Design of Higher-Order Fractional Filters With Fully Controllable Frequency Characteristics

Higher-order fractional filters with fully controllable frequency characteristics are realized in this work, after fitting the filter’s magnitude response data using a minimum-phase state-space model. Subsequently, rational integer-order transfer functions are derived and implemented using as active elements: a) Operational Transconductance Amplifiers (OTAs) and b) a Field Programmable Analog Array (FPAA) device. The realized filters enjoy electronic adjustability of their type, order, and characteristic frequencies while being easily validated on the digitally programmable FPAA platform.


I. INTRODUCTION
Fractional-order filtering is an attractive technique for performing fine adjustment of the slope of the transition from the pass-band to the stop-band of the filter, as well as for performing scaling of the corresponding time-constants [1]. The former feature originates from the fact that the slope is expressed by the formula ±20(n + α) dB/dec., with n being the integer part of the order and 0 < α < 1 being its noninteger part. The latter feature is the result of the dependence of the filter's cutoff frequency on both the pole frequency and the order.
In the literature, fractional-order filters of orders in the range ]0,1[ have been developed through the following ways: 1) on the circuit level by substitution of capacitors in the conventional integer-order filters with RC networks (e.g. Foster or Cauer types) approximating the behavior of the required fractional-order capacitors [2], or The associate editor coordinating the review of this manuscript and approving it for publication was Venkata Rajesh Pamula .
2) on the system level by employing suitable approximation tools (e.g. Oustaloup, continued fraction expansion, etc.) for approximating the Laplacian operator s α = (jω) α , used for the transposition of the integer-order transfer functions to the fractional-order domain. The first procedure is straightforward, in the sense that the derivation of the circuitry is a one-step procedure and the calculation of the passive element values can be easily performed via a systematic way [3]. The second one is slightly more complicated, because additional algebraic operations are required for deriving the rational integer-order transfer function. On the other hand, it offers design flexibility because the designer can freely choose between multiple filter design and implementation techniques [4].
In the present work, a procedure for deriving the approximation transfer functions, which realize the pre-determined frequency characteristics of the fractional-order prototype filters of order greater than two, is introduced. It is based on the fit of the frequency response magnitude data by a minimumphase state-space model, using Log-Chebyshev magnitude design. This means that all well-known filters types, such as Butterworth and Chebyshev, can be implemented in any standard form (e.g. low-pass, high-pass or band-pass). Here, the implementation of the resulting transfer functions is performed in two different ways: (i) using Operational Transconductance Amplifiers (OTAs) as active elements, and (ii) using a Field Programmable Analog Array (FPAA) device. In both cases, full controllability of the frequency characteristics is achieved without altering the core structure that implements the integer-order approximation functions, hence offering design flexibility and versatility. The paper is organized as follows: detailed overview of the relevant literature is performed in Section II, while the proposed procedure is also introduced. Possible implementations are demonstrated in Section III, and their behavior is evaluated through simulation and experimental results in Section IV.

A. BACKGROUND
The expression of the magnitude response of a Butterworth low-pass transfer function of order (n + α), with n ∈ N and 0 < α < 1, is given by where the maximum gain is equal to one, and the half-power frequency (or equivalently the −3 dB cutoff frequency) is equal to ω 0 . The first attempt for implementing fractionalorder Butterworth filters of order (n + α) > 2 was reported in [5], [6], where the (approximated) transfer function of order (1 + α) Butterworth low-pass filter is divided by an (n − 1) order Butterworth polynomial, in order to realize an (n + α) order filter. The work in [22] introduces a new technique to optimally design the fractional-order Butterworth low-pass filters in the complex F-plane using the constrained composite differential evolution (C 2 oDE) algorithm. The presented 1.5 order fractional low-pass filter realization is performed by substituting the capacitors in the well-known Sallen-Key circuit by fractional-order ones, approximated by the Valsa network [28]. The work in [25] describes the design of analog pseudo-differential fractional Butterworth filters of the order (2+α) operating in a mixed-transadmittance mode (voltage input, current output) and was implemented using second generation Current Conveyors (CCIIs), Differential Difference Current Conveyors (DDCCs) or Differential Voltage Current Conveyors (DVCCs) as active elements. The procedure followed was: (i) a (2 + α) order transfer function was considered and its coefficients were determined as a function of the order of the filter using a numerical optimization algorithm to minimize the error between magnitude in dB of the considered transfer function and that in (1). For this purpose, the MATLAB function fminsearch with a suitable fitness function was used. (ii) The approximation of the fractional-order capacitors was performed through the utilization of Foster type-II RC networks. This work was further extended in [26], where transfer functions of low-pass/highpass filters with Butterworth maximally flat magnitude and fractional-orders higher than two were realized. The implementation presented in [26] was based on the employment of OTAs as active elements, and the required fractional-order capacitors were approximated using the Valsa network. The magnitude frequency response of a (n + α) order Chebyshev low-pass filter is given by with 0 < ε ≤ 1 being the ripple factor, C n+α (ω) = cos (n + α) · cos −1 ω ω 0 being the corresponding Chebyshev polynomial, and ω 0 being the cutoff frequency where the gain becomes equal to 1/ √ 1 + ε 2 . A design procedure for realizing Chebyshev low-pass filters of order (n + α) was introduced in [7] where the MATLAB lsqcurvefit function was used to implement the non-linear least squares optimization. No implementation of a filter with order greater than two was presented in [7]. In the work of [29], metaheuristic evolutionary algorithms were employed for approximating the response of (n + α) order Chebyshev low-pass filters. Operational amplifier (op-amp) and OTA based implementations of orders in the range ]1, 2[ were reported in this work. Finally, in [27] a genetic algorithm was also employed to optimize the parameters of the fractional-order transfer function so that its magnitude characteristics approximate the ideal response of the fractional-order Chebyshev low-pass filter. It provided a design of 2.2 order low-pass filter realized using op-amps, while the Foster type-I RC network was employed for approximating the behavior of the fractionalorder capacitors. 43206 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

B. FREQUENCY RESPONSE MAGNITUDE APPROXIMATION 1) LOW-PASS FILTER FUNCTIONS
The gain responses described by (1)-(2) can be approximated through the following steps: • Collect the frequency response data of (1) or (2), within the frequency range of interest, using the MATLAB frd function of the Control System Toolbox.
• Assuming an approximation order (N ), obtain the rational integer-order transfer function of the filter using the MATLAB commands fitmagfrd, which performs the fit of the frequency response magnitude data by a minimum-phase state-space model, and tf, which converts the model into a transfer function. Both commands are included in the Robust Control Toolbox. These steps are summarized in the following program listing, where a Butterworth low-pass filter of order equal to 2.4 is approximated. It must be mentioned at this point that only the line describing the variable mag_ideal of the provided code must be changed for obtaining different types of filter functions.
Listing 1. MATLAB code for approximating a Butterworth low-pass filter.
The resulting transfer function will have the form of (3) with A i and B j (i = 0, 1 . . . N − 1, j = 0, 1 . . . N ) being positive and real coefficients. For demonstration purposes, let us consider Butterworth low-pass filter functions with cutoff frequency ω 0 = 1 rad/s, which will be approximated in the frequency range [10 -1 , 10 1 ] rad/s. Employing a 4 th -order approximation, the resulting values of the coefficients A i and B j are summarized in Table 1. The corresponding values for approximating Chebyshev low-pass filter functions, with a 3 dB ripple and the same ω 0 , under the same conditions of approximation are provided in Table 2. The resulting frequency responses are depicted in Figs. 1 and 2, respectively, where the validity of the presented concept is verified.

2) HIGH-PASS FILTER FUNCTIONS
The magnitude response of high-pass fractional filters with Butterworth characteristics is given in (4)  and the corresponding expression of the Chebyshev high-pass filters is with C n+α (ω) = cos (n + α) · cos −1 ω 0 ω . Following the same procedure in the case of high-pass filters described by (4)

3) BAND-PASS FILTER FUNCTIONS
The expression of the magnitude of a Butterworth band-pass filter of order (n + α) is given by (6) with ω 0 being the center frequency and BW is the bandwidth of the filter (BW = ω high − ω low ). Considering a band-pass frequency response with ω 0 = 1 rad/s, ω low = ω o /2 and ω high = 2ω 0 , the values of coefficients which are associated with filters of orders {2.2, 2.4, 2.6} are given in Table 5.   The corresponding plots of the frequency responses plots are demonstrated in Fig. 5.
From the above, it is obvious that the proposed method offers the following advantages: • It is versatile, in the sense that it is applicable in various types of filters functions (i.e., low-pass, high-pass, bandpass) and is independent of the type of the polynomial that is used for the prototype filter (e.g. Butterworth and Chebyshev).
• It is very simple because, having available the frequency response of the filter, one step is required to obtain the rational integer-order transfer function which approximates the fractional-order filter function.
• It offers design flexibility and programmability since the transfer function in (3) is used for implementing the derived transfer functions of all types, i.e. the same core can be used for implementing different filters.

A. ELECTRONICALLY CONTROLLED IMPLEMENTATION
The transfer function in (3) can be realized using an Inverse Follow the Leader Feedback (IFLF) topology, described by the functional block diagram in Fig. 6 [30]. The realized transfer function in this case is (7) H (s) = G n s N and the values of time-constants as well of scaling factors are calculated by equalizing the coefficients of (3) and (7). The resulting design equations are thus (8)-(9) The corresponding OTA-C structure, which implements the block diagram in Fig. 6, in the case of a 4 th -order approximation, is depicted in Fig. 7(a). The circuitry shown in Fig. 7(b) could be utilized for realizing the required OTAs, due to the offered improved linearity compared to that offered by conventional implementations [31]. Its smallsignal transconductance parameter is given by g m = 5I B 9nV T with n being the slope factor of a MOS transistor in subthreshold region (1 < n < 2), V T being the thermal voltage (26 mV at 27 • C), and I B being the dc bias current. Consequently, the time-constant of each integration stage τ i = C i /g mi (i = 1,2. . . 4) is controlled by the dc bias current of the associated OTAs; in addition, the scaling factors G j (j = 0,1. . . 4) are implemented by scaling the dc bias currents of the OTAs employed for performing this operation.
Using the values of Tables 1-4, the design equations in (8)- (9) and denormalizing frequency to ω 0 = 1 krad/s, the derived values of the bias currents of the integration sections are given in Table 6 in the case of Butterworth filters, and in Table 7 in the case of Chebyshev filters. In both cases the main bias current of the summation stages is equal to 98.3 nA.  Fig. 7.

B. DIGITALLY CONTROLLED IMPLEMENTATION
Another possible implementation of the transfer function in (3) is that based on the functional block diagram in Fig. 8, which represents a Follow the Leader Feedback (FLF) structure [32]. The realized transfer function is the same as that in (7) and, therefore, the design equations (8)-(9) are still valid. Its implementation can be performed using the FPAA AN231E04 device provided by Anadigm [33]. The internal intermediate blocks are interconnected through programmable buses and this offers attractive design flexibility and versatility [34], [35].
The Moreover, the values of the inherit gain of the inputs of the summation stages will become equal to that of the associated extra gain stage (i.e., G j ), for ensuring that the total gain across the path is equal to that of the associated scaling factor. Following this, the resulting design obtained using the Anadigm Designer ® 2 EDA software is demonstrated in Fig. 15, where the corresponding experiment implementation is also provided. The interface is used for performing the single-to-differential conversion of the input signal, as well as, the differential-to-single-ended conversion of the output signal. The parameters of ''GainHold'' (gain stage) and ''Integrator'' Configurable Analogue Modules (CAMs) for implementing low and high-pass filters with cutoff frequency equal to 100 krad/s, as well as band-pass filter with the same center frequency employing a 4 th -order approximation, are provided in Tables 8-9, respectively.

A. SIMULATION RESULTS
The performance of the topology in Fig. 7(a) will be evaluated using the Cadence IC design suite and the Austria Mikro Systeme (AMS) CMOS 0.35 μm design kit. Considering a dc bias voltages scheme V DD = −V SS = 0.75 V and that the MOS transistors are biased at the sub-threshold region, their chosen aspect ratios are provided in Table 10. The layout design of the OTA is demonstrated in Fig. 9.     in Fig. 12, while their important frequency characteristics are summarized in Table 11.  Fig. 7(b).
The time-domain behavior of the filters has been evaluated for the case of a Butterworth low-pass filter of order equal FIGURE 9. Layout design of the OTA structure in Fig.7(b).     Fig. 13(a), where the measured value of the gain is −3.01 dB, close to the theoretically predicted value of −3 dB.

demonstrated in
The linearity of the filter has been evaluated through a 62.8 rad/s stimulus of variable amplitude, and the Total Harmonic Distortion (THD) versus amplitude plot are depicted in Fig. 13(b). THD levels of 1% and 2% are observed at 52 mV and 73.5 mV amplitudes, respectively. Integrating the noise over the pass-band of the filter, the rms value of the input referred noise was 42.6 μV. Consequently, the predicted value of the Dynamic Range (DR) of the filter (at 1% THD level) is equal to 58.72 dB.
The sensitivity of the filters has been evaluated through the utilization of Monte-Carlo analysis for N = 500 runs. The obtained statistical distribution of the cutoff frequency and bandwidth of the Butterworth low-pass and band-pass filters of order 2.4 are demonstrated in Figs. 14(a, b). The corresponding values of the standard deviation are 104.32 rad/s and 24.1 rad/s, respectively, and taking into account that the associated mean values are 1.02 krad/s and 1.496 krad/s, it is concluded that the introduced topology offers reasonable sensitivity characteristics.

B. EXPERIMENTAL RESULTS
The frequency responses of the Butterworth band-pass filters of orders {2.2, 2.4, 2.6}, and of low/high-pass filters of orders {4.2, 4.5} obtained using the HP4395A Network/Impedance analyzer are given in Figs. 16(a, b). The     Table 12.
The time-domain behavior of the filters has been evaluated as follows: the band-pass filter of order 2.6 is stimulated by a 1Vp-p sinusoidal signal with frequency equal to the upper cutoff frequency (i.e., 197.55 krad/s), while the low-pass filter was stimulated by a 1 Vp-p sinusoidal signal with frequency corresponding to the cutoff frequency of the filters (i.e. 98.2 krad/s). Using a DSO6034A oscilloscope, the resulting screenshots are demonstrated in Figs. 17(a, b).
The values of the measured gains were −2.8dB and −2.85dB, close to the theoretical −3dB value.
Finally, the linear performance of the implemented filters was evaluated in the case of a low-pass filer of order 4.5, stimulated by a 6.28 krad/s signal with the amplitude set to the maximum allowable by the FPAA (i.e., 3 Vp-p). The second harmonic was found at 77.65 dB below the fundamental. With regards to the presented implementations which both offer adjustability of the derived frequency characteristics of the realized filter functions, the OTA-C structure offers the following benefits: a) capability of monolithic implementation on silicon, and b) greater maximum frequency of operation than that offered by the FPAA implementation. This is due to the absence of a compensation network in the internal structure of the OTA, by contrast to the FPAA device where internally compensated op-amps are used as active elements.
Another imposed restriction on the maximum frequency of operation of the FPAA structure is due to the fact that the switched-capacitor technique is used for implementing the signal processing. The maximum allowed clock frequency in the presented design was 2 MHz, restricting the maximum frequency of (accurate) operation to the order of few kHz. On the other hand, the employment of the small-signal transconductance parameter (g m ) of OTAs for realizing the required time-constants and scaling factors imposes restriction on the maximum amplitude of the signal that can be handled by the system with a low level of distortion. This is obvious from the provided results, where THD levels of 1% and 2% are observed at 52 mV and 73.5 mV, respectively. The FPAA implementation does not suffer from this obstacle, because of the employment of op-amps. This is evident from the provided results where, for a 1.5 V amplitude, the second harmonic was found to be 77.65 dB below the fundamental tone.
With regards to the dc power dissipation, considering a band-pass filter of order equal to 2.4, the corresponding values are 3.88 μW for the OTA-C realization and 430 mW for the FPAA implementation. The increased power dissipation of the FPAA structure is the price paid for its better linear performance, compared to that offered by the OTA-C filter.

V. CONCLUSION
A systematic way for deriving rational integer-order transfer functions which approximate the behavior of fractional filters of order greater than two was presented in this paper. This is actually a two-step procedure based on the fitting of the magnitude response of the desired fractional-order filter function. In this work, the chosen order of approximation was equal to four in order to achieve an accurate level of approximation. The procedure is versatile in the sense that it is independent from the order as well as from the type of filter. For demonstration purposes, Butterworth and Chebyshev filter responses have been implemented in this work, but the procedure is also applicable to other types such as inverse Chebyshev, Bessel or Elliptic filters. It is noticeable that this procedure preserves the cutoff frequency of the approximated filters unlike the methods of [5] and [9] and without using any complex algorithms as in [7], [22], [25], [26], and [27]. We have provided robust implementations to validate the designed higher-order filters on the integrated circuit level (using CMOS OTAs) and experimentally using a switched-C based FPAA platform.

ACKNOWLEDGMENT
The publication of the article in OA mode was financially supported by HEAL-Link. His research interests include circuit theory, nonlinear dynamics, chaos theory, and fractional-order circuits and systems with diverse applications ranging from the modeling of oscillatory networks and nonlinear behavior in electronic circuits and plasma physics to modeling of energy storage devices, bio-materials, and biological tissues. He has authored or coauthored more than 350 publications in these areas From 1997 to 1999, he was with the Institute of Signal and Information Processing, Swiss Federal Institute of Technology (ETH), Zürich, Switzerland. Since 2008, he has been visiting the Faculty of Engineering, Bar-Ilan University, Israel, and doing research in the field of analog circuits and filters. He is currently a Full Professor with the Faculty of Electrical Engineering and Computing (FER), University of Zagreb. He lectures in the field of electrical circuits, signals and systems, and analog and mixed-signal processing circuitry. His research interests include analog and digital signal processing and filter designs, integrated circuit designs, and the study and analysis of fractional-order systems. He has been a MC Member of the COST Action CA15225 ''Fractional-Order Systems: Analysis, Synthesis and Their Importance for Future Design.'' He is a member of the Croatian Society for Communications, Computing, Electronics, Measurement and Control and the IEEE-CAS Society. He was awarded the Silver Plaque Josip Loncar for his Ph.D. thesis and an IEEE Best Paper Finalist Award for a conference paper.