Further Generalization and Approximation of Fractional-Order Filters and Their Inverse Functions of the Second-Order Limiting Form

  Citation: Mahata, S.; Herencsar, N.; Kubanek, D. Further Generalization and Approximation of Fractional-Order Filters and Their Inverse Functions of the Second-Order Limiting Form. Fractal Fract. 2022,6, 209. https://doi.org/ 10.3390/fractalfract6040209 Academic Editor: Da-Yan Liu Received: 10 February 2022 Accepted: 5 April 2022 Published: 8 April 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fractal and fractional Article Further Generalization and Approximation of Fractional-Order Filters and Their Inverse Functions of the Second-Order Limiting Form Shibendu Mahata 1, Norbert Herencsar 2,* and David Kubanek 2 1Department of Electrical Engineering, Dr. B. C. Roy Engineering College, Durgapur 713206, West Bengal, India; [email protected] 2Department of Telecommunications, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 12, 61600 Brno, Czech Republic; [email protected] *Correspondence: her[email protected]g; Tel.: +420-541-146-981 Abstract: This paper proposes a further generalization of the fractional-order filters whose limiting form is that of the second-order filter. This new filter class can also be regarded as a superset of the recently reported power-law filters. An optimal approach incorporating constraints that restricts the real part of the roots of the numerator and denominator polynomials of the proposed rational approximant to negative values is formulated. Consequently, stable inverse filter characteristics can also be achieved using the suggested method. Accuracy of the proposed low-pass, high-pass, band-pass, and band-stop filters for various combinations of design parameters is evaluated using the absolute relative magnitude/phase error metrics. Current feedback operational amplifier-based circuit simulations validate the efficacy of the four types of designed filters and their inverse functions. Experimental results for the frequency and time-domain performances of the proposed fractionalorder band-pass filter and its inverse counterpart are also presented. Keywords: analog filter approximation; current feedback operational amplifier; fractional-order filter; inverse filter; optimization; power-law filter; second-order filter 1. Introduction The concepts of fractional calculus [ 1 ], the branch of mathematics which generalizes the integration and differentiation operations, have seen widespread applications in various fields of science and engineering [ 2 ]. The Grunwald–Letnikov definition of a fractional derivative of order αfor a function f(t) is given by (1) [3]. aDtαf(t) = lim h→0 1 hα "t−a h# ∑ j=0 (−1)jα jf(t−jh)(1) where [x] denotes the integer part of x; α j=Γ(α+1) Γ(j+1)Γ(α−j+1) represents the binomial coefficients; aand tare the bounds of the operation; and α∈( 0, 1 ) . Under zero initial conditions, the Laplace transformation of (1) is given by (2). L{0Dα tf(t)}=sαF(s)(2) The presence of the additional tuning parameter α provides several fundamental advantages to fractional-order (FO) filters when compared against the traditional (integerorder) filters: (i) exact meeting of design specifications, which implies precise control of filter roll-off characteristics. For instance, the fractional-order low-pass filter (FLPF) exhibits Fractal Fract. 2022,6, 209. https://doi.org/10.3390/fractalfract6040209 https://www.mdpi.com/journal/fractalfract Fractal Fract. 2022,6, 209 2 of 25 a roll-off rate of − 20 (N+α) decibel/decade (dB/dec), where N= 0, 1, . . . , and α∈( 0, 1 ) ; the same for the traditional filter is − 20 n dB/dec (n= 1, 2, . . .) [ 3 ]. The roll-off rate of − 20 α dB/dec, which finds application in the music industry [ 4 ], cannot be obtained using the integer-order filters. Similarly, a fractional-order band-pass filter (FBPF), in addition to exhibiting an asymmetric slope, can attain a roll-off extending down to 0 dB/dec [ 5 ]. In contrast, the minimum slope yielded for the classical band-pass filter, which always exhibits a symmetric magnitude-frequency behavior about the center frequency, is ± 20 dB/dec. Generalization of the traditional filters, such as the Butterworth [ 6 ], Chebyshev [ 7 ], shadow filter [ 8 ], etc., exploits the afore-mentioned advantages; (ii) various filter characteristics such as bandwidth, center frequency, quality factor, etc., are dependent on α which provides an additional design parameter to the circuit designer, and, hence, better flexibility [ 9 ]; and (iii) FO filters (for example, the FO Sallen-Key filter [ 10 ]) may achieve improved stability range compared to their classical counterparts. The FO filters can be modeled using the fractional-order transfer function (FTF), where the use of constant phase element for practical circuit implementation is required [ 11 ]. These constant phase elements, also known as the FO elements or fractance devices, provide an impedance phase characteristic which is independent of the frequency of operation and whose value is not limited to only + 90 or − 90 degrees, as is usual for classical inductors and capacitors [ 12 ]. Realization of FO filter circuits using the fractors can lead to reduced component count compared to the alternative technique involving the approximation of FO filters using the integer-order transfer function (ITF). For example, a 1.5 th -order Butterworth filter can be realized using a single operational amplifier (op-amp), one FO capacitor, one conventional capacitor, and three resistors [ 13 ]. In contrast, a third-order approximant of the same filter for six decades of design bandwidth requires four current feedback op-amps (CFOAs), three traditional capacitors, and eight resistors [ 14 ]. Due to the absence of commercial fractance elements, the impedance characteristics of the FO capacitor can be emulated using passive RC ladder networks or active components [ 15 ]. For example, the Valsa network [ 16 ] emulates the FO capacitor behavior for three decades of frequency using a ladder structure comprising five resistors and five capacitors. Hence, the passive component count for the FTF-based FO filter circuit can also increase substantially. Certain FO filters may be more conveniently modeled by determining their rational approximations. Two techniques exist in the literature for conversion of the FTF into its integer-order counterpart: (a) substitution of the rational approximation of the sα operator [ 17 ] in the FTF, as reported in [ 18 ], or further optimizing the obtained integerorder model to enhance the design bandwidth [ 14 ], and (b) direct determination of the ITF which approximates the magnitude and phase-frequency characteristics of the FO filter. For this purpose, either optimization techniques or curve-fitting procedures have been employed for the design of FO transitional filter [19], power-law filter (PLF) [20–23], etc. The contributions of this paper are the following: 1. Further generalization of the FO filter of the second-order limiting form is proposed. It is demonstrated that the PLFs reported in [ 20 – 23 ] are also a small subset of the class of filters proposed in this work; 2. This new class of filters of the low-pass (LP), high-pass (HP), band-pass (BP), and band-stop (BS) type, which attains more generic magnitude and phase-frequency characteristics, is then approximated as a rational transfer function. The coefficients of the ITF models are optimally determined such that the poles and zeros of the approximant are restricted to reside in the left-half s-plane; 3. As reported in [ 21 ], the frequency-domain-based curve-fitting using the Sanathanan– Koerner least-squares method presented in [ 20 , 22 , 23 ] can lead to unstable inverse LP and inverse HP type PLFs for some design cases. The proposed approach to formulate the constraints, which is different from the method adopted in [ 21 ], helps convert the stable filter into its stable inverse counterpart also through a simple inversion. To the best of the authors’ knowledge, these inverse filters, which further generalizes the models reported in [21], are also presented for the first time in the literature; Fractal Fract. 2022,6, 209 3 of 25 4. Circuit implementations for all the four filter types as well as their inverse counterparts are demonstrated on Simulation Program with Integrated Circuit Emphasis (SPICE) platform based on the CFOA being employed as an active component. Experimental measurements for the proposed FBPF and its corresponding inverse filter are also presented, which confirm their practical feasibility. In the rest of the paper, the new theoretical filter transfer functions are introduced in Section 2along with the proposed design technique. In Section 3, MATLAB simulations are conducted to investigate the performance of the proposed models; circuit simulations and experimental results are also presented in this section. Finally, the conclusions and future scope of research are outlined in Section 4. 2. Problem Formulation 2.1. Generalization of Power-Law Filter Transfer Function The generalized FO filter transfer function whose characteristic polynomial depends on two FO capacitors of order α, is given by (3) [24]. H(s) = cs2α+dsα+h s2α+2asα+b(3) where a,b,c,d, and hare constant coefficients. H(s) is always stable provided the conditions a2≥b , a< 0, or b< 0 are avoided. Fractional step magnitude and phase behaviors of the LP, HP, BP, and BS types may be obtained from (3) by setting c=d= 0, d=h=0 , c=h= 0, and d= 0, respectively. An optimal approximant for the transfer function of the form h/(s2α+ 2 asα+b) was obtained using a multi-objective optimization routine [ 25 ]. In [ 26 ], three different transfer functions, such as h/(s1+α+asα+b) , h/(s1+α+as +b) , and h/(s1+α+as0.5(1+α)+b) , were formulated and their coefficients were optimally determined to approximate the frequency-domain characteristics of the second-order LP filter 1/(s2+s/Q+1), where the quality factor Qcan be adjusted to any arbitrary value. Optimal rational approximations for FO filters of the LP, HP, and BP types, such as h/R(s) , hsα+β/R(s) , and hsβ/R(s) , respectively, where R(s) = sα+β+asβ+b , and β∈( 0, 1 ) , were also reported [27]. A new class of FO filter, as given by (4) , can be proposed by introducing a FO exponent βin the filter transfer function defined in (3): Hα,β D(s) = cs2α+dsα+h s2α+2asα+bβ (4) where β∈( 0, 1 ] and β∈[− 1, 0 ) leads to the standard and inverse filter characteristics, respectively. The transfer function, magnitude, and phase of the LP, HP, BP, and BS type proposed theoretical filters are presented in Table 1. H(s) is a special case of Hα,β D(s) when β= 1, which implies further generalization of the filter forms reported in [ 24 ]. The classical second-order filter can be obtained from (4) by choosing α=β= 1. As a representative, Figure 1shows the different magnitude and phase-frequency characteristic curves exhibited for the theoretical FLPF ( c= 0, d= 0, h= 1, a= 1, b= 1), with α= 0.6 and β={ 0.6, 0.8, 1 } . The introduction of the additional tuning parameter β in the proposed FO filter transfer function given by (4) can yield a much wider variety of magnitude and phase-frequency characteristics which is not possible to attain using only a single tuning knob αin (3). Fractal Fract. 2022,6, 209 4 of 25 Table 1. Transfer function, magnitude, and phase of the proposed theoretical filters. Filter Type Transfer Function Parameter Expression Low-pass h s2α+2asα+bβMagnitude hβ [ω4α+4a2ω2α+b2+2bω2αcos(απ) + 4abωαcos(απ/2) + 4aω3αcos(απ/2)]β/2 Phase −βtan−1hω2αsin(απ) + 2aωαsin(απ/2) ω2αcos(απ) + 2aωαcos(απ/2) + bi High-pass cs2α s2α+2asα+bβMagnitude cβω2αβ [ω4α+4a2ω2α+b2+2bω2αcos(απ) + 4abωαcos(απ/2) + 4aω3αcos(απ/2)]β/2 Phase βαπ −tan−1hω2αsin(απ) + 2aωαsin(απ/2) ω2αcos(απ) + 2aωαcos(απ/2) + bi Band-pass dsα s2α+2asα+bβMagnitude dβωαβ [ω4α+4a2ω2α+b2+2bω2αcos(απ) + 4abωαcos(απ/2) + 4aω3αcos(απ/2)]β/2 Phase β0.5απ −tan−1hω2αsin(απ) + 2aωαsin(απ/2) ω2αcos(απ) + 2aωαcos(απ/2) + bi Band-stop cs2α+h s2α+2asα+bβMagnitude hc2ω4α+h2+2hcω2αcos(απ) ω4α+4a2ω2α+b2+2bω2αcos(απ) + 4abωαcos(απ/2) + 4aω3αcos(απ/2)iβ/2 Phase βtan−1hcω2αsin(απ) cω2αcos(απ) + hi−tan−1hω2αsin(απ) + 2aωαsin(απ/2) ω2αcos(απ) + 2aωαcos(απ/2) + bi 10−2 10−1 100101102 −60 −40 −20 0 Frequency (rad/s) Magnitude (dB) 10−2 10−1 100101102 −100 −50 0 Frequency (rad/s) Phase (deg) β = 0.6 β = 0.8 β = 1 Figure 1. Magnitude and phase responses of the theoretical FLPF for different values of β with α=0.6. A different form of double FO-exponent LP and HP filter transfer functions, such as { 1 /(sα+ 1 )}β and {sα/(sα+ 1 )}β , respectively, where α , β∈(− 2, 2 ) , were reported in [ 28 ]. Two FO elements-based band-stop filter of the form (sp+k1sq+k2)/(sp+k3sq+k2) , where, p= 1 +β1+β2 , q= 1 +β2 , and β1 , β2∈( 0, 1 ) , was presented in [ 29 ], which is also unlike the proposed fractional-order band-stop filter (FBSF) model. Note that the transfer function of the PLF reported in [ 20 – 23 ] can be obtained from (4) by setting α= 1. Therefore, the proposed FO filter transfer function also provides a further generalization of the PLF models. As an example, the magnitude and phase-characteristics of the theoretical LP type PLF [ 20 – 23 ] and that of the proposed filter, with ( c=d= 0, h=a=b= 1, β= 0.7), are compared in Figure 2. It can be seen that for the considered case (i.e., β= 0.7), the PLF attains only a single characteristic curve since α is fixed at 1, whereas the proposed filter can achieve various other responses (viz., α=0.6, 0.8). Fractal Fract. 2022,6, 209 5 of 25 10−2 10−1 100101102 −60 −40 −20 0 Frequency (rad/s) Magnitude (dB) α = 0.6 α = 0.8 α = 1 (PLF [20, 21, 22, 23]) 10−2 10−1 100101102 −150 −100 −50 0 Frequency (rad/s) Phase (deg) Figure 2. Magnitude and phase responses of the theoretical FLPF for different values of α with β=0.7. 2.2. Proposed Technique Defining an ITF as per (5) , where Nis a positive integer, the frequency-domain characteristics of the theoretical FO filter may be approximated in the optimal sense by minimizing the mean absolute relative magnitude and phase errors between Hα,β D(jω) and Hα,β,N P(jω) . Hα,β,N P(s) = A(s) B(s)= N ∑ i=0 aisi sN+ N−1 ∑ k=0 bksk (5) For this purpose, the objective function for the proposed optimization (minimization) routine is formulated according to (6) , subject to the nonlinear inequality constraints which ensure that the zeros and poles of Hα,β,N P(s)lie strictly on the left-half s-plane. f=1 L L ∑ i=1  1−Hα,β,N P(jωi,X) Hα,β D(jωi)  + 1−∠Hα,β,N P(jωi,X) ∠Hα,β D(jωi) (6) Subject to: (i) ak>0 (k=0, 1, . .., N); (ii) bk>0 (k=0, 1, . .. , N−1); (iii) Real part of roots of A(s) and B(s)<0. where Xdenotes the vector of decision variables, i.e., X= [a N a N−1 ... a 0 b N−1 b N−2 ... b 0 ]; Ldenotes the number of log-spaced sample points in the bandwidth [ωmin , ωmax] rad/s; and the dimension (D) of the problem is 2 N+ 1. The application of metaheuristics for the optimal approximation of FO filters and systems has shown promising performances [ 30 ]. The constrained composite differential evolution (C 2 oDE) algorithm [ 31 ] can be employed as the optimization problem solver for this work. C 2 oDE integrates the basic framework of composite differential evolution [ 32 ], with the constraint-handling mechanisms based on the feasibility rule [ 33 ] and the ε -constrained method [ 34 ]. The detailed discussion of the C 2 oDE algorithm can be found in [ 31 ]. The proposed objective function is minimized by C 2 oDE, where the FO exponents { α , β } of the theoretical filter transfer function, order of the proposed rational approximant ( N ), length of data sample points ( L ), lower limit ( ωmin ), and upper limit ( ωmax ) of the desired bandwidth are the user-defined inputs to the optimization routine. The metaheuristic search procedure of C 2 oDE determines the Fractal Fract. 2022,6, 209 6 of 25 best feasible solution ( X∗ ), i.e., the vector of decision variables (coefficients of Hα,β,N P(s) ) which achieves the smallest value of f while also satisfying the design constraints, after maxrun number of independent trial runs of the algorithm are conducted. Therefore, X∗ is declared as the near-global optimal solution at the end of the proposed search optimization procedure. In Figure 3, the flowchart of the proposed analog filter design technique is presented. As highlighted in [ 35 ], the issue of design stability for the inverse filters is an important topic. A different set of constraints have been proposed here to ensure the approximant’s stable and minimum phase response as compared to the Hurwitz determinant method employed in [ 21 ]. While forcing the coefficients a k and b k to be positive, the proposed constraints also ensure that the real part of all the roots of the numerator and denominator polynomials of Hα,β,N P(s) attains a negative value. Since all the zeros of the proposed approximant are constrained to lie in the left-half s-plane, inverting the transfer function Hα,β,N P(s) , i.e., [Hα,β,N P(s)]−1 , provides the stable inverse filter characteristics for the proposed FLPF, fractional-order high-pass filter (FHPF), FBPF, and FBSF. The inverse filter transfer functions proposed in this work are also different from those published in the literature [ 36 – 41 ]. Henceforth, the FO inverse LP, HP, BP, and BS filters will be abbreviated as FILPF, FIHPF, FIBPF, and FIBSF, respectively. Start Inputs: maxmin,,,,,  NL Set run = 1, maxrun = 20 Minimize f based on the algorithmic framework of C2oDE Store the best solution Xrun and the corresponding fitness function value frun Is run = = maxrun ? run = run + 1 No Determine the decision variables vector )( * X with the minimum fitness function value (min{ frun}) from all runs Output: * X End Yes Figure 3. Flowchart of the proposed filter design technique. 3. Results and Discussions In this work, the population size and the maximum number of function evaluations for C 2 oDE are chosen as 100 and 10,000D, respectively; the remaining control parameters are set according to the recommendations provided in [ 31 ]. The lower bound of the decision Fractal Fract. 2022,6, 209 7 of 25 variables is chosen as 10 −6 while the upper bound is set as 20,000. For all the design cases presented in this paper, L= 100, ωmin = 10 −2 rad/s, ωmax = 10 2 rad/s, and maxrun = 20 , are chosen. The proposed optimization routine is implemented in MATLAB, where the roots of A(s) and B(s) for each search agent are determined using the roots() function. To evaluate the design accuracy, the absolute relative magnitude error (ARME) and the absolute relative phase error (ARPE) metrics are used. These error indices are defined as per (7) and (8), respectively: ARME = Hα,β D(jω)−Hα,β,N P(jω) Hα,β D(jω)  (7) ARPE = ∠Hα,β D(jω)−∠Hα,β,N P(jω) ∠Hα,β D(jω) (8) 3.1. Performance Analysis 3.1.1. Fractional-Order Low-Pass Filter The optimal model coefficients of the FLPFs ( c=d= 0, h=a=b= 1) for different combinations of α , β , and Nare presented in Table 2. The stability criteria are satisfied for all the design cases since the system poles are always located on the left-half s-plane. For instance, the poles of the proposed FLPF with N= 4 and (α , β) = ( 0.6, 0.8 ) are located at s={− 0.0271, − 0.2303, − 1.2976, − 9.5261 } . To achieve a stable FILPF, the zeros of the corresponding FLPF must reside on the left-half s-plane. It is noteworthy that the proposed design constraints in this regard are also satisfied. For example, the zeros of the afore-mentioned design case are located at s={−0.0315, −0.3949, −5.6393, −1054.7}. Table 2. Optimal coefficients of the proposed FLPFs for different values of α,β, and N. α β N[aNaN–1 ... a0] [bN–1bN–2 ... b0] 0.6 0.6 3 0.0142 2.5299 6.6189 0.4942 11.4218 9.2670 0.5131 4 0.0114 3.0287 25.5717 13.7481 0.4771 24.2389 55.7542 16.8974 0.4922 5 0.0092 3.4666 61.4831 110.5999 22.6230 0.4607 41.6920 208.3549 178.6875 26.2131 0.4730 0.6 0.8 3 0.0013 0.9827 1.6323 0.0935 4.7579 2.3646 0.0980 4 0.0010 1.0608 6.4002 2.5499 0.0741 11.0810 15.1524 3.2481 0.0770 5 0.0008 1.1094 15.6780 21.3117 3.5163 0.0625 21.2276 62.7773 37.1129 4.2024 0.0646 0.7 0.6 3 0.0051 1.6591 4.5010 0.3897 7.7235 5.9418 0.3968 4 0.0041 1.8637 16.5030 9.4477 0.3705 17.7793 34.5354 11.0523 0.3761 5 0.0031 2.0251 38.1221 70.6279 15.5349 0.3518 32.3963 132.0535 107.9572 17.2623 0.3562 0.9 0.5 3 0.0021 1.4230 6.3582 0.9979 7.7812 7.7812 1.0000 4 0.0018 1.5155 18.6555 16.4228 0.9982 17.9383 37.3110 17.9383 1.0000 5 0.0014 1.5881 38.2769 97.7768 30.9821 0.9986 32.5702 136.0537 136.0537 32.5702 1.0000 The effect due to variations of Non the accuracy of the proposed FLPFs is presented in Table 3by considering the maximum (max) and mean values of ARME and ARPE metrics. In total, 1000 log-spaced data sample points are considered for evaluating the mean error. This is due to the fact that the mean error remains nearly the same if a higher number of data points are used; while the error marginally increases if a lower number of sample points are chosen. For example, the values of {mean ARME (dB), mean ARPE (dB)} attained by the proposed FLPF with α = 0.7, β = 0.6, and N = 4 for 100, 1000, and 10,000 sample points are { − 36.36, − 32.69}, { − 36.53, − 32.82}, and { − 36.54, − 32.83}, respectively. Similar results are also obtained for the other design cases, which justify the selection of 1000 sample points for the evaluation of the mean errors. It is also found that the accuracy increases (i.e., error reduces) with an increase in the design order (N) for all the cases. For instance, considering Fractal Fract. 2022,6, 209 8 of 25 the FLPF for (α , β) = ( 0.9, 0.5 ) , the max and mean values of ARME for N = {3, 4, 5} are { − 20.25, − 25.36, − 31.21} dB and { − 35.53, − 43.34, − 51.13} dB, respectively; similarly, the max and mean ARPE values are { − 20.13, − 25.31, − 31.51} dB and { − 31.91, − 39.78, − 47.24} dB, respectively. These results are in accordance with the fact that the accuracy of approximating any FTF with an integer-order one improves as Nincreases; the modeling error being theoretically zero when the integer-order model has an order of infinity. Since such infinite-dimensional systems are impractical, a trade-off between accuracy and order (design/hardware complexity) is a pertinent issue in the rational approximation of any FO system. Table 3. Performance indices of the designed FLPFs for different values of N. α β NARME (dB) ARPE (dB) MC(dB) θC(deg) ωM(rad/s) ωθ(rad/s) Max Mean Max Mean 0.6 0.6 3−15.00 −26.51 −13.04 −21.96 −5.937 −35.72 1.0210 0.7918 4−19.00 −34.16 −18.72 −29.74 −6.054 −30.85 0.9886 1.1610 5−24.33 −41.81 −26.29 −37.09 −6.013 −32.99 1.0030 0.9484 0.6 0.8 3−17.93 −28.88 −15.09 −25.73 −8.348 −45.08 0.9380 0.8406 4−23.49 −36.76 −21.59 −33.59 −7.887 −42.35 1.0380 1.0530 5−29.09 −44.52 −29.88 −41.41 −8.092 −43.52 0.9867 0.9768 0.7 0.6 3−15.98 −28.08 −14.46 −24.99 −5.539 −40.24 1.0060 0.8658 4−20.75 −36.53 −19.84 −32.82 −5.569 −36.66 0.9995 1.0820 5−26.28 −44.21 −26.75 −40.12 −5.564 −38.24 0.9999 0.9723 0.9 0.5 3−20.25 −35.53 −20.13 −31.91 −3.555 −41.18 1.0220 0.9725 4−25.36 −43.34 −25.31 −39.78 −3.685 −40.19 0.9895 1.0150 5−31.21 −51.13 −31.51 −47.24 −3.628 −40.63 1.0040 0.9949 The magnitude ( MCT ) of the theoretical FLPF at the frequency of 1 rad/s for ( α , β ) = (0.6, 0.6), (0.6, 0.8), (0.7, 0.6), and (0.9, 0.5) is − 6.023 dB, − 8.031 dB, − 5.565 dB, and − 3.643 dB, respectively. The phase ( θCT ) of the theoretical filters at 1 rad/s for the same combinations of ( α , β ) is –32.40 ◦ , − 43.21 ◦ , − 37.81 ◦ , and − 40.51 ◦ , respectively. The magnitude and phase of the proposed filters at 1 rad/s, denoted by MC and θC , respectively, are presented in Table 3. Results reveal that both these values approach the theoretical ones for all the design cases as N is increased. The frequency values attained by the proposed FLPF at MCT and θCT are denoted by ωM and ωθ , respectively. As demonstrated in Table 3, these performance indices also approach the theoretical frequency value of 1 rad/s with an increase in N. Figure 4a highlights the improved accuracy in the magnitude and phase responses of the optimal FLPFs for the design case (α= 0.7, β= 0.6 ) when Nis increased from 3 to 5. Further confirmation is provided in Figure 4b, which shows the reduction in ARME and ARPE with an increase in N. The magnitude responses of the proposed FLPFs ( N= 4) for (α , β)=( 0.6, 0.8 ) and (0.9, 0.5) and their corresponding inverse counterparts, i.e., FILPFs ( N= 4) for (α,β) = (0.6, −0.8) and ( 0.9, − 0.5 ) , are presented in Figure 5a, and the phase responses are shown in Figure 5b. Both the design cases exhibit proximity with the theoretical responses in the desired bandwidth. The zeros and poles of the FLPF for (α , β) = ( 0.9, 0.5 ) are located at { − 0.0657, − 0.8830, − 11.5331, − 829.4627}, and { − 0.0640, − 0.6091, − 1.6418, − 15.6234}, respectively, in the s-plane, which implies that both the FLPF and its inverse counterpart are stable. Fractal Fract. 2022,6, 209 9 of 25 10−2 10−1 100101102 −40 −30 −20 −10 0 Frequency (rad/s) Magnitude (dB) Ideal N = 3 N = 4 N = 5 10−2 10−1 100101102 −80 −60 −40 −20 0 Frequency (rad/s) Phase (deg) (a) 10−2 10−1 100101102 0 0.1 0.2 0.05 0.15 Frequency (rad/s) ARME N = 3 N = 4 N = 5 10−2 10−1 100101102 0 0.05 0.1 0.15 0.2 Frequency (rad/s) ARPE (b) Figure 4. ( a ) Magnitude, phase and ( b ) ARME, ARPE responses of the proposed FLPF ( α= 0.7, β=0.6) for different values of N. 10−2 10−1 100101102 −40 −30 −20 −10 0 10 20 30 40 Frequency (rad/s) Magnitude (dB) (α, β) = (0.6, −0.8) (α, β) = (0.9, −0.5) (α, β) = (0.9, 0.5) (α, β) = (0.6, 0.8) (a) 10−2 10−1 100101102 −100 −80 −60 −40 −20 0 20 40 60 80 100 Frequency (rad/s) Phase (deg) (α, β) = (0.6, −0.8) (α, β) = (0.6, 0.8) (α, β) = (0.9, −0.5) (α, β) = (0.9, 0.5) (b) Figure 5. ( a ) Magnitude and ( b ) phase responses of the proposed FLPFs and FILPFs for N= 4. The theoretical response is shown in solid red. 3.1.2. Fractional-Order High-Pass Filter The well-known LP-to-HP transformation technique [42], which involves replacing s with 1/s in the FLPF transfer function, can convert the FLPF models presented in Table 2 into the FHPFs of the same order. The proposed technique can also allow the optimal design of FHPFs directly without obtaining the FLPF model in the first stage. In Table 4, the stable FHPF approximants for (α , β)=( 0.8, 0.5 ) and ( 0.7, 0.7 ) with ( d=h= 0, c=a=b= 1) are obtained by carrying out the proposed optimization technique for different values of N. For instance, the poles and zeros of the FHPF with (α , β , N) = ( 0.8, 0.5, 4 ) are obtained as { − 0.0530, − 0.4949, − 2.0205, − 18.8688} and { − 0.0025, − 0.0972, − 1.2887, − 17.8685}, respectively, in the s-plane. Such pole-zero locations justify that both the FHPF and its inverse model are stable. The max and mean indices of the ARME and ARPE metrics are evaluated for the proposed FHPF transfer functions, as shown in Table 5. A similar finding regarding the improvement in accuracy with an increase in the design order is also noted here. The {MCT , θCT} values of the theoretical FHPF for ( α , β ) = (0.8, 0.5) are {–4.178 dB, 35.99 ◦ }; for ( α , β ) = (0.7, 0.7), these are {–6.488 dB, 44.09 ◦ }. The MC , θC , ωM , and ωθ parameters attained Fractal Fract. 2022,6, 209 16 of 25 FHPF are evaluated as 10 M Ω and 1.78 M Ω , respectively; for practical purposes, these values are set as ∞ Ω (i.e., open circuit). Circuit simulations are carried out in OrCAD PSPICE software with the AD844A/AD IC SPICE model used as the CFOA. The SPICE simulated magnitude and phase responses of the proposed filters are shown in Figure 10a,b; the same for their inverse functions are presented in Figure 11a,b;. It is found that the frequency-domain behavior of the proposed filters designed using the nominal values of components attain good agreement with the theoretical characteristics. Quantitatively, the absolute maximum deviations in { magnitude, phase } of the designed FLPF, FILPF, FHPF, FIHPF, FBPF, FIBPF, FBSF, and FIBSF from the theoretical characteristics are { 0.0374, 6.72 ◦} , { 1.9915, 6.76 ◦} , { 0.0360, 11.65 ◦} , { 3.1215, 3.36 ◦} , { 0.0170, 2.87 ◦} , { 1.3074, 8.48 ◦} , { 0.0633, 4.74 ◦} , and { 0.1128, 5.76 ◦} , respectively. The ARME and ARPE indices with L= 100 log-spaced data sample points, yielded for the SPICE-simulated filters based on the nominal values of passive components are presented in Table 13; in Figure 12a,b, the broken line graphs of these error indices for each design case are also presented. Monte-Carlo simulations are conducted in PSPICE to determine the performance of the filters due to 5% and 10% deviations (following a Gaussian distribution) from the nominal values of resistors and capacitors, respectively. In total, 100 Monte-Carlo runs are carried out for each case and the magnitude and phase plots are shown in green in Figures 10 and 11. The minimum (min), max, mean, and standard deviation (SD) indices for the magnitude and phase at 1000 rad/s of the practical filters are presented in Table 14. The small values of SD justify good agreement with the theoretical (ideal) characteristics. Table 12. Values of components to realize the proposed filters. [Note: Numbers in parenthesis represent (α,β)]. Component FLPF FILPF FHPF FIHPF FBPF FIBPF FBSF FIBSF (0.6, 0.8) (0.6, 0.8) (0.8, 0.5) (0.8, 0.5) (0.65, 0.85) (0.65, 0.85) (0.75, 0.65) (0.75, 0.65) R(kΩ)10 100 10 100 7.5 100 75 100 RF(kΩ)10 1 10 100 7.5 100 75 100 Rout (kΩ)10 100 10 11 7.5 10 75 110 Rin (kΩ)10 10 10 10 7.5 10 75 100 R1(kΩ)∞1 10 100 220 3.3 75 100 R2(kΩ)100 1 11 91 47 16 91 82 R3(kΩ)24 4.3 20 51 20 39 160 47 R4(kΩ)13 8.2 91 11 47 16 91 82 R5(kΩ)10 10 ∞0.56 220 3.3 75 100 C1(nF) 10 1 4.7 0.56 3.3 0.047 0.56 0.47 C2(nF) 68 1.8 47 8.2 33 1 4.7 6.8 C3(nF) 470 27 220 100 560 100 33 15 C4(µF) 3.9 0.33 2.2 3.9 5.6 2.2 0.33 0.22 Table 13. ARME and ARPE performances of the SPICE-simulated proposed filters based on nominal values of components. Metric Index FLPF FILPF FHPF FIHPF FBPF FIBPF FBSF FIBSF (0.6, 0.8) (0.6, 0.8) (0.8, 0.5) (0.8, 0.5) (0.65, 0.85) (0.65, 0.85) (0.75, 0.65) (0.75, 0.65) ARME (dB) Max −15.84 −15.52 −20.50 −22.19 −16.05 −16.66 −22.85 −23.72 Mean −27.80 −22.10 −30.05 −29.94 −28.24 −24.11 −26.94 −31.91 ARPE (dB) Max −11.77 −11.32 −0.56 7.88 −8.07 2.90 6.97 8.67 Mean −21.61 −20.20 −15.92 −14.56 −26.31 −16.47 −12.57 −12.87 Fractal Fract. 2022,6, 209 17 of 25 Table 14. Statistical indices about the magnitude/phase at the shift frequency of the designed filters based on 100 Monte-Carlo simulation runs. Parameter Index FLPF FILPF FHPF FIHPF FBPF FIBPF FBSF FIBSF @ 1 krad/s (0.6, 0.8) (0.6, 0.8) (0.8, 0.5) (0.8, 0.5) (0.65, 0.85) (0.65, 0.85) (0.75, 0.65) (0.75, 0.65) Magnitude Min −10.56 5.60 −5.98 1.68 −10.75 5.99 −9.10 4.95 Max −6.22 9.79 −1.72 6.11 −6.18 10.15 −4.58 8.98 Mean −7.99 7.55 −4.05 3.78 −8.50 8.03 −6.88 7.08 (dB) SD 0.932 0.921 0.972 0.962 0.898 0.851 0.902 0.850 Ideal −8.02 8.02 −4.18 4.18 −8.22 8.22 −7.25 7.25 Phase Min −52.42 34.16 31.03 −51.45 −4.19 −4.02 −14.24 −11.79 Max −31.51 54.64 49.47 −27.64 4.24 5.99 9.63 9.86 Mean −41.07 42.07 38.10 −36.59 −0.13 2.15 −1.07 −0.56 (deg) SD 4.154 3.875 4.287 4.587 1.875 1.747 4.393 4.692 Ideal −43.20 43.20 36.00 −36.00 0.00 0.00 0.00 0.00 105 101103 −60 −40 −20 0 20 Frequency (rad/s) Magnitude (dB) 105 101103 −40 −20 0 20 105 101103 −30 −20 −10 0 105 101103 −10 −5 0 5 (a) 105 101103 −100 −50 0 Frequency (rad/s) Phase (deg) 105 101103 0 50 100 −25 105 101103 −70 0 70 105 101103 −40 −20 0 20 40 (b) Figure 10. SPICE simulated ( a ) magnitude and ( b ) phase responses of the proposed FLPF, FHPF, FBPF, and FBSF. [Note: Theoretical responses are shown in dashed red; responses based on nominal values of components are shown in solid black; Monte-Carlo simulation-based responses are shown in green]. 3.2.2. Experimental Validation The hardware circuit realizations of the proposed FBPF and FIBPF, whose transfer functions are defined by (14) and (15) , respectively, are demonstrated as representative cases in this section. The commercial Analog Devices AD844AN-type CFOAs were employed as the active elements. The ICs were provided with the supply voltage from the Agilent E3630A power supply. The OMICRON Lab Bode 100 network analyzer was used to measure the frequency response (magnitude and phase) of the practical filter circuits. The level of the testing harmonic signal was set to 1 V and 100 mV (peak-to-peak values) for the FBPF and FIBPF circuits, respectively. An Agilent InfiniiVision DSO-X 2002A digital storage oscilloscope was used to observe the time-domain response of the filters. The FBPF and FIBPF circuits were subjected to a peak-to-peak signal of 1 V and 100 mV, respectively, from the Agilent 33521A function/arbitrary waveform generator. The photograph of the experimental set-up for the FIBPF is illustrated in Figure 13. Fractal Fract. 2022,6, 209 18 of 25 105 101103 0 25 50 −10 Frequency (rad/s) Magnitude (dB) 105 101103 0 20 40 −10 105 101103 0 10 20 30 105 101103 −5 0 5 10 (a) 105 101103 0 50 100 −20 Frequency (rad/s) Phase (deg) 105 101103 −100 −50 0 20 105 101103 −70 0 70 105 101103 −40 −20 0 20 40 (b) Figure 11. SPICE simulated ( a ) magnitude and ( b ) phase responses of the proposed FILPF, FIHPF, FIBPF, and FIBSF. [Note: Theoretical responses are shown in dashed red; responses based on nominal values of components are shown in solid black; Monte-Carlo simulation-based responses are shown in green]. F L P F F I L P F F H P F F I H P F F B P F F I B P F F B S F F I B S F - 4 0 - 3 0 - 2 0 - 1 00 1 0 M a x M e a n A R M E ( d B ) ( 0 . 6 , 0 . 8 ) ( 0 . 6 , 0 . 8 ) ( 0 . 8 , 0 . 5 ) ( 0 . 8 , 0 . 5 ) ( 0 . 6 5 , 0 . 8 5 ) ( 0 . 6 5 , 0 . 8 5 ) ( 0 . 7 5 , 0 . 6 5 ) ( 0 . 7 5 , 0 . 6 5 ) - 2 4 - 2 2 - 2 0 - 1 8 - 1 6 - 1 4 (a) F L P F F I L P F F H P F F I H P F F B P F F I B P F F B S F F I B S F - 3 0 - 2 0 - 1 00 1 0 2 0 M a x M e a n A R P E ( d B ) ( 0 . 6 , 0 . 8 ) ( 0 . 6 , 0 . 8 ) ( 0 . 8 , 0 . 5 ) ( 0 . 8 , 0 . 5 ) ( 0 . 6 5 , 0 . 8 5 ) ( 0 . 6 5 , 0 . 8 5 ) ( 0 . 7 5 , 0 . 6 5 ) ( 0 . 7 5 , 0 . 6 5 ) - 2 4 - 2 2 - 2 0 - 1 8 - 1 6 - 1 4 (b) Figure 12. Broken line plots of the ( a ) ARME and ( b ) ARPE responses for the SPICE simulated proposed FLPF, FHPF, FBPF, FBSF, and their inverse counterparts realized using the nominal values of components. Fractal Fract. 2022,6, 209 19 of 25 Figure 13. Photograph of the experimental set-up. 101102103104105 −30 −20 −10 0 Frequency (rad/s) Magnitude (dB) Theoretical Measurement 101102103104105 −50 0 50 Frequency (rad/s) Phase (deg) (a) 101102103104105 0 10 20 30 Frequency (rad/s) Magnitude (dB) Theoretical Measurement 101102103104105 −50 0 50 Frequency (rad/s) Phase (deg) (b) Figure 14. Magnitude and phase responses of the proposed ( a ) FBPF and ( b ) FIBPF based on experimental measurements. The experimentally measured magnitude and phase responses of the FBPF and FIBPF are presented in Figure 14a,b, respectively. Comparisons with the theoretical characteristics reveal that: (i) the {max ARME (dB), mean ARME (dB), max ARPE (dB), and mean ARPE (dB)} values for the FBPF and FIBPF are { − 15.19, − 27.20, 5.82, − 22.53} and { − 7.28, − 22.07, 10.54, − 13.37}, respectively; (ii) the responses of the FBPF stay in proximity to the theoretical plots throughout the design range; (iii) for the FIBPF, the magnitude and phase responses deviate from the theoretical ones in the frequency range [10, 48.49] rad/s and [10, 129] rad/s, respectively; (iv) the magnitude of the FBPF at the center frequency ( ω0 = 952.53 rad/s), lower half-power frequency ( ωH,low = 164.18 rad/s), and upper half-power frequency ( ωH,high = 5.52 krad/s) is − 8.55 dB, − 11.61 dB, and − 11.61 dB, respectively. The corresponding theoretical magnitude values at these frequencies are − 8.22 dB, − 11.25 dB, and − 10.96 dB; and (v) the magnitude at ω0 (=912.44 rad/s), ωH,low (=171.53 rad/s), and ωH,high (=5.26 krad/s) for the theoretical FIBPF is 8.23 dB, 11.12 dB, and 10.81 dB, respectively. The corresponding measured values from the practical filter are 8.14 dB, 10.96 dB, and 11.13 dB, which demonstrate closeness with the theoretical anticipations. Figure 15a–c present the time–domain response measurements of the FBPF when the input signal frequency is ω0 , ωH,low , and ωH,high , respectively. The peak-to-peak output voltage ( VOUT,P-P ) for these considered cases is obtained as 370 mV, 260 mV, and 260 mV, Fractal Fract. 2022,6, 209 20 of 25 respectively, which matches closely with the theoretical values of 384.27 mV, 271.10 mV, and 280.30 mV. The time-domain waveforms of the FIBPF for these same frequencies are illustrated in Figure 16a–c. It is found that VOUT,P-P of 258 mV, 357 mV, and 364 mV is yielded by the practical filter, which agrees with the theoretical values of 260.51 mV, 363.34 mV, and 350.61 mV, at the excitation frequencies ω0,ωH,low, and ωH,high, respectively. VIN VOUT (a) VOUT VIN (b) VOUT VIN (c) Figure 15. Time-domain input-output waveforms observed in oscilloscope for the proposed FBPF with an input frequency of ( a ) ω0= 952.53 rad/s, ( b ) ωH,low = 164.18 rad/s, and ( c ) ωH,high =5.52 krad/s. Fractal Fract. 2022,6, 209 21 of 25 VIN VOUT (a) VIN VOUT (b) VIN VOUT (c) Figure 16. Time-domain input-output waveforms observed in oscilloscope for the proposed FIBPF with an input frequency of ( a ) ω0= 912.44 rad/s, ( b ) ωH,low = 171.53 rad/s, and ( c ) ωH,high =5.26 krad/s. Fractal Fract. 2022,6, 209 22 of 25 In Figure 17a–c, the Fast Fourier Transform spectrum measurements displayed up to the sixth harmonic above − 92 dBV of the FBPF at ω0 , ωH,low , and ωH,high , are presented. The Spurious-Free Dynamic Range (SFDR) for these three cases are obtained as 54.80 dBc, 51.12 dBc, and 54.63 dBc, respectively; the Total Harmonic Distortion (THD) is determined from the first six harmonics as 0.25%, 0.34%, and 0.29%, respectively. The experimentally obtained Fourier spectrums up to the sixth harmonic above − 92 dBV for the FIBPF at frequencies of ω0 , ωH,low , and ωH,high are shown in Figure 18a–c. Experimental results reveal that the {SFDR, THD} for these three cases are {68.40 dBc, 0.06%}, {58.40 dBc, 0.17%}, and {57.73 dBc, 0.16%}, respectively. 0 . 0 3.0x103 6.0x103 - 1 2 0 - 9 0 - 6 0 - 3 0 0 - 1 8 . 5 0 - 7 8 . 8 27 3 . 3 0 - 8 0 . 6 87 6 . 4 9 F B P F - w 0: T H D = 0 . 2 5 % VO U T ( d B V ) F r e q u e n c y ( r a d / s ) ( a ) S F D R = 54.80 dBc (a) 0 . 0 5.0x102 1 0 3 - 1 2 0 - 9 0 - 6 0 - 3 0 0 - 2 1 . 4 1 - 8 6 . 5 2 - 7 2 . 5 3 - 7 6 . 9 9 - 8 3 . 2 8 F B P F - w H , l o w : T H D = 0 . 3 4 % VO U T ( d B V ) F r e q u e n c y ( r a d / s ) ( b ) S F D R = 51.12 dBc (b) 0 . 0 1.8x104 3.6x104 - 1 2 0 - 9 0 - 6 0 - 3 0 0 - 2 1 . 4 7 - 8 2 . 5 17 6 . 0 9 - 7 6 . 7 0 - 8 1 . 8 1 - 9 0 . 4 6 F B P F - w H , h i g h : T H D = 0 . 2 9 % VO U T ( d B V ) F r e q u e n c y ( r a d / s ) ( c ) S F D R = 54.63 dBc (c) Figure 17. Experimentally obtained Fourier spectrums of the proposed FBPF at frequency of ( a ) ω0=952.53 rad/s, (b)ωH,low =164.18 rad/s, and (c)ωH,high =5.52 krad/s. 0 . 0 3.0x103 6.0x103 - 1 2 0 - 9 0 - 6 0 - 3 0 0 - 2 1 . 9 2 - 9 0 . 3 2 F I B P F - w 0: T H D = 0 . 0 6 % VO U T ( d B V ) F r e q u e n c y ( r a d / s ) ( d ) S F D R = 68.40 dBc (a) 0 . 0 6.0x102 1.2x103 - 1 2 0 - 9 0 - 6 0 - 3 0 0 - 1 8 . 7 5 - 7 8 . 0 1 - 7 7 . 1 5 - 8 7 . 6 1 - 8 5 . 3 6 F I B P F - w H , l o w : T H D = 0 . 1 7 % VO U T ( d B V ) F r e q u e n c y ( r a d / s ) ( e ) S F D R = 58.40 dBc (b) 0 . 0 1.8x104 3.6x104 - 1 2 0 - 9 0 - 6 0 - 3 0 0 - 1 8 . 7 8 - 8 2 . 0 77 6 . 5 1 - 8 7 . 3 08 3 . 8 1 F I B P F - w H , h i g h : T H D = 0 . 1 6 % VO U T ( d B V ) F r e q u e n c y ( r a d / s ) ( f ) S F D R = 57.73 dBc (c) Figure 18. Experimentally obtained Fourier spectrums of the proposed FIBPF at frequency of ( a ) ω0=912.44 rad/s, (b)ωH,low =171.53 rad/s, and (c)ωH,high =5.26 krad/s. 4. Conclusions Further generalization of the fractional-order filters exhibiting the low-pass, high-pass, band-pass, and band-stop behavior of the second-order limiting form is presented along with their optimal and stable rational approximation in this paper. It is demonstrated that the power-law filters [ 20 – 23 ] can be treated as a particular case of the proposed filters, since the proposed model introduces an additional degree-of-freedom (viz. the new tuning parameter α ) in the transfer function of the power-law filter. A different approach is adopted to formulate the design constraints compared to the published literature [ 21 ]. The proposed strategy also allows the attainment of stable inverse filters since the zeros and poles of the approximant pertaining to the standard filter are constrained to lie in the left-half s-plane. The performance of the proposed models in approximating the frequency-domain characteristics of the theoretical filter is investigated using different error indices. Cur- Fractal Fract. 2022,6, 209 23 of 25 rent feedback operational amplifiers are employed as active components to realize the discrete components based circuits for the proposed filters and their inverse counterparts. Monte-Carlo simulations conducted in SPICE environment highlight good agreement in the magnitude and phase responses of the designed models with the theory. Hardware implementations of the proposed fractional-order band-pass (both normal and inverse) filters and their magnitude-frequency, phase-frequency, AC transients, and Fourier analysis are also presented to demonstrate the practical viability. Future work will investigate the effectiveness of the proposed transfer function in improving the design performances of the power law compensator [ 45 ] and the bioimpedance models [46]. Author Contributions: Conceptualization, S.M. and N.H.; methodology, S.M.; software, S.M.; validation, S.M., N.H., and D.K.; formal analysis, S.M.; investigation, S.M.; resources, N.H.; data curation, S.M.; writing—original draft preparation, S.M. and N.H.; writing—review and editing, S.M., N.H., and D.K.; visualization, S.M. and N.H.; supervision, N.H.; project administration, N.H.; funding acquisition, N.H. All authors have read and agreed to the published version of the manuscript. Funding: The research results described in this paper are supported by The Czech Science Foundation, project No. 19-24585S. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: All the data are presented in the paper. Conflicts of Interest: The authors declare no conflicts of interest. Abbreviations The following abbreviations are used in this manuscript: ARME Absolute relative magnitude error ARPE Absolute relative phase error BP Band-pass BS Band-stop CFOA Current feedback operational amplifier C2oDE Constrained composite differential evolution FBPF Fractional-order band-pass filter FBSF Fractional-order band-stop filter FHPF Fractional-order high-pass filter FIBPF Fractional-order inverse band-pass filter FIBSF Fractional-order inverse band-stop filter FIHPF Fractional-order inverse high-pass filter FILPF Fractional-order inverse low-pass filter FLPF Fractional-order low-pass filter FO Fractional-order FTF Fractional-order transfer function HP High-pass ITF Integer-order transfer function LP Low-pass PLF Power-law filter SD Standard deviation SFDR Spurious-free dynamic range SPICE Simulation program with integrated circuit emphasis THD Total harmonic distortion Fractal Fract. 2022,6, 209 24 of 25 References 1. 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