Post on 19-Jan-2023
June 16, 2008 14:9 WSPC/123-JCSC 00416
Journal of Circuits, Systems, and ComputersVol. 17, No. 1 (2008) 55–66c© World Scientific Publishing Company
FIRST-ORDER FILTERS GENERALIZED TOTHE FRACTIONAL DOMAIN
A. G. RADWAN∗,§, A. M. SOLIMAN†,¶ and A. S. ELWAKIL‡,‖
∗Department of Engineering Mathematics,Faculty of Engineering, Cairo University, Egypt
†Department of Electronics and Communications,Faculty of Engineering, Cairo University, Egypt
‡Department of Electrical and Computer Engineering,University of Sharjah, P. O. Box 27272, Emirates
§ahmedgom@yahoo.com¶asoliman@ieee.org‖elwakil@ieee.org
Revised 8 July 2007
Traditional continuous-time filters are of integer order. However, using fractional cal-culus, filters may also be represented by the more general fractional-order differentialequations in which case integer-order filters are only a tight subset of fractional-orderfilters. In this work, we show that low-pass, high-pass, band-pass, and all-pass filters canbe realized with circuits incorporating a single fractance device. We derive expressionsfor the pole frequencies, the quality factor, the right-phase frequencies, and the half-power frequencies. Examples of fractional passive filters supported by numerical andPSpice simulations are given.
Keywords: Analog filters; fractional calculus; fractional-order circuits; filter design.
1. Introduction
Filter design is one of the very few areas of electrical engineering for which a com-plete design theory exists. Whether passive or active, filters necessarily incorporateinductors and capacitors, the total number of which dictates the filter order. How-ever, an inductor or capacitor is not but a special case of the more general so-calledfractance device; which is an electrical element whose impedance in the complexfrequency domain is given by Z(s) = asα ⇒ Z(jω) = aωαej(πα/2).1–5 For the spe-cial case of α = 1 this element represents an inductor while for α = − 1 it representsa capacitor. In the range 0 < α < 2, this element may generally be considered torepresent a fractional-order inductor while for the range − 2 < α < 0, it may beconsidered to represent a fractional-order capacitor. At α = − 2, it represents thewell-known frequency-dependent negative resistor (FDNR). Although a physical
55
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56 A. G. Radwan, A. M. Soliman & A. S. Elwakil
fractance device does not yet exist in the form of a single commercial device, it maybe emulated via higher-order passive RC or RLC trees, as described in Refs. 1–3 forsimulation purposes.a However, very recently the authors of Ref. 6 have describedand demonstrated a single apparatus which preforms as a physical fractional-ordercapacitor. It is not easy to reconstruct the apparatus described in Ref. 6, but it isan indication that a fractance device might soon be commercially available. Hence,it is important to generalize the filter design theory to the fractional-order domain.This work is a contribution in this direction.
Fractional calculus is the field of mathematics which is concerned with the inves-tigation and application of derivatives and integrals of arbitrary (real or complex)order.7–10 The Riemann–Liouville definition of a fractional derivative of order α isgiven by
Dαf(t) :=
1Γ(m − α)
dm
dtm
∫ t
0
f(τ)(t − τ)α+1−m dτ m − 1 < α < m ,
dm
dtmf(t) α = m .
(1)
A more physical interpretation of a fractional derivative is given by the Grunwald–Letnikov approximationb
Dαf(t) (∆t)−αm∑
j=0
Γ(j − α)Γ(−α)Γ(j + 1)
f((m − j)∆t) , (2)
where ∆t is the integration step. Applying the Laplace transform is widely usedto describe electronic circuits in the complex frequency s-domain. Hence, applyingthe Laplace transform to Eq. (1), assuming zero initial conditions, yields7–10
L 0dαt f(t) = sαF (s) , (3)
where 0dαt f(t) = df(t)/dt with zero initial conditions. In this work, we explore
the characteristics of a filter which includes a single fractance device. We deriveexpressions for the filter center frequency and quality factor and also for the half-power and right-phase frequencies. PSpice simulations of three passive fractional-order filters are shown using higher-order emulation trees.1–3 Finally, impedanceand frequency scaling are discussed in the case of fractional-order filters. It is worthnoting that a fractional-order Wien bridge oscillator, which includes two equal-value equal-order fractional capacitors, was studied in Ref. 11.
aThe use of a high integer-order transfer function to emulate a fractional-order transfer functionwhose order is less than 1 was also explained in Chap. 3 of Ref. 5. This approximation is basedon a Bode-plot approximation but does not imply equivalence in the state space.bNumerical simulations in this paper are carried out using a backward difference method basedon the Grunwald–Letnikov approximation.
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First-Order Filters Generalized to the Fractional Domain 57
2. Single Fractional Element Filters
The general transfer function of a filter with one fractional element is
T (s) =bsβ + d
sα + a. (4)
From the stability point of view, this system is stable if and only if a > 0 and α < 2while it will oscillate if and only if a > 0 and α = 2; otherwise it is unstable.12
The location of the poles is important to determine the filter center frequency ωo
and its quality factor Q. From Eq. (4) it is seen that the poles in the s-plane arelocated at sp = a1/αe±j(2m+1)π/α, m ∈ I+. The possible range of the angle inphysical s-plane is |θs| < π and hence there are no poles in the physical s-planefor α < 1. For 1 < α < 2, there are only two poles located at s1,2 = a1/αe±j π
α .
Comparing with a classical second-order system whose poles are located s1,2 =(−ωo/2Q) ± jωo
√1 − (1/4Q2) = ωoe
±jδ where δ = cos−1(− 1/2Q), it can be seenthat ωo and Q are given, respectively, by
ωo = a1/α , Q =− 1
2 cos(π/α). (5)
From the above equation, Q is negative for α < 1, and positive for α ≥ 1. At α = 1(classical first-order filter), it is seen that Q = 0.5 as expected. It is important nowto define the following critical frequencies
(1) ωm is the frequency at which the magnitude response has a maximum or aminimum and is obtained by solving the equation (d/dω)|T (jω)|ω=ωm = 0.
(2) ωh is the half-power frequency at which the power drops to half the passbandpower, i.e., |T (jωh) = (1/
√2)|T (jωpassband)|.
(3) ωrp is the right-phase frequency at which the phase ∠T (jωrp) = ± π/2.
2.1. Fractional-order low-pass filter
Consider the fractional low-pass filter (FLPF) whose transfer function is
TFLPF(s) =d
sα + a. (6)
The magnitude and phase of this transfer function are
|TFLPF(jω)| =d√
ω2α + 2aωα cos(απ/2) + a2,
∠TFLPF(jω) = − tan−1 ωα sin(απ/2)ωα cos(απ/2) + a
.
(7)
The important critical frequencies for this FLPF are found as ωm =ωo(− cos(απ/2))1/α, ωrp = ωo/(− cos(απ/2))1/α, and ωh = ωo(
√1 + cos2(απ/2) −
cos(απ/2))1/α, where ωo is as given by Eq. (5). From these expressions it is seen thatboth ωm and ωrp exist only if α > 1, in agreement with the quality-factor expres-sion (5). Table 1 summarizes the magnitude and phase values at some important
June 16, 2008 14:9 WSPC/123-JCSC 00416
58 A. G. Radwan, A. M. Soliman & A. S. Elwakil
Table 1. Magnitude and phase values at important frequencies for the FLPF.
ω = |TFLPF(jω)| ∠TFLPF(jω)
0d
a0
ωod
2a cos(απ/4)− απ
4
→ ∞ 0 − απ
2
ωmd
a sin(απ/2)
(1 − α)π
2
ωhd√2a
tan−1 sin(απ/2)
2 cos(0.5απ) +p
1 + cos2(απ/2)
ωrpd
acot(απ/2)
π
2
frequencies. Figure 1(a) plots the values of ωm, ωh, and ωrp (normalized with respectto ωo) for different values of the fractional-order α. Figure 1(b) is a plot of transferfunction magnitude (normalized with respect to |T (j0)|) at different critical fre-quencies versus α. Finally, Fig. 1(c) shows numerically simulated filter magnitudeand phase responses for two different cases α = 0.4 and α = 1.6, respectively. Notethe peaking in the magnitude response at α = 1.6 which is classically only possibleto observe in a second-order (α = 2) low-pass filter.
2.2. Fractional-order high-pass filter
Consider the fractional high-pass filter (FHPF) whose transfer function is
TFHPF(s) =bsα
sα + a. (8)
The magnitude and phase of this transfer function are
|TFHPF(jω)| =bωα√
ω2α + 2aωα cos(απ/2) + a2,
∠TFHPF(jω) =απ
2+ ∠TFLPF(jω) .
(9)
The important critical frequencies are found as ωm = (− a/cos(απ/2))1/α, ωrp =(− a cos(απ/2))1/α, and ωh = ωo[cos(απ/2) +
√1 + cos2(απ/2)]1/α. Figure 2 plots
the filter magnitude and phase responses for the two cases α = 0.4 and α = 1.6.
2.3. Fractional-order band-pass filter
Consider the fractional system
TFBPF(s) =bsβ
sα + a, (10)
June 16, 2008 14:9 WSPC/123-JCSC 00416
First-Order Filters Generalized to the Fractional Domain 59
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1
2
3
4
5
6
7
8
9
10
ωrp
o ω
ωh
o ωω m ωo
α0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
0.5
1
1.5
2
2.5
3
|T(jω )| o |T(j0)|
|T(jω )| h |T(j0)|
|T(jω )| m |T(j0)|
|T(jω )| rp |T(j0)|
α
(a) (b)
10−2
100
102
104
0
0.5
1
1.5
ω
|T(j
ω)| α = 0.4
10−2
100
102
104
−150
−100
−50
0
ω
Ph
ase(
T(j
ω))
α = 0.4
10−2
100
102
104
0
0.5
1
1.5
ω
|T(j
ω)|
α = 1.6
10−2
100
102
104
−150
−100
−50
0
ω
Ph
ase(
T(j
ω)) α = 1.6
ω h
ω h ω h
ωh
(c)
Fig. 1. Numerical simulation for the FLPF representing (a) normalized critical frequencies versusα, (b) normalized magnitude response versus α, and (c) filter magnitude and phase responses atα = 0.4 and α = 1.6 assuming a = d = 4.
whose magnitude and phase functions are
|TFBPF(jω)| =bωβ√
ω2α + 2aωα cos(απ/2) + a2,
∠TFBPF(jω) =π
2− ∠TFLPF(jω) .
(11)
June 16, 2008 14:9 WSPC/123-JCSC 00416
60 A. G. Radwan, A. M. Soliman & A. S. Elwakil
10−2
100
102
104
0
0.5
1
1.5
ω
|T(j
ω)| α = 0.4
10−2
100
102
104
0
50
100
150
ωP
has
e(T
(jω
))
α = 0.4
10−2
100
102
104
0
0.5
1
1.5
ω
|T(j
ω)|
α = 1.6
10−2
100
102
104
0
50
100
150
ω
Ph
ase(
T(j
ω)) α = 1.6
ω h ω h
ω h ω h
Fig. 2. Magnitude and phase response of the FHPF when a = 4 and b = 1 for α = 0.4 andα = 1.6, respectively.
Table 2. Magnitude and phase values atimportant frequencies for the FBPF.
ω = |TFBPF(jω)| ∠TFBPF(jω)
0 0π
2
ωobaβ/α
2a cos`
απ4
´ βπ
2− απ
4
→ ∞ bω(β−α) (β − α)π
2
Table 2 summarizes the magnitude and phase values at important frequencies forthis system. It is seen from Table 2 that for β < α ⇒ limω→∞ |T (jω)| = 0 andhence the filter can be a band-pass filter (BPF). For β = α ⇒ limω→∞ |T (jω)| = b
which makes the filter a HPF. The maxima frequency ωm is equal to ωo · (X)1/α
where X is given by
X =cos(απ/2)[(2β − α) +
√α2 + 4β(α − β) tan2(απ/2)]
2(α − β). (12)
June 16, 2008 14:9 WSPC/123-JCSC 00416
First-Order Filters Generalized to the Fractional Domain 61
The case α = 2β yields ωm = ωo. It is easy to see that there is always a maximumpoint in the magnitude response if α > β. Figure 3(a) shows the value of X versusα for different ratios of β and α. Figure 3(b) shows the magnitude response forthe filter when a = b = 1. Note from Fig. 3(b) that the center frequency ωo is not
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1
2
3
4
5
6
7
8
9
10
X
α
β=α
β=0.7α
β=0.5α
β=0.2α
(a)
10−2
100
102
104
0
0.2
0.4
0.6
0.8
1
ω
|T(j
ω)|
α = 1β = 0.3
10−2
100
102
104
0
0.2
0.4
0.6
0.8
1
ω
|T(j
ω)|
α = 1β = 0.5
10−2
100
102
104
0
0.2
0.4
0.6
0.8
1
ω
|T(j
ω)|
α = 1β = 0.8
10−2
100
102
104
0
0.2
0.4
0.6
0.8
1
ω
|T(j
ω)|
α = 1β = 1
ωo
ωo ωo
ωo
(b)
Fig. 3. Numerical simulations for the FBPF (a) values of X versus α and (b) magnitude responsefor different values of α and β (a = b = 1).
June 16, 2008 14:9 WSPC/123-JCSC 00416
62 A. G. Radwan, A. M. Soliman & A. S. Elwakil
necessarily equal to ωm (where the maxima occurs); which is significantly differentfrom what is known in integer-order filters.
2.4. Fractional-order all-pass filter
Consider the fractional all-pass filter system
TAPF(s) =b(sα − a)sα + a
. (13)
The magnitude and phase of this system are, respectively,
|TFAPF(jω)| = b
√ω2α − 2aωα cos(απ/2) + a2√ω2α + 2aωα cos(απ/2) + a2
,
∠TFAPF(jω) = ∠b(ωα cos(απ/2)− a) + jωα sin(απ/2)(ωα cos(απ/2) + a) + jωα sin(απ/2)
.
(14)
Table 3 summarizes the magnitude and phase values at important frequencies forthis filter. It is seen here that ωm = ωrp = ωo and that at this frequency a minimaoccurs if α < 1 and a maxima occurs if α > 1 while the magnitude remains flat whenα = 1 (classical integer-order all-pass filter). The half-power frequency is given byωh = ωo[2 cos(απ/2) +
√4 cos2(απ/2) − 1]1/α. Figure 4 shows the magnitude and
phase responses for the two different cases α = 0.4 and α = 1.6.
3. PSpice Simulations
Passive filters are chosen for simulations. In all simulations, we fix the fractancedevice to a fractional capacitor of order α = 0.4, 1, or 1.6. Figure 5(a) is the structureproposed in Ref. 1 to simulate a fractional capacitor of order 0.5 (YCF = CFs0.5;CF =
√C/R) while the circuit in Fig. 5(b), proposed in Ref. 3, is used to simulate
a fractional capacitor of arbitrary order α (YCF = CFsα). To realize α = 0.4,
for example, we need n = 31 branches with Rn+1/Rn 0.5686 and Cn+1/Cn 0.4287.3 To simulate a capacitor of order α = 1.6, a floating GIC circuit13 is used.The input impedance of a GIC is Zi = Z1Z2Z3/Z4Z5; taking Z3 = Z4 = R, Z1 =Z2 = 1/sCF, and Z4 = 1/s0.4CF results in Zi = 1/s1.6CF.
Figures 5(c) and 5(d) show a passive FLPF and PSpice simulation of its magni-tude response in three cases α = 0.4, 1, and 1.6, respectively. Figures 5(e) and 5(f)show a passive FHPF and its PSpice magnitude response while Figs. 5(g) and 5(h)
Table 3. Magnitude and phase values atimportant frequencies for the FAPF.
ω = |TFAPF(jω)| ∠TFAPF(jω)
0 b π
ωo b tan“ απ
4
” π
2→ ∞ b 0
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First-Order Filters Generalized to the Fractional Domain 63
10−2
100
102
104
0
0.5
1
1.5
ω
|T(j
ω)|
α = 0.4
10−2
100
102
104
0
50
100
150
ωP
has
e(T
(jω
)) α = 0.4
10−2
100
102
104
0
0.5
1
1.5
2
2.5
3
ω
|T(j
ω)|
α = 1.6
10−2
100
102
104
0
50
100
150
ω
Ph
ase(
T(j
ω)) α = 1.6
ωo ωo
ωo ωo
Fig. 4. FAPF magnitude and phase responses when α = 0.4 and α = 1.6 (a = 4, b = 1).
show a passive FAPF and its PSpice simulation results for the same three values ofα. Note the peaking in the response for α > 1 which is expected to increase as thefilter approaches α = 2, i.e., second-order filter.
We have constructed the fractional low-pass filter of Fig. 5(c) in the labora-tory using one of the fractional capacitors donated by the authors of Ref. 6. Thisfractional capacitor has α ≈ 0.5 up to approximately 30 kHz. Results are shown inFig. 6 compared with a normal capacitor.
4. Scaling
Impedance and frequency scaling can be used to adjust the filter component valuesor operating frequency. A fractional-order filter is similar to an integer-order onein terms of impedance scaling. However, for frequency scaling and assuming allcritical frequencies are to be scaled by a factor λ in which the new frequenciesequal λ times the old ones, then the components may be scaled according to thefollowing equations
Rnew =1λα
R or Cnew =1λα
Cold . (15)
June 16, 2008 14:9 WSPC/123-JCSC 00416
64 A. G. Radwan, A. M. Soliman & A. S. Elwakil
R
C
R
C
R
C
R
C
R
C
R
C
R
C
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
CF
CFp
R1
C1
R2
C2
R3
C3
Rn
Cn
.........
.........
(a) (b)
R
C α
+ +
_ _
ViVo
(c) (d)
R
+ +
_ _
Vi Vo
αC
(e) (f)
Fig. 5. (a) Realization of a fractional capacitor of order 0.5,1 (b) realization of a fractionalcapacitor of order α < 1,3 (c) FLPF circuit with transfer function T (s) = d/(sα + a), d = a =1/RC = 4, (d) PSpice simulation of the FLPF (R = 6.74 kΩ, C = CF = 37 µF), (e) FHPF circuitwith transfer function T (s) = dsα/(sα + a), d = 1, a = 1/RC = 4, (f) PSpice simulation ofthe FHPF (R = 6.74 kΩ, C = CF = 37 µF), (g) FAPF circuit with transfer function T (s) =− (1/2)(sα − a)/(sα + a), a = 1/RC = 4, and (h) PSpice simulation of the FAPF (R = 6.74 kΩ,C = CF = 37 µF).
June 16, 2008 14:9 WSPC/123-JCSC 00416
First-Order Filters Generalized to the Fractional Domain 65
R
C α
+
+
_
_Vi
Vo
R1
R1
(g) (h)
Fig. 5. (Continued)
Low-Pass filter
0
500
1000
1500
2000
2500
3000
3500
1 10 100 1000 10000 100000
Frequency (Hz)
Ou
tpu
t (m
V)
NOrmal Capacitance
Prob Capacitance
Fig. 6. Experimental results of a fractional-order low-pass filter with α ≈ 0.5 compared with anormal capacitor.
5. Conclusion
In this work, we have generalized classical first-order filter networks to be offractional-order. We have shown simulation results for filters of order 0 < α < 1
June 16, 2008 14:9 WSPC/123-JCSC 00416
66 A. G. Radwan, A. M. Soliman & A. S. Elwakil
and 1 < α < 2. It is clear that more flexibility in shaping the filter response canbe obtained via a fractional-order filter. It is also clear that the band-pass filter,classically known to be realizable only through a second-order system, can actuallybe realized by a fractional filter of order 1 < α < 2.
References
1. M. Nakagawa and K. Sorimachi, Basic characteristics of a fractance device, IEICETrans. Fundam. Electron. Commun. Comput. Sci. E75 (1992) 1814–1819.
2. K. Saito and M. Sugi, Simulation of power-law relaxations by analog circuits: Fractaldistribution of relaxation times and non-integer exponents, IEICE Trans. Fundam.Electron. Commun. Comput. Sci. E76 (1993) 205–209.
3. M. Sugi, Y. Hirano, Y. F. Miura and K. Saito, Simulation of fractal immittanceby analog circuits: An approach to the optimized circuits, IEICE Trans. Fundam.Electron. Commun. Comput. Sci. E82 (1999) 1627–1634.
4. A. Abbisso, R. Caponetto, L. Fortuna and D. Porto, Non-integer-order integration byusing neural networks, Proc. Int. Symp. Circuits and System ISCAS’01, Vol. 3 (2001),pp. 688–691.
5. P. Arena, R. Caponetto, L. Fortuna and D. Porto, Nonlinear Noninteger Order Cir-cuits and Systems (World Scientific, 2002).
6. K. Biswas, S. Sen and P. Dutta, Realization of a constant phase element and itsperformance study in a differentiator circuit, IEEE Trans. Circuits Syst.-II 53 (2006)802–806.
7. K. B. Oldham and J. Spanier, Fractional Calculus (Academic Press, New York, 1974).8. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives:
Theory and Application (Gordon & Breach, 1987).9. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional
Differential Equations (John Wiley & Sons, 1993).10. T. T. Hartley and C. F. Lorenzo, Initialization, conceptualization, and application
in the generalized fractional calculus, National Aeronautics and Space Administration(NASA/TP-1998-208415) (1998).
11. W. Ahmad, R. El-Khazali and A. S. Elwakil, Fractional-order Wien-bridge oscillator,Electron. Lett. 37 (2001) 1110–1112.
12. D. Matignon, Stability results in fractional differential equations with applications tocontrol processing, Proc. Multiconf. Computional Engineering in Systems and Appli-cation IMICS IEEE-SMC, Vol. 2 (1996), pp. 963–968.
13. A. Budak, Passive and Active Network Analysis and Synthesis (Waveland Press Inc.,Illinois, 1991).