A Fractional-Order Transitional Butterworth-Butterworth Filter and Its Experimental Validation

This paper introduces the generalization of the classical Transitional Butterworth-Butterworth Filter (TBBF) to the Fractional-Order (FO) domain. Stable rational approximants of the FO-TBBF are optimally realized. Several design examples demonstrate the robustness and modeling efficacy of the proposed method. Practical circuit implementation using the current feedback operational amplifier employed as an active element is presented. Experimental results endorse good agreement (<inline-formula> <tex-math notation="LaTeX">$\text {R}^{2} = 0.999968$ </tex-math></inline-formula>) with the theoretical magnitude-frequency characteristic.


I. INTRODUCTION
The modeling techniques and realization of classical (integerorder) analog filters are well-established. To further improve the performance of such filters (e.g., reduction in passband error, sharper transition-band characteristic), the use of graphical methods [1] and optimal procedures [2]- [4] have been adopted.
Recently, the theoretical concept of fractional calculus, which deals with the generalization of the classical definitions of differentiation and integration, has been applied to achieve a more precise attenuation behavior of analog filters [5]. This is possible due to the generalization of the classical Laplacian operator s to the Fractional-Order (FO) form s α , where α ∈ (0, 1), which causes additional degrees of freedom in system modeling. The impedance function containing the s α operator may be realized using fractance devices or Constant Phase Elements (CPE) [6]. Due to the commercial unavailability of these devices, CPE emulators in the integrated form [7] or discrete-components-based [8] have been reported. The s α operator forms the basic building The associate editor coordinating the review of this manuscript and approving it for publication was Qi Zhou. block of the FO transfer functions, which can lead to generalizations of classical Butterworth filter [9], oscillators [10], and resonators [11]. Both active and passive elements have been employed to realize the FO impedances [12], [13]. Another popular method is to approximate the FO system using the integer-order transfer function [14]. The exact dynamics of a FO system can be theoretically achieved by a system of infinite integer order. For practical purposes, the characteristics of the FO filter need to be approximated using a finite-order rational approximant. An integer-order model of lower-order is desirable since it results in smaller hardware overhead. The rational approximation of s α may be achieved using frequency-domain-based curve fitting [15], a weighted sum of first-order optimal high-pass filter sections [16], etc.
Transitional filters merge the frequency responses of various classical filters (e.g., Butterworth, Chebyshev, Bessel, Legendre, Thomson) to attain conciliation between the amplitude and group delay characteristics [17]. Transitional filters may be designed by combining different filter poles using the arithmetic or geometric interpolation, as exemplified by the transitional Legendre-Thomson filter [18], and the transitional ultraspherical-ultraspherical filter [19]. An alternative design technique involves combining the classical filter polynomials [20]. The magnitude squared function of the classical Transitional Butterworth-Butterworth Filter (TBBF) is given by (1) [20]: where n and k are integers, 0 ≤ k ≤ n; ε is the ripple constant; and ω is the angular frequency in radians per second (rad/s). For n = k, and rewriting the ripple constant as ε/ √ 2, the magnitude characteristics of the n th order Butterworth filter can be also obtained from (1). The response of the TBBF comprises the arithmetic interpolation between two classical Butterworth filters. It may be inferred from (1) that for n > k, the dominating responses in the passband and stopband regions are due to the k th order and the n th order Butterworth filters, respectively. Hence, the passband and stopband responses of the TBBF can be nearly independently adjusted.
Optimization techniques were employed to approximate the characteristics of the FO Butterworth Filter (FOBF) [21], [22]. However, to the best of the authors' knowledge, no literature exists on the FO modeling of TBBFs. This paper introduces the definition of FO-TBBF characteristic by removing the restrictions of integer values for n and k imposed in (1). Optimal rational approximations are proposed, which can meet the theoretical magnitude-frequency behavior of the FO-TBBF. Design stability is ensured by representing the denominator polynomial of the proposed model as a cascade of first-order and second-order terms comprising positive coefficients. Thus, inequality constraints are avoided to meet the s-domain stability criteria. Table 1 compares the advantages and limitations of the proposed method with those of the FOBF [9], [21], [22], and TBBF [20] design techniques. Several design cases are considered to evaluate the performance of the proposed technique. Current Feedback Operational Amplifier (CFOA) [23] based hardware circuit implementation of the proposed FO-TBBF approximant is demonstrated. Simulation and experimental results confirm excellent agreement with the ideal magnitude characteristics.
In the rest of the paper, the proposed technique is presented in Section II. MATLAB simulations are carried out to highlight the modeling efficiency in Section III. Section IV presents the circuit implementation and measurement results, while conclusions are drawn in Section V.

II. DESIGN METHODOLOGY A. DEFINITION
The theoretical squared-magnitude function for the FO-TBBF is proposed according to (2): where n 1 and n 2 are integer numbers, α, β ∈ [0, 1], and (n 1 + α) ≥ (n 2 + β). For α = β = 0 and 1, the TBBF can be treated as a special case of the FO-TBBF. Note that (2) may yield the definition of a FOBF when (n 1 +α) = (n 2 +β). The proposed definition also allows the exponents of ω in (2) to attain any value between 0 and 2, which is not possible using the classical TBBF.

B. PROPOSED TECHNIQUE
The proposed FO-TBBF approximant G(s) is modeled according to (3): if n 1 is odd; , if n 1 is even.
The resulting integer order of the approximant G(s), as defined by (3), is determined as n 1 + 3. The approximation of the magnitude-frequency response of the normalized FO-TBBF is formulated as an optimization problem by minimizing the cost function f, as proposed in (4): Subject to: if n 1 is odd; where L denotes the total number of frequency points logarithmically distributed in the interval ω ∈ [ω min , ω max ] rad/s; and X represents the vector of decision variables. For odd values of if n 1 is even, then The constraints can be satisfied by choosing a positive value for the lower bound of the decision variables. Hence, the proposed optimization problem can be solved using any unconstrained global search optimization technique.

C. ALGORITHM IMPLEMENTATION
Algorithm 1 presents the pseudocode to implement the proposed optimization routine for a single trial run. In order to guarantee the generation of a stable rational approximant, the lower bound (Lb) for all decision variables (except k) is set as 10 −4 ; in the case of k, Lb may be fixed as 0. A large value of upper bound (Ub) needs to be avoided since it may result in a large dispersion of the decision variables. A wide variation in the coefficients of the FO-TBBF transfer function will lead to larger spreading (ranging from a few ohms to several mega-ohms) in the values of passive components, which is undesirable for the practical implementation. To attain the passive components values within practical limits, Ub may be chosen as 1000. The initial point is randomly varied iter times between Lb and (Lb + c), where c ∈ Z + . A single trial run of the optimization algorithm generates an iter number of solutions; the best solution (X best ) is the one that attains the smallest value of the error fitness function (f min ). Thirty independent trial runs of the algorithm are executed for each design case to identify the most accurate model.

III. SIMULATION RESULTS
The MATLAB based optimization routine uses the solver fmincon (algorithm: 'active-set') with the following arguments: MaxFunEvals = 50000; MaxIter = 5000; TolFun = 1E-10; and TolX = 1E-10. The optimal values of the decision variables for 15 design examples, with [ω min , ω max ] = [10 −2 , 10 2 ] rad/s, ε 2 = 0.5, L = 50, iter = 100, c = 10, and Ub = 1000, are presented in Table 2. To quantify the effectiveness of the modeling accuracy, the coefficient of determination (R 2 ) index (evaluated for L magnitude-frequency data sample points) is also shown in Table 2 Set ω min , ω max , L, iter, c, Lb, Ub Minimize (4) and store f i 15 Store X best ← X i corresponding to f min other designs achieve R 2 > 0.9999, which highlights a good agreement in the magnitude responses of the optimal model with the ideal FO-TBBF. The proposed method can also attain the same solution quality for other values of c, such as 100 and 1000. Table 3 presents the minimum (min), maximum (max), mean, and standard deviation (SD) indices of f min for all considered design cases based on 30 runs. Out of the 15 examples, 13 cases yield the same fitness values for min, max, and mean indices. This implies that the same solution quality is obtained irrespective of the number of independent trial runs of the optimization technique. The excellent robustness of the proposed technique is further highlighted by the small value of the SD index.
The magnitude plot of the proposed model for no. 1 [(n 1 , n 2 , α, β) = (0, 0, 0.8, 0.5)] attains agreement with the    [20]. Group delay comparisons of the proposed approximant for no. 5 with the classical TBBFs cited in [20] for (n = 2, k = 1) and (n = 3, k = 2) are shown in Figure 2 (bottom). Results reveal that the group delay behavior of the optimal model lies in-between the responses of the classical filters.
The effectiveness of the proposed models in attaining a smaller group delay in the passband as compared to the FOBF is also demonstrated. For this purpose, the magnitude (top) and group delay (bottom) responses of the 1.6 th -order FOBF reported in [22] are compared with the proposed FO-TBBF model no. 2 [(n 1 , n 2 , α, β) = (1, 0, 0.6, 0.8)], as presented in Figure 4. It may be noted that since the stopband attenuation characteristic for the FO-TBBF is dominated by the (n 1 + α) th -order Butterworth filter, hence, the magnitude roll-off rate for the proposed FO-TBBF is similar to that of the FOBF of order 1.6. The magnitudes of the FO-TBBF at the frequencies of 10 rad/s and 100 rad/s are −29.06 dB and −61.08 dB, respectively. Therefore, the roll-off rate for the FO-TBBF is −32.02 decibel/decade (dB/dec), which is close to the theoretical value of −32.0 dB/dec obtained for  [22]. Note that the numbers within the parenthesis represent (n 1 , n 2 , α, β). the 1.6 th -order FOBF. However, the maximum group delay achieved for the proposed FO-TBBF (1.217 s) is substantially smaller as compared to the reported FOBF model [22] (1.392 s). This is due to the fact that the dominating response in the passband for the FO-TBBF depends on the (n 2 + β) thorder Butterworth filter, which is of order 0.8 in the present case. Thus, the proposed design achieves an improved group delay performance as compared to the FOBF without compromising the stopband behavior.

IV. CIRCUIT IMPLEMENTATION AND EXPERIMENTAL VERIFICATION
The circuit realization of the FO-TBBF approximants is demonstrated using the CFOA employed in a follow-theleader feedback topology [24]. The complete circuit can be constructed using the nodal connections shown in Figure 5. For e.g., nodes P and X are respectively represented as P and X in the figure. Source input and signal output voltages of the circuit are denoted by V I and V O , respectively. The series connections of the CFOAs between the nodes P and P+1 are carried out for P varying from 1 to N − 1, where N = n 1 + 3. The total number of amplifiers, resistors, and capacitors needed to realize the FO-TBBF model are N + 1, 2N + 4, and N , respectively. Thus, 3N + 4 number of design variables is required to realize the circuit. The transfer function of the proposed circuit is given by (5): The values of the R-C components are determined by comparing (3) and (5), which results in (N + 3) independent equations. Hence, the values for (2N + 1) number of passive components can be initially chosen. As a representative, the circuit realization steps of the proposed FO-TBBF model no. 5 for (n 1 , n 2 , α, β) = (2, 1, 0.5, 0.5) are presented as follows: (i) Set n 1 = 2. Therefore, N = 5. The circuit comprises 6 CFOAs, 14 resistors, and 5 capacitors. The value of P is incremented from 1 to 4, (ii) Set the desired cut-off frequency of the filter, such as f c =1 kHz, (iii) Eight modeling equations relate the circuit transfer function to the coefficients of G(s). The number of design variables is 19. Therefore, the values of 11 passive elements are initially set, (iv) The passive components are selected from the E-24 standard industrial series for the resistors and the E-12 series for the capacitors. The following resistor values are set as: The values of the other R-C components are derived as follows: R G 6 = 8.2 k , C 1 = 0.12 nF, C 2 = 10 nF, C 3 = 4.7 nF, C 4 = 15 nF, C 5 = 27 nF, R 2 = 7.5 k , and R 3 = 750 . The circuit for the proposed FO-TBBF no. 5 was assembled on a breadboard using the above listed R-C component values. The supply voltage for Analog Devices AD844AN amplifiers was provided by the Agilent E3630A power supply. The frequency responses of the FO-TBBF were measured by the OMICRON Lab Bode 100 network analyzer. 401 logarithmically spaced frequency points in the range 10 Hz to 100 kHz were considered. The level of the testing harmonic signal was set to 10 dBm (0.7071 V RMS ). The receiver bandwidth of the analyzer was fixed at 10 Hz to obtain precise results. The THRU calibration of the analyzer was performed before the measurement to eliminate the influence of the measurement setup. After connecting the proposed FO-TBBF circuit to the analyzer, the frequency responses were measured and displayed by the connected computer with the Bode Analyzer Suite software. The photograph of the hardware setup is presented in Figure 6.  The magnitude-frequency response measurements of the proposed FO-TBBF are compared with the ideal and simulated ones in Figure 7 (top). The practical filter demonstrates excellent agreement with the ideal characteristic up to nearly 70 kHz. The magnitude of the approximant at f c = 1 kHz for measurement (-3.100 dB) demonstrates conformity with the ideal (-3.010 dB) and MATLAB simulations (-3.029 dB). R 2 of 0.999968 is achieved for the measured magnitude response data compared to the theoretical one. Figure 7 also depicts the experimental results for the phase (middle) and group delay frequency responses (bottom) of the FO-TBBF. Comparisons with the simulated plots highlight excellent matching of the phase plot for nearly 3 decades and the group delay for the entire design bandwidth.

V. CONCLUSION
Optimal and robust modeling of several frequency characteristics for the FO-TBBF is introduced. The generalization of the classical TBBF results in more precise control of the magnitude, phase, and group delay behaviors. The efficient modeling performance of the proposed technique is validated through numerical simulations and experiments made on CFOA-based circuit implementation.