of FOIFfor selected configuration are given in section III. Section IVpresents the simulation and experimental results for selectedcases. Finally, the conclusion is drawn in section V.

II. INVERSE FILTERS WITH ORDER 2αThe general FOIF transfer function (TF) using two equal

orders of FOCs can be represented as follows:

T (s) =N(s)

D(s)=

s2α + asα + b

ds2α + hsα + k, (2)

where a, b, d, h, and k are constants. Various FOIFs responsescan be obtained based on D(s). The IFLPF transfer functionis defined as follows:

T (s) =s2α + asα + b

k. (3)

The effect of the fractional parameters α on the IFLPFresponse is presented in Fig.2. At d = k = 0, the IFBPFcan be realized where the TF is defined as follows:

T (s) =s2α + asα + b

hsα. (4)

The magnitude and phase response for different FO α arepresented in Fig. 3. The TF of IFHPF can be obtained as

-200

20406080

Mag

nitu

de (d

B)

10-2 100 102 104-180

-135

-90

-450

Phas

e (d

eg)

=0.1 =0.3 =0.5 =0.7 =0.9

f, Hz

Fig. 4. Numerical simulation of IFHPF at different values of α

-20

-10

0

10

Mag

nitu

de (d

B)

10-2 100 102 104-90

-45

0

45

90

Phas

e (d

eg)

=0.4 =0.8 =1.2 =1.4

f, Hz

Fig. 5. Numerical simulation of IFNPF at different values of α

- Rm+ +

-Rm

Y2

Y7Y6

Y4

Y8

Y5

Y3

VoVin

Y1

Fig. 6. The FOIF prototype topology

follows:T (s) =

s2α + asα + b

ds2α, (5)

where h = k = 0 in Eqn.(2). Figure 4 presents the magnitudeand phase response for different values of α. At h = 0, theTF of IFNF can be obtained as follows:

T (s) =s2α + asα + b

ds2α + k. (6)

Figure 5 presents the magnitude and phase response for variousvalues of α. The FO parameters increase the flexibility and addextra degrees of freedom on the FOIF.

III. THE INVERSE FILTER TOPOLOGYThe generalized topology is depicted in Fig. 6 using two

OTRA blocks with eight generalized admittances. The general

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TABLE I. ALL POSSIBLE GENERALIZED FOIFS TOPOLOGIES

Configuration No. TF Condition1. Y3Y7+Y2Y5Y7Y8 Y1= Y6= Y4= 02. Y3Y7+Y2Y5−Y6Y7Y4Y5 Y1= Y8= 03. Y3Y7+Y2Y5−Y1Y5Y7Y8 Y4= Y6= 04. Y2Y5Y7Y8 Y1= Y6= Y3= Y4= 05. Y2Y5−Y1Y5−Y6Y7Y7Y8 Y3= Y4= 06. Y2Y5−Y1Y5−Y6Y7Y4Y5 Y3= Y8= 07. Y3Y7+Y2Y5Y7Y8+Y 4Y5 Y1= Y6= 0

transfer function of the circuit is given as follows [1]:

T (s) =Y3Y7+ Y2Y5− Y1Y5− Y6Y7

Y7Y8+ Y4Y5. (7)

The generalization of the eight admittances increases theflexibility to extract seven different configurations, as sum-marized in Table I. Each configuration in Table I can beemployed to realize the IFLPF, IFBPF, and IFHPF based ondifferent admittance combinations. Also, the configuration 7provides various realizations for IFNF. Configuration 1 usesfive different admittance which increases the flexibility todesign different FOIF as summarized in Table II. There arefour, three, and six different realizations for the IFLPF, IFBPF,and IFHPF, respectively. The same possible realizations canbe achieved in configuration 2 with the specific conditionfor each case, as presented in Table III. Moreover, differentrealizations can be achieved by configurations 5 and 6 havingnegative gain. In addition, configuration 7 can be used to real-ized IFNF where one of possible admittance combinations isY3= sαC3+G3, Y7= sαC7, Y8= sαC8, Y4= G4, Y5= G5,and Y2 = G2. The low number of used admittances are inconfiguration 4 which provides one, two, and one realizationfor IFLPF, IFBPF, and IFHPF as summarized in Table IV,respectively.

IV. SIMULATIONS AND EXPERIMENTAL RESULTSSelected cases from different configurations are simulated

using PSPICE with the macrocmodel of AD844. The circuitis biased with ±15v DC power supply, and the input voltageis taken as 1V AC. The RC constant phase element circuit isemployed to simulate the FOC [11]. The IFLPF of order αconfiguration 1 (case 1 and case 2) is selected to be discussed,and the same procedure can be applied to the other presentedcases. The simulation is implemented using four AD844 andthe parameters are chosen to be (R3 = R8 = 2.4kΩ, R5 =1.6kΩ, R7= 6.5kΩ, and C2= C5= 100 ∗ 10−9) for α = 0.8at f = 5kHz. The simulation result is presented in Fig.7.Figure 8(a) shows experimental set-up for case 1 where thesimulations and experimental results are presented in Fig.8(b).

V. CONCLUSIONA generalized FOIF topology using OTRA block was intro-

duced given seven different configurations involves general ad-mittances. Each configuration was employed to realize IFLPF,

100 103 106Frequency (Hz)

-100

10203040506070

Mag

nitu

de (D

B) Case 1 Case 2

Fig. 7. PSPICE simulation for IFLPF at f = 5kHz at α = 0.8

(a)

100 101 102 103 104Frequency (Hz)

01020304050607080

Mag

nitu

de (D

B)

Simulation Experimental results

(b)

Fig. 8. (a) Circuit connection (b) experimental and PSPICE simulation forIFLPF at f = 5kHz at α = 0.8

IFHPF, IFNPF, and IFBPF using different admittances com-binations. The numerical analysis demonstrated the additionaldegrees of flexibility yielded by the fractional-order parame-ters. Finally, PSPICE simulations results were demonstratedto approve the reliability of the presented inverse filters andconfirmed experimentally.

REFERENCES

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TABLE II. ALL POSSIBLE INVERSE FILTERS FROM CONFIGURATION 1

No. Y2 Y7 Y5 Y3 Y8 Type1. sαC2 G7 G5+ sαC5 G3 G8

IFLPF2. G2+ sαC2 G7 sαC5 G3 G8

3. G2+ sαC2 G7 G5+ sαC5 G3 G84. sαC2 G7 sαC5 sαC3+G3 G85. sαC2+G2 sαC7 G5+ sαC5 G3 G8

IFBPF6. G2+ sαC2 G7 sαC5+G5 G3 sαC87. G2 sαC7 G5 sαC3+G3 G88. G2 sαC7 G5+ sαC5 sαC3 sαC8

IFHPF

9. G2+ sαC2 sαC7 G5+ sαC5 sαC3 sαC810. G2+ sαC2 sαC7 G5 sαC3 sαC811. G2 sαC7 G5 sαC3+G3 sαC812. G2 sαC7 G5+ sαC5 sαC3+G3 sαC813. G2+ sαC2 sαC7 G5 sαC3+G3 sαC8

TABLE III. ALL POSSIBLE INVERSE FILTERS FROM CONFIGURATION 2

No. Y3 Y5 Y7 Y2 Y4 Y6 Condition Type1. sαC3 G5 G7+ sαC7 G2 G4 G6 C3G7> C7G6, G2G5> G6G7

IFLPF2. G3+ sαC3 G5 sαC7 G2 G4 G6 G3> G6

3. G3+ sαC3 G5 G7+ sαC7 G2 G4 G6G3G5+G3G7> G6G7,C3G7+G3C7> G6C7

4. sαC3 G5 sαC7 sαC2+G2 G4 sαC6+G6 C3> C6, C2G5> G6C75. sαC3+G3 sαC5 G7+ sαC7 G2 G4 G6 G3> G2, G2C5+ C3G7+G3C7> G2C7

IFBPF6. G3+ sαC3 G5 sαC7+G7 G2 sαC4 G6G2G5+G3G7> G6G7,C3G7+ C7G3> C7G6

7. G3 sαC5 G7 sαC2+G2 G4 sαC6+G6 G3> G6, G2C5> G7C6

8. G3 sαC5 G7+ sαC7 sαC2 sαC4 sαC6G3C7> G7C6,C2C5> C6C7

IFHPF9. G3+ sαC3 sαC5 G7+ sαC7 sαC2 sαC4 sαC6

G7C3+G3C7> G7C6,C2C5+ C3C7> C6C7

10. G3+ sαC3 sαC5 G7 sαC2 sαC4 sαC6 G3> G611. G3 sαC5 G7 sαC2+G2 sαC4 sαC6+G6 G3> G6, G2C5> G7C612. G3 sαC5 G7+ sαC7 sαC2+G2 sαC4 sαC6+G6

G2C5+G3C7> G7C6+ C7G6,C2C5> C6C7, G3> G6

13. G3+ sαC3 sαC5 G7 sαC2+G2 sαC4 sαC6+G6 G3> G6, G2C5+ C3G7> G7C6

TABLE IV. ALL POSSIBLE INVERSE FILTERS FOR CONFIGURATION 3

No. Y2 Y7 Y5 Y8 Type1. G2+ sαC2 G7 sαC5+G5 G8 IFLPF2. G2+ sαC2 sαC7 G5+ sαC5 G8 IFBPF3. G2+ sαC2 G7 G5+ sαC5 sαC84. G2+ sαC2 sαC7 sαC5+G5 sαC8 IFHPF

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