This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution

and sharing with colleagues.

Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party

websites are prohibited.

In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information

regarding Elsevier’s archiving and manuscript policies areencouraged to visit:

http://www.elsevier.com/copyright

Author's personal copy

On the practical realization of higher-order filterswith fractional stepping

B. Maundy a, A.S. Elwakil b,�, T.J. Freeborn a

a Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta, Canadab Department of Electrical and Computer Engineering, University of Sharjah, P.O. Box 27272, Emirates

a r t i c l e i n f o

Article history:

Received 22 November 2009

Received in revised form

16 June 2010

Accepted 17 June 2010Available online 23 June 2010

Keywords:

Fractional-order filters

Filter design

Analog filters

a b s t r a c t

We propose the use of a compact integer-order transfer function approximation of

the fractional-order Laplacian operator sa to realize fractional-step filters. Lowpass

and bandpass filters of orders ðnþaÞ and 2ðnþaÞ, where n is an integer and 0oao1,

can, respectively, be designed. A 5th-order lowpass filter with fractional steps from

0.1 to 0.9 (i.e. 5:1-5:9) is given as an example with its characteristics compared to

5th- and 6th-order Butterworth filters. Spice simulations and experimental results

are shown.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Filters are electronic circuits which are traditionallyclassified as 1st-, 2nd- or n th-order circuits with n beingan integer number. The describing transfer functions offilters are usually of the form T(s)=N(s)/D(s) where N(s)and D(s) are polynomials in the Laplacian operator s raisedto an integer exponent; i.e. s, s2 or sn. However, a Laplacianof non-integer exponent sa where 0oao1 is mathema-tically valid and is representative of a fractional-ordersystem. Recall that a fractional derivative may accordingto the Riemann–Liouville definition [1–3] be given by

da

dtaf ðtÞ �Daf ðtÞ ¼

1

Gð1�aÞd

dt

Z t

0ðt�tÞ�af ðtÞdt ð1Þ

and Gð�Þ is the gamma function. Zero initial conditionsyield the Laplace transform of the above derivative as

Lf0dat f ðtÞg ¼ saFðsÞ ð2Þ

Migrating the concepts of fractional calculus andfractional-order systems into circuit design has slowlybeen gaining momentum [4]. Since solving fractional-order

differential equations in the time domain is mathemati-cally demanding [5,6] and since design and analysis toolsdeveloped under the control systems theory and linearcircuit theory are based on integer-order differentialequations, several researchers have proposed differentmethods to approximate the fractional Laplacian operatorsa via integer-order transfer functions [7–11].

In this work, we use a fixed second-order approximationfor the fractional Laplacian sa with an acceptable phase errorto realize fractional-step filters. We particularly extend thecharacteristics of Butterworth filters to the advantage offractional-stepping. An example of a lowpass filter of order5:1-5:9 is constructed, simulated using Spice and verifiedexperimentally. Fractional-order lowpass transfer functionshave an inherent peaking problem in the pass-band. Wefurther show how this problem can be solved.

It is worth noting that fractional-order filters arefinding increasing need in the measurements of biochem-ical and biomedical applications [12–14] and as accuratesensors in microbiology [15]. It is also worth noting thatthere is on going research to physically implement anaccurate fractional-order capacitor whose impedance isgiven by Zc ¼ 1=Csa [16,17] as well as to propose activerealizations (e.g. switched-capacitor circuits) of the frac-tional-order integro-differential operator [18,19].

Contents lists available at ScienceDirect

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

Signal Processing

0165-1684/$ - see front matter & 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.sigpro.2010.06.018

� Corresponding author.

E-mail address: elwakil@ieee.org (A.S. Elwakil).

Signal Processing 91 (2011) 484–491

Author's personal copy

2. Second-order approximation of sa

A second-order approximation for GðsÞ ¼ sa may beobtained using the continued fraction expansions (CFEs)[20,21] of the two functions

GhðsÞ ¼ ð1þsÞa �2þð1þaÞs2þð1�aÞs

ð3Þ

and

GlðsÞ ¼ 1�1

sþ1

� �a�

sþð1�aÞ=2

sþð1þaÞ=2ð4Þ

where Gh(s) is the approximation for high frequenciesoc1, and Gl(s) the approximation for o� 1. Noting thatthe product GhðsÞ � GlðsÞ ¼ sa, a second-order approxima-tion to sa can therefore be written as

sa �2ð1þaÞs2þð5�a2Þsþ2ð1�aÞ2ð1�aÞs2þð5�a2Þsþ2ð1þaÞ

¼a0s2þa1sþa2

a2s2þa1sþa0ð5Þ

A 4th-order approximation to sa can likewise be shown tobe

saFb4s4þb3s3þb2s2þb1sþb0

b0s4þb1s3þb2s2þb3sþb4ð6Þ

where b4 ¼ 12a2þ36aþ24, b3 ¼�6a3�6a2þ96aþ168,b2 ¼ a4�41a2þ292, b1 ¼ 6a3�6a2�96aþ168, and b0 ¼

12a2�36aþ24.Fig. 1 is a plot of the magnitude and phase of both the

2nd- and 4th-order approximations of s0.5. It can beobserved that for o 2 ½0:1,10�; over two decades, themagnitude error does not exceed 1.05 dB for bothapproximations. For o 2 ½0:373,2:683� and o 2½0:132,7:586� a phase error of not more than 51 isobtained using the two approximations, respectively.Although the 4th-order approximation has smallerphase error over a wider frequency range, its overheadwhen considering a practical realization is excessive. Wewill therefore adopt the second-order approximation.

3. Lowpass fractional-step filters

Consider the fractional-order lowpass filter (LPF)transfer function HLP

nþaðsÞ ¼ k1=ðsnþaþk2Þ where n is an

integer, k1,2 are positive constants and 0oao1. Extend-ing the results of [22], it can be shown that this filterresponse has a half power frequency at which themagnitude drops to k1=

ffiffiffi2p

k2 located at

o3 dB ¼ k2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þcos2

ðnþaÞp2

r�cos

ðnþaÞp2

!" #1=ðnþaÞ

ð7Þ

Also, the magnitude response always has a maximum(peak) equal to k1=½k2 sinðnþaÞp=2� located at the fre-quency op ¼ ½�k2 cosðnþaÞp=2��ðnþaÞ. This inherent peak-ing of HLP

nþaðsÞ may cause a problem in the pass-band ofthe filter and will be dealt with in a later section. Moreimportant is the stability analysis of HLP

nþaðsÞ [23] whichindicates that this transfer function is always unstable forany ðnþaÞ42. Therefore, such a filter is not realizable ifðnþaÞ42; i.e. only HLP

1þaðsÞ is stable. This means that it isonly possible to realize a fractional-step filter up tosecond-order. To overcome this stability constraint forhigher-order filters, we shall use (5) to approximateHLP

1þaðsÞ, which is always stable, and then divide byhigher-order normalized Butterworth polynomials. Inparticular, using (5) it can be shown that

HLP1þaðsÞ �

k1

a0

ða2s2þa1sþa0Þ

s3þ½ðk2a2þa1Þ=a0�s2þ½ða1k2þa2Þ=a0�sþk2

ð8Þ

indicating that the filter of order ð1þaÞ may be approxi-mated by a stable 3rd-order transfer function. A filter oforder ðnþaÞ may then be obtained as

HLP2þaðsÞ �

HLP1þaðsÞ

sþ1and HLP

nþaðsÞ �HLP

1þaðsÞ

Bn�1ðsÞ; nZ2 ð9Þ

where Bn(s) is a standard Butterworth polynomial of ordern [24]. Note that the necessary overhead to realize afractional filter of order ðnþaÞ would be to realize aninteger one of order n+2. This overhead can be toleratedfor applications where a precise fractional-step filter isneeded particularly in microbiological sensor applications[15] and biomedical measurements [14].

Fig. 2 is a simulation of the magnitude response for anexample H5+0.5

LP (s) (a filter of order 5.5) with k1=k2=1. Theresponse of 1/(s5.5+1) whose magnitude peaks atom � 0:939 rad=s reaching a maximum of 1.414 (3 dB)while having a bandwidth o3 dB � 1:13 rad=s is comparedto that of the stable 7th-order approximation HLP

1þ0:5ðsÞ=B4ðsÞ.Note that the magnitude response of the integer-orderapproximation HLP

1þ0:5ðsÞ=B4ðsÞ has a lower peak of 2.614 dBat om � 0:721 rad=s and has a bandwidth o3 dB � 1:14 rad=s.We also note that the stop-band attenuation is precisely�ð20� 5:5ÞdB=dec¼�110 dB=dec. It is important toremember that the filter 1/(s5.5+1) is unstable and henceunrealizable while its approximation HLP

1þ0:5ðsÞ=B4ðsÞ isstable.

In Fig. 3, the magnitude response of HLP5þaðsÞ for

a¼ ð0:1,0:5,0:9Þ is plotted. The response of this 5th-order fractional step filter is compared with maximally

Fig. 1. Magnitude and phase of s0.5 using the proposed 2nd- and 4th-

order approximations compared to ideal response.

B. Maundy et al. / Signal Processing 91 (2011) 484–491 485

Author's personal copy

flat Butterworth low pass filters of 5th and 6th orders,respectively (see Fig. 3) where it is clearly seen that thefractional step filter offers the advantage of preciseattenuation in the stop-band in steps of �2 dB/dec.However, the peaking in the pass-band reaches amaximum of 16 dB at order 5.9 which may pose aproblem.

4. High/band-pass fractional-step filters

Obtaining a fractional step highpass filter from itslowpass counterpart is straightforward and implies repla-cing s with 1/s as usual. A bandpass filter may likewise beobtained making use of (8) and the standard LP to BPtransformation s-ðs2þo2

oÞ=Bs where oo is the BP centerfrequency and B is the desired bandwidth. Assuming

oo ¼ 1, (8) then transforms into the bandpass transferfunction

HBP1þa ¼

k1B

a0

s½a2s4þa1Bs3þð2a2þa0B2Þs2þa1Bsþa2�

s6þD1s5þD2s4þD3s3þD4s2þD5sþ1ð10Þ

where D1 ¼ Bða1þa2k2Þ=a0, D2 ¼ ½B2ða2þa1k2Þþ3a0�=a0,

D3 ¼ Bða0B2k2þ2a2k2þ2a1Þ=a0, D4 ¼ ð3þB2þB2a2=a0Þ

and D5 ¼ Bða1þa2Þ=a0. Note that due to the LP to BPtransformation, HBP

1þa as given by (10) represents abandpass filter of order 2ð1þaÞ which is effectivelyapproximated by 6th-order equivalent transfer function.It can be shown that stability of (10) is guaranteed for any0oao1 if k2=1 or otherwise if

k2

2a4þðk2

2þ1Þa3�ðk22þ5k2þ1Þa2

�ð5k22þ8k2þ5Þaþ5k2

2þ17:5k240 ð11Þ

Using (10) along with Butterworth polynomials, higher-order fractional-step band pass filters of order 2ðnþaÞmaybe approximated by

HBPnþaðsÞ �

HBP1þaðsÞ

BBPn�1ðsÞ

; nZ2 ð12Þ

where BnBP(s) is obtained by applying the LP to BP

transformation s-ðs2þ1Þ=Bs to the standard Butterworthpolynomial Bn(s). Fig. 4 shows the magnitude response of afractional-step bandpass filter of order 2ð3þaÞ withB=1 rad/s for values of a¼ 0:3,0:5 and 0.7 compared tothe response of 6th- and 8th-order Butterworth filters.Note that the inherent peaking observed earlier in thelowpass filter also appears here as a increases with theresult that the response resembles that of a doubly tunedfilter.

5. Simulations and experimental results

Circuits wise, a canonical form of a ð5þaÞ lowpassfilter was realized using the implementation shown inFig. 5(a). The coefficients given within Fig. 5(a) aredemonstrative of an order 5.3 filter. Spice simulationwas carried out after implementing all blocks using

Fig. 2. Magnitude response of the stable H1+ 0.5LP (s)/B4(s) compared to the

unstable 1/(s5.5+1).

Fig. 3. Magnitude response of a fractional-step 5th-order filter com-

pared to 5th- and 6th-order Butterworth filters.

Fig. 4. Magnitude response of a fractional-step bandpass filter of order

2ð3þaÞ.

B. Maundy et al. / Signal Processing 91 (2011) 484–491486

Author's personal copy

MC1458 op amps and scaling all integrator time constantsto 0.1 ms using unit resistors of 1 kO and capacitors of0:1mF. Spice simulation results are shown in Fig. 5(b). Thecanonical realization of Fig. 5(a) was also implemented as

shown in Fig. 5(c). Components were set to realize a LPF oforder 5.3 and then re-adjusted to realize one of order 5.7.All integrators were implemented using 1 kO resistors and0:1mF capacitors. The normalized transfer function

Fig. 5. Implementation of a ð5þaÞ lowpass filter (a) canonical form with coefficients representing a 5.3 filter, (b) Spice simulations of an op amp-based

implementation and (c) actual constructed circuit for experimental results of the 5.3 filter.

B. Maundy et al. / Signal Processing 91 (2011) 484–491 487

Author's personal copy

realized for the order 5.3 and implemented by the circuitof Fig. 5(c) is

Vo

VinðsÞ ¼

0:54s2þ1:88sþ1

s7þ5:04s6þ12:18s5þ18:24s4þ18:24s3þ12:18s2þ5:04sþ1

ð13Þ

with the appropriate resistor values equal to the reciprocalof the numerator and denominator coefficients scaled-upby 1000.

The measured gain response of the circuit in Fig. 5(c) isshown in Fig. 6(a) alongside the ideal response, shown assolid lines. To ensure stability, the step response of theorder 5.3 filter was examined and is shown in Fig. 6(b).The MATLAB simulation of the step response of thecorresponding transfer function (13), adjusted to a timeconstant of 0.1 ms is shown for comparison in Fig. 6(c).The stop-band attenuation of the 5.7 filter was also

measured as �34 dB/octave which is very close to itstheoretically value of ð5:7� 6Þ ¼ �34:2 dB=Octave.

6. Maximally flat response

To obtain a maximally flat response of the fractionalfilter, the peaking problem needs to be solved. This is doneby considering the modified transfer functionHLP

1þaðsÞ ¼ 1=ðsaðsþk2Þþk3Þ where k2,3 are positive con-stants. Via appropriate selection of k2,3, we can maintainthe fractional stepping through the stopband whileeliminating the undesired passband peaking. The valuesof k2,3 for minimum passband error were numericallycomputed with the cumulative error compared to theButterworth response up until the �3 dB frequencyfor each value of a. Fig. 7 shows the optimum values ofk2,3 versus a computed both for the ideal sa and for its

Fig. 6. (a) Experimental response of canonical 5.3 and 5.7 lowpass filters, (b) experimental step response for the filter of order 5.3 and (c) corresponding

step response from (13).

B. Maundy et al. / Signal Processing 91 (2011) 484–491488

Author's personal copy

integer-order approximation. The boundaries of stabilitywere also numerically computed [23] and are shown inFig. 8. The maximally flat response of a ð5þaÞ fractionalLPF obtained using this modified HLP

1þaðsÞ is now shown inFig. 9. We have also constructed the circuit shownin Fig. 10(a) to realize this modified transfer function. Inparticular, using the integer-order approximation weobtain

HLP1þaðsÞ ¼

1

saðsþk2Þþk3�

1

a0

a2s2þa1sþa0

s3þb0s2þb1sþb2ð14Þ

where a0 ¼ a2þ3aþ2, a1 ¼ 8�2a2, a2 ¼ a2�3aþ2,b0 ¼ ða1þa0k2þa2k3Þ=a0, b1 ¼ ½a1ðk2þk3Þþa2�=a0 andb2 ¼ ða0k3þa2k2Þ=a0. This 3rd-order approximation can

then be decomposed into a cascade of 1st- and 2nd-ordersections realizable via a parallel RC network and a single-amplifier-biquad (SAB) [24], as seen in Fig. 10(a).

Spice simulations and experimental results ofFig. 10(a) using MC1458 op amps and 0:1mF capacitors

Fig. 7. Values of k2 and k3 that yield minimum passband error.

Fig. 8. Stability boundaries in the k2�a plane for different k3.

Fig. 9. Magnitude response of a maximally flat ð5þaÞ LPF.

Fig. 10. (a) Circuit realizing the transfer function (14) and (b) Spice

(dotted) and experimental (solid) results.

B. Maundy et al. / Signal Processing 91 (2011) 484–491 489

Author's personal copy

are shown together in Fig. 10(b). The resistor values usedare listed in Table 1. The characteristics are those of 1.1,1.5 and 1.9 maximally flat lowpass filters. The measuredstopband attenuations were respectively �22.93, �29.74and �36.44 dB/dec which are very close to the theoreticalvalues of �20ð1þaÞdB=dec. It is worth noting that theeffect of the op amp finite gain bandwidth product as wellas sensitivity analysis for Single Amplifier Biquads hasbeen studied in the literature [25,26] and can be appliedto the circuit in Fig. 10(a) if needed. Fig. 11 shows a SpiceMonte Carlo simulation for Fig. 10(a) assuming 20%tolerance (which is significantly large) for the threecapacitors used when the circuit realizes a filter of order1.5 and again a filter of order 1.9.

7. Conclusion

We have proposed a method of realizing fractionallystepped filters. Although the proposed method approx-imates a fractional-order filter by an integer-order onewhich is two degrees higher than the order of the actualfilter being realized, this is to the best of the author’sknowledge currently the only way which leads to a robustcircuit implementation. Alternatively, one would have to

wait until fabricated fractional-order capacitors becomereliable and commercially available [15–17]. This work isalso a continuation of our efforts to import concepts fromfractional calculus into circuit design [27,28].

References

[1] K.B. Oldham, J. Spanier, Fractional Calculus, Academic Press, NewYork, 1974.

[2] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals andDerivatives: Theory and Application, Gordon & Breach, 1987.

[3] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus andFractional Differential Equations, John Wiley & Sons, 1993.

[4] M.D. Ortigueira, An introduction to the fractional continuous-timelinear systems: the 21st century systems, IEEE Circuits and SystemsMagazine 8 (2008) 19–26.

[5] J. Wu, C. Chen, A new operational approach for solving fractionalcalculus and fractional differential equations numerically, IEICETransactions on Fundamentals E87 (2004) 1077–1082.

[6] C. Hwang, J. Leu, S. Tsay, A note on time-domain simulation offeedback fractional-order systems, IEEE Transactions on AutomaticControl 47 (2002) 625–631.

[7] A. Charef, H. Sun, Y. Tsao, B. Onaral, Fractal system as representedby singularity function, IEEE Transactions on Automatic Control 37(1992) 1465–1470.

[8] I. Podlubny, B. Vinagre, P. O’Leary, L. Dorcak, Analogue realizations offractional-order controllers, Nonlinear Dynamics 29 (2002) 281–296.

[9] I. Podlubny, Fractional-order systems and PID-controllers, IEEETransactions on Automatic Control 44 (1999) 208–214.

Table 1

Resistor values in kO used in the realization of Fig. 10(a).

Order Ra Rb Rc R1 R2 R3 R4 R5 R6 R7

1.1 38.3 1 13.7 60.4 0.0768 1.27 1.62 1 3.24 0.931

1.5 6.19 1 24.9 6.8 0.105 7.87 1.74 0.866 2.21 0.953

1.9 0.102 1 4.99 4.32 0.249 1 0.825 0.0324 1.43 1.05

Frequency

100Hz 1.0KHz 10KHz 100KHz

db (V(U2A:OUT)/ V(Vin:+))db (V(U2B:OUT)/ V(Vin:+))

-40

-20

-0

-50

filter order = 1.9

filter order = 1.5

Fig. 11. Spice Monte Carlo simulation assuming 20% tolerance for the three capacitors used in Fig. 10(a).

B. Maundy et al. / Signal Processing 91 (2011) 484–491490

Author's personal copy

[10] A. Oustaloup, F. Levron, B. Mathieu, Frequency-band complexnoninteger differentiator: characterization and synthesis, IEEETransactions on Circuits and Systems—I 47 (2000) 25–39.

[11] A. Arena, R. Caponetto, L. Fortuna, D. Porto, Nonlinear Nonintegerorder circuits and systems, World Scientific Series on NonlinearScience, Series A 38 (2002).

[12] A. Inaba, T. Manabe, H. Tsuji, T. Iwamoto, Electrical impedance analysisof tissue properties associated with ethylene induction by electriccircuits in cucumber, Plant Physiology 107 (1995) 195–205.

[13] I. Jesus, A. Tenreiro, J. Cunha, Fractional electrical impedances inbotanical elements, Journal of Vibration & Control 14 (2008)1389–1402.

[14] D. Gupta, C. Lammersfeld, P. Vashi, J. King, S. Sahlk, J. Grutsch, C. Lis,Bioelectrical impedance phase angle as a prognostic indicator inbreast cancer, BMC Cancer 8 (2008) 249–256.

[15] K. Biswas, S. Sen, P. Dutta, A constant phase element sensor formonitoring microbial growth, Sensors & Actuators B 119 (2006)186–191.

[16] K. Biswas, S. Sen, P. Dutta, Realization of a constant phase elementand its performance study in a differentiator circuit, IEEE Transac-tions on Circuits and Systems—II 53 (2006) 802–806.

[17] C. Haba, G. Loum, T. Zoueu, G. Ablart, Use of a component withfractional impedance in the realization of an analog regulator oforder 1/2, Journal of Applied Sciences 8 (2008) 59–67.

[18] P. Varshney, M. Gupta, G. Visweswaran, New switched capacitorfractional order integrator, Journal of Active and Passive ElectronicDevices 2 (2007) 187–197.

[19] G. Santamarıa, J. Valverde, R. Perez-Aloe, B. Vinagre, Microelec-tronic implementations of fractional-order integro-differential

operators, Journal of Computational and Nonlinear Dynamics 3(2008) (Available online doi:10.1115/1.2833907).

[20] Y. Chen, B. Vinagre, I. Podlubny, Continued fraction expansionapproaches to discretizing fractional order derivatives-an exposi-tory review, Nonlinear Dynamics 38 (2004) 155–170.

[21] Y. Chen, K. Moore, Discretization schemes for fractional-orderdifferentiators and integrators, IEEE Transactions on Circuits andSystems—I 49 (2002) 363–367.

[22] A.G. Radwan, A.M. Soliman, A.S. Elwakil, First order filters general-ized to the fractional domain, Journal of Circuits Systems &Computers 17 (2008) 55–66.

[23] A.G. Radwan, A.M. Soliman, A.S. Elwakil, A. Sedeek, On the stabilityof linear systems with fractional order elements, Chaos, Solitons &Fractals 40 (2009) 2317–2328.

[24] T.L. Deliyannis, Y.Y. Sun, J. Fidler, Continuous Time Active FilterDesign, CRC Press LLC, New York, 1999.

[25] P.E. Fleischer, Sensitivity minimization in a single amplifier biquadcircuit, IEEE Transactions on Circuits and Systems—I 23 (1976)45–55.

[26] A. Sedra, L. Brown, A refined classification of single amplifier filters,International Journal of Circuit Theory and Applications 7 (1979)127–137.

[27] A.G. Radwan, A.S. Elwakil, A.M. Soliman, Fractional-ordersinusoidal oscillators: design procedure and practical examples,IEEE Transactions on Circuits and Systems—I 55 (2008)2051–2063.

[28] B. Maundy, A.S. Elwakil, S. Gift, On a multivibrator that employs afractional capacitor, Journal of Analog Integrated Circuits & SignalProcessing 62 (2010) 99–103.

B. Maundy et al. / Signal Processing 91 (2011) 484–491 491