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

: 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 fractional-order band-pass ﬁlter and its inverse counterpart are also presented.


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 α for a function f (t) is given by (1) [3].
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 1.
Further generalization of the FO filter of the second-order limiting form is proposed. It is demonstrated that the PLFs reported in [20][21][22][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;

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 2 along 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.

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].
where a, b, c, d, and h are constant coefficients. H(s) is always stable provided the conditions a 2 ≥ 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/(s 2α + 2as α + b) was obtained using a multi-objective optimization routine [25].
A new class of FO filter, as given by (4), can be proposed by introducing a FO exponent β in the filter transfer function defined in (3): (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 H α,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 1 shows 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 α in (3).

Proposed Technique
Defining an ITF as per (5), where N is 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 H α,β 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 H α,β,N P (s) lie strictly on the left-half s-plane.
Subject to: Real part of roots of A(s) and B(s) < 0.
where X denotes the vector of decision variables, i.e., L denotes the number of log-spaced sample points in the bandwidth [ω min , ω max ] rad/s; and the dimension (D) of the problem is 2N + 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 (ω min ), and upper limit (ω max ) of the desired bandwidth are the user-defined inputs to the optimization routine. The metaheuristic search procedure of C 2 oDE determines the best feasible solution (X * ), i.e., the vector of decision variables (coefficients of H α,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 H  [36][37][38][39][40][41]. Henceforth, the FO inverse LP, HP, BP, and BS filters will be abbreviated as FILPF, FIHPF, FIBPF, and FIBSF, respectively.

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 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, ω min = 10 −2 rad/s, ω max = 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:

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 N are 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.  The effect due to variations of N on the accuracy of the proposed FLPFs is presented in Table 3 by 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 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 N increases; 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.  The magnitude (M CT ) 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 (θ CT ) 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 M C and θ C , 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 M CT and θ CT are denoted by ω M and ω θ , 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 N is 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.

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. 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 {M CT , θ CT } 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 M C , θ C , ω M , and ω θ parameters attained by the designed approximants for these cases are shown in Table 5, which highlights an improved proximity with the theoretical values for an increasing N. Figure 6a,b present the magnitude and phase plots, respectively, of both the optimal FHPFs and their inverse functions, for N = 4. The frequency-domain characteristics of the approximants closely match with those of the theoretical responses.
However, cancellation of a pair of pole-zero occurs for odd values of N of the designed FBPFs. For example, the zeros and poles of the FBPF for (α, β, N) = (0.65, 0.85, 5) are located at s = {-0.0048, -0.0792, -1.0000, -12.6216, -210.4950} and s = {-0.0233, -0.1947, -1.0000, -5.1365, -42.8770}, respectively. Therefore, cancellation of the pole-zero pair occurring at s = −1 converts the fifth-order FBPF into a fourth-order one. Hence, it may be inferred that the design of FBPFs for even values of N is only appropriate. This issue of order truncation is also reflected in the modeling performance, as shown in Table 7, where the ARME and ARPE values for N = 5 and 7 are close to those obtained for N = 4 and 6, respectively. However, a large improvement in accuracy is exhibited between two even values of N. The M CT values of the theoretical FBPF for (α, β) = (0.65, 0.85) and (α, β) = (0.7, 0.4) are −8.221 dB and −3.710 dB, respectively, whereas, θ CT = 0 • for both the cases. The bandwidth (BW) of the theoretical filter, which represents the difference between the upper and lower half-power frequencies, is 5.858 rad/s and 12.289 rad/s for (α, β) = (0.65, 0.85) and (0.7, 0.4), respectively. The M C , θ C , and BW yielded by the proposed approximants for various values of N are presented in Table 7. It is found that (i) θ C for all the cases is close to the theoretical value; (ii) the error index δ M = |M CT − M C | reduces as N (considering the even values) is increased. For example, δ M attained by the proposed FBPF with (α, β) = (0.7, 0.4) are 0.712 dB and 0.014 dB for N = 4 and N = 6, respectively; and (iii) the difference in the bandwidth between the theoretical and proposed models also reduces with an increased design order. For instance, the BW yielded by the proposed FBPF for the case (α, β) = (0.65, 0.85) with N = 4 is 6.073 rad/s, whereas the same for N = 6 is obtained as 5.835 rad/s. These findings indicate an improved accuracy for the sixth-order approximant with respect to the theoretical anticipation (BW = 5.858 rad/s). Figure 7a,b show the magnitude and phase plots of the proposed fourth-order FBPFs and their corresponding inverse transfer functions. Good agreement with the theoretical responses is obtained for both the test cases.    Hence, designs for only even values of N are presented here. The improvement in modeling accuracy with increasing N is justified using the max and mean values of ARME and ARPE, as presented in Table 9.
The magnitude and phase plots of the fourth-order FBSFs and FIBSFs are illustrated in Figure 8a,b, respectively. The responses of the proposed approximants remain in proximity with the theoretical characteristics throughout the design bandwidth.

Comparison with the Literature
Comparisons regarding the modeling accuracy for the designed filters with the PLFs reported in the literature [20] are carried out. For demonstration purposes, the value of β is fixed as 0.3, whereas, two different values of α (viz., 0.7 and 1), are considered for each type of filter. The designs reported in the published literature can model (4) for only α = 1. Hence, such a method is not applicable for the design of generalized PLFs (where fractional value of α such as 0.7 occurs); in contrast, the proposed method exhibits no such limitation. For comparison purposes, the values of quality factor and pole frequency for the PLFs reported in [20] are chosen as 0.5 and 1 rad/s, respectively. Therefore, the coefficients of the theoretical PLF transfer function in [20] are the same as that of the values of the filter coefficients a, b, c, d, and h chosen in Sections 3.1.1-3.1.4. The coefficients of the FLPF, FHPF, FBPF, and FBSF approximants based on the proposed method and the cited literature are presented in Table 10. Results presented in Table 11 show that the proposed filters attain comparable or lower ARME and ARPE values than the designs published in the literature for α = 1. Practical implementations of FO filters using integrated forms [18,22], discrete components [5,13], and field-programmable analog arrays [23,43] have been reported. The use of CFOAs for electronic filter realization has gained prominence due to several reasons, such as improved gain-bandwidth product, lower component count, etc. [44]. Previous works on the applicability of CFOAs for FO filter implementation can be found in [21,39]. The CFOA-based circuit topology reported in [21] can be used to realize the proposed filters and their inverse counterparts. The circuit (see Figure 9) and its transfer function are repeated here for the sake of completeness. Figure 9. CFOA-based circuit to realize the proposed filters.
The transfer functions of the proposed filters, as given by (10) The component values to realize the proposed filters, as shown in Table 12, considering a shift frequency of 1000 rad/s, are chosen from the industrial E24 and E12 series for the resistors and capacitors, respectively. The actual values of R 1 and R 5 for the FLPF and  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.

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.   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 ( Figure 15a-c present the time-domain response measurements of the FBPF when the input signal frequency is ω 0 , ω H,low , and ω H,high , respectively. The peak-to-peak output voltage (V OUT,P-P ) for these considered cases is obtained as 370 mV, 260 mV, and 260 mV, 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 V OUT,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 ω 0 , ω H,low , and ω H,high , respectively.  In Figure 17a-c, the Fast Fourier Transform spectrum measurements displayed up to the sixth harmonic above −92 dBV of the FBPF at ω 0 , ω H,low , and ω H,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 ω 0 , ω H,low , and ω H,high are shown in Figure 18a-

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][21][22][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-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]. 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: