A 1 + α Order Generalized Butterworth Filter Structure and Its Field Programmable Analog Array Implementation

Abstract

Fractional-order Butterworth filters of order 1 + $\alpha$ (0 < $\alpha$ < 1) can be implemented by a unified structure, using the method presented in this paper. The main offered benefit is that the cutoff frequencies of the filters are fully controllable using a very simple method and, also, various types of filters (e.g., low-pass, high-pass, band-pass, and band-stop) could be realized. Thanks to the employment of a Field Programmable Analog Array device, the implementation of the introduced method is fully reconfigurable, in the sense that various types of filter functions as well as their order are both programmable.

1. Introduction

Fractional-order filters offer a fine adjustment of the slope of the attenuation from the pass-band to the stop-band, and this is the result of the fact that it is equal to $\pm 20·(n+\alpha )$ dB/dec, where n is the integer part of the order and 0 < $\alpha$ < 1 is its non-integer counterpart [1,2,3,4]. In addition, they offer the scaling of the cutoff frequencies due to their dependence on both the pole frequency and the order of the filter. In the case of filters of order 0 < $\alpha$ < 1, there are simple expressions describing the dependence of the cutoff frequencies on the aforementioned parameters. This is not the case for filters with the order 1 + $\alpha$, where complicated non-linear equations must be solved to find the value of the cutoff frequency. In addition, the realization of well-known filter functions such as the Butterworth transfer functions is an even more difficult task [5,6].

A significant research effort has been dedicated to overcoming this obstacle, where appropriate cost functions error minimization metaheuristic and genetic algorithms were reported. A first attempt for approximating fractional-order Butterworth filters has been performed in [7], where a polynomial fitting-based method was used for expressing the coefficients of the fractional-order transfer function as a function of the order of the filter. This method is oriented to the realization of low-pass filters, and the high-pass filter functions have been derived through a frequency transformation. In [8], the approximation of the fractional-order Butterworth low-pass filters is performed using the pole placement in the W-plane. In [9], the Flower Pollination Algorithm (FPA) with an appropriate cost function is employed for approximating Butterworth low-pass filters, while in [10], a nature-inspired optimization technique called the Gravitational Search Algorithm (GSA) has been employed for the same purpose. In [11], the Symbiotic Organisms Search (SOS) algorithm is presented for approximating Butterworth high-pass filters. In [12], a genetic algorithm is introduced for approximating Butterworth low-pass filters.

From the above, it is readily obtained that the solutions which are available in the literature are based on the employment of relatively complicated optimization algorithms. In addition, they do not offer a simultaneous approximation of different types of Butterworth filter functions, such as the low-pass, high-pass, band-pass, and band-stop.

In the present work, a simple procedure for realizing various types of 1 + $\alpha$ order Butterworth transfer functions, with fully controllable cutoff frequencies, is introduced. It is based on the fit of the frequency response magnitude data by a minimum-phase state-space model, using a Log-Chebyshev magnitude design. The main important benefits are as follows: (a) The procedure is independent of the type of the filter; low-pass, high-pass, band-pass, and band-stop filter functions can be implemented. (b) The resulting structure has the capability of implementing all the aforementioned filter functions without altering its core.

This paper is organized as follows: The description of the basic characteristics of the fractional filters of order 1 + $\alpha$ is performed in Section 2. The proposed procedure is introduced in Section 3, where a possible implementation of the resulting approximation functions using a Field Programmable Array (FPAA) device is also demonstrated. The performance of the designed filters is experimentally verified in Section 4.

2. Fractional-Order Filters of Order 1 + $\alpha$ (0 < $\alpha$ < 1)

Let us consider the transfer function of a low-pass filter of order $0<\alpha <1$ and pole frequency ${\omega }_0=1/\tau$, with the variable $\tau$ being a time constant

$$H\left(s\right)={\displaystyle \frac{1}{{\left(\tau s\right)}^{\alpha }+1}}.$$

(1)

Setting $s^{\alpha }={\omega }^{\alpha }·[cos\left(0.5\alpha \pi \right)+jsin\left(0.5\alpha \pi \right)]$ in (1), the following expression of the magnitude response is derived:

$$\mid H\left(\omega \right)\mid ={\displaystyle \frac{1}{{\left[{\left({\displaystyle \frac{\omega }{{\omega }_0}}\right)}^{2\alpha }+2{\left(\frac{\omega }{{\omega }_0}\right)}^{\alpha }cos\left(\frac{\alpha \pi }{2}\right)+1\right]}^{1/2}}}.$$

(2)

The cutoff or half-power frequency of the filter (${\omega }_h$), defined as the frequency where there is a 3 dB drop in the magnitude from its maximum value, is determined using (2) and setting $\mid H\left(\omega \right)\mid =0.707$. The resulting expression is given by (3)

$${\omega }_h={\omega }_0{\left[\sqrt{1+{cos}^2\left({\displaystyle \frac{\alpha \pi }{2}}\right)}-cos\left({\displaystyle \frac{\alpha \pi }{2}}\right)\right]}^{MM1/\alpha }.$$

(3)

In a similar way, the behavior of a high-pass filter is described by (4) and (5) and the associated cutoff frequency is given by the expression in (6)

$$H\left(s\right)={\displaystyle \frac{{\left(\tau s\right)}^{\alpha }}{{\left(\tau s\right)}^{\alpha }+1}},$$

(4)

$$\mid H\left(\omega \right)\mid ={\displaystyle \frac{{\left(\frac{\omega }{{\omega }_0}\right)}^{\alpha }}{{\left[{\left(\frac{\omega }{{\omega }_0}\right)}^{2\alpha }+2{\left(\frac{\omega }{{\omega }_0}\right)}^{\alpha }cos\left(\frac{\alpha \pi }{2}\right)+1\right]}^{1/2}}},$$

(5)

$${\omega }_h={\omega }_0{\left[\sqrt{1+{cos}^2\left({\displaystyle \frac{\alpha \pi }{2}}\right)}+cos\left({\displaystyle \frac{\alpha \pi }{2}}\right)\right]}^{1/\alpha }.$$

(6)

Inspecting (3) and (6), it is readily obtained that, having available the pole frequency and the order of the filter, the cutoff frequency can be fully determined.

In the case of a 1 + $\alpha$ order low-pass filter, its transfer function is given by (7)

$$H\left(s\right)={\displaystyle \frac{1}{{\left(\tau s\right)}^{1+\alpha }+\frac{1}{Q}{\left(\tau s\right)}^{\alpha }+1}},$$

(7)

with Q being a factor equal to the quality factor of the corresponding second-order filter. The half-power frequency is the solution of the following non-linear equation:

$${\left({\displaystyle \frac{{\omega }_h}{{\omega }_0}}\right)}^{2\left(1+\alpha \right)}-2{\left({\displaystyle \frac{{\omega }_h}{{\omega }_0}}\right)}^{1+\alpha }sin\left({\displaystyle \frac{\alpha \pi }{2}}\right)+\frac{1}{Q^2}{\left({\displaystyle \frac{{\omega }_h}{{\omega }_0}}\right)}^{2\alpha }+\frac{2}{Q}{\left({\displaystyle \frac{{\omega }_h}{{\omega }_0}}\right)}^{\alpha }cos\left(\frac{\alpha \pi }{2}\right)-1=0.$$

(8)

The corresponding non-linear equation for obtaining the half-power frequency of a high-pass filter, described by (9),

$$H\left(s\right)={\displaystyle \frac{{\left(\tau s\right)}^{1+\alpha }}{{\left(\tau s\right)}^{1+\alpha }+\frac{1}{Q}{\left(\tau s\right)}^{\alpha }+1}},$$

(9)

is provided by (10)

$${\left({\displaystyle \frac{{\omega }_h}{{\omega }_0}}\right)}^{2\left(1+\alpha \right)}+2{\left({\displaystyle \frac{{\omega }_h}{{\omega }_0}}\right)}^{1+\alpha }sin\left({\displaystyle \frac{\alpha \pi }{2}}\right)-\frac{1}{Q^2}{\left({\displaystyle \frac{{\omega }_h}{{\omega }_0}}\right)}^{2\alpha }-\frac{2}{Q}{\left({\displaystyle \frac{{\omega }_h}{{\omega }_0}}\right)}^{\alpha }cos\left(\frac{\alpha \pi }{2}\right)-1=0.$$

(10)

For band-pass and band-stop filters described by (11) and (12), respectively,

$$H\left(s\right)={\displaystyle \frac{\frac{1}{Q}{\left(\tau s\right)}^{\alpha }}{{\left(\tau s\right)}^{1+\alpha }+\frac{1}{Q}{\left(\tau s\right)}^{\alpha }+1}},$$

(11)

$$H\left(s\right)={\displaystyle \frac{{\left(\tau s\right)}^{1+\alpha }+1}{{\left(\tau s\right)}^{1+\alpha }+\frac{1}{Q}{\left(\tau s\right)}^{\alpha }+1}}.$$

(12)

The peak (center) frequency is calculated by solving the differential equation:

${\displaystyle \frac{\partial }{\partial \omega }}{\left(\mid H\left(\omega \right)\mid \right)}_{\omega ={\omega }_{peak}}=0$, while the associated lower (${\omega }_{low}$) and upper (${\omega }_{high}$) cutoff frequencies are calculated by setting $\mid H\left(\omega \right){\mid }_{\omega ={\omega }_{low,high}}=0.707·\mid H\left(\omega \right){\mid }_{\omega ={\omega }_{peak}}$.

From the above, it is clear that the determination of the characteristic frequencies in the case of 1 + $\alpha$ order filters is a difficult procedure, which is the reason for using the aforementioned algorithmic methods. This obstacle can be overcome through the utilization of the curve fitting-based procedure which will be described in the next section.

3. Proposed Procedure for Approximating 1 + $\alpha$ Order Filters

3.1. Derivation of the Rational Integer-Order Approximation Function

According to the proposed method, the starting point is the consideration of the gain responses of the filters. For example, the corresponding expressions of low-pass, high-pass, band-pass, and band-stop Butterworth-type filters are those in (13)–(16).

$$\left|H_{LP,1+\alpha }\left(\omega \right)\right|=\frac{1}{\sqrt{1+{\left(\frac{\omega }{{\omega }_0}\right)}^{2(1+\alpha )}}},$$

(13)

$$\left|H_{HP,1+\alpha }\left(\omega \right)\right|=\frac{1}{\sqrt{1+{\left(\frac{{\omega }_0}{\omega }\right)}^{2(1+\alpha )}}},$$

(14)

$$\left|H_{BP,1+\alpha }\left(\omega \right)\right|=\frac{1}{\sqrt{1+{\left(\frac{\left|{\omega }_0^2-{\omega }^2\right|}{\omega ·({\omega }_{high}-{\omega }_{low})}\right)}^{2(1+\alpha )}}},$$

(15)

$$\left|H_{BS,1+\alpha }\left(\omega \right)\right|=\frac{1}{\sqrt{1+{\left(\frac{\omega ·({\omega }_{high}-{\omega }_{low})}{\left|{\omega }_0^2-{\omega }^2\right|}\right)}^{2(1+\alpha )}}}.$$

(16)

The next step is to collect the values of the gain of the filter transfer function of interest within the desired range (${\omega }_{min},{\omega }_{max}$). This can be performed through the utilization of the MATLAB frd built-in function. The last steps include the fit of the collected gain data with a minimum-phase state-space model using the Log-Chebyshev magnitude design, realized using the MATLAB built-in command fitmagfrd, and the derivation of the corresponding rational integer-order transfer function using the tf MATLAB command.

The resulting transfer function will have the form of (17)

$$H\left(s\right)=\frac{B_n{s}^n+B_{n-1}{s}^{n-1}+\dots +B_{1}s+B_0}{s^n+A_{n-1}{s}^{n-1}+\dots +A_{1}s+A_0},$$

(17)

with $A_i$ and $B_j$$\left(i=0,1...n-1,j=0,1...n\right)$ being positive and real coefficients, and n being the order of the approximation.

Considering Butterworth low-pass and high-pass filters of order 1.2, 1.4, and 1.6, with cutoff frequency ${\omega }_h$=1 rad/s and approximated by a 4th-order approximation within the range of [10${}^{-1}$, 10¹] rad/s, the resulting values of the coefficients $A_i$ and $B_j$ are summarized in

Table 1. Values of coefficients for approximating Butterworth low-pass filters of order 1.2, 1.4, 1.6 (${\omega }_h$ = 1 rad/s).

Coefficient		Order
	1.2	1.4	1.6
$B_4$	0.005867	0.00228	0.001421
$B_3$	0.6167	0.3336	0.1768
$B_2$	5.149	3.841	2.674
$B_1$	6.85	6.96	6.071
$B_0$	1.704	1.769	1.64
$A_3$	7.196	7.014	6.537
$A_2$	12.66	12.4	10.97
$A_1$	8.606	8.877	7.976
$A_0$	1.701	1.77	1.644

and

Table 2. Values of coefficients for approximating Butterworth high-pass filters of order 1.2, 1.4, 1.6 (${\omega }_h$ = 1 rad/s).

Coefficient		Order
	1.2	1.4	1.6
$B_4$	1.002	0.9991	0.9974
$B_3$	4.028	3.932	3.693
$B_2$	3.028	2.17	1.627
$B_1$	0.3626	0.1884	0.1075
$B_0$	0.00345	0.001288	0.0008641
$A_3$	5.06	5.015	4.852
$A_2$	7.446	7.003	6.676
$A_1$	4.231	3.962	3.976
$A_0$	0.588	0.565	0.6083

. In the case of a band-pass filter, the same order of approximation is considered. While the peak and cutoff frequencies are ${\omega }_{peak}$ = 1 rad/s, ${\omega }_{low}$ = 0.5 rad/s, and ${\omega }_{high}$ = 2 rad/s, for the band-stop filter, the corresponding values are ${\omega }_{peak}$ = 1 rad/s, ${\omega }_{low}$ = 0.2 rad/s, and ${\omega }_{high}$ = 5 rad/s. The values of the coefficients are provided in

Table 3. Values of coefficients for approximating Butterworth band-pass filters of order 1.2, 1.4, 1.6 (${\omega }_{peak}$ = 1 rad/s, ${\omega }_{low}$ = 0.5 rad/s, ${\omega }_{high}$ = 2 rad/s).

Coefficient		Order
	1.2	1.4	1.6
$B_4$	0.002795	0.002358	0.004134
$B_3$	0.9857	0.632	0.4071
$B_2$	4.828	4.016	3.325
$B_1$	0.9857	0.632	0.4071
$B_0$	0.002795	0.002358	0.004134
$A_3$	4.448	3.73	3.118
$A_2$	6.772	5.939	5.246
$A_1$	4.448	3.73	3.118
$A_0$	1	1	1

and

Table 4. Values of coefficients for approximating Butterworth band-stop filters of order 1.2, 1.4, 1.6 (${\omega }_{peak}$ = 1 rad/s, ${\omega }_{low}$ = 0.2 rad/s, ${\omega }_{high}$ = 5 rad/s).

Coefficient		Order
	1.2	1.4	1.6
$B_4$	1.271	1.815	2.373
$B_3$	0.1948	0.1959	0.1668
$B_2$	2.542	3.63	4.747
$B_1$	0.1948	0.1959	0.1668
$B_0$	1.271	1.815	2.373
$A_3$	8.074	13.59	18.94
$A_2$	4.52	8.957	19.05
$A_1$	8.074	13.59	18.94
$A_0$	1	1	1

.

The derived plots are demonstrated in Figure 1 and Figure 2, accompanied by their associated error plots.

3.2. A Possible Implementation

A possible realization of the transfer function in (17) is using the Follow-the-Leader Feedback (FLF) structure in Figure 3, where the realized transfer function is given by (18) [13].

$$H\left(s\right)=\frac{G_4{s}^4+\frac{G_3}{{\tau }_1}{s}^3+\frac{G_2}{{\tau }_1{\tau }_2}{s}^2+\frac{G_1}{{\tau }_1{\tau }_2{\tau }_3}s+\frac{G_0}{{\tau }_1{\tau }_2{\tau }_3{\tau }_4}}{s^4+\frac{1}{{\tau }_1}{s}^3+\frac{1}{{\tau }_1{\tau }_2}{s}^2+\frac{1}{{\tau }_1{\tau }_2{\tau }_3}s+\frac{1}{{\tau }_1{\tau }_2{\tau }_3{\tau }_4}}.$$

(18)

Equalizing the coefficients of (17) and (18), the resulting design equations are given by (19)

$${\tau }_{i+1}=\frac{A_{4-i}}{A_{3-i}}(i=0...3)G_j=\frac{B_j}{A_j}(j=0...4).$$

(19)

The implementation of the integration blocks in Figure 3 can be performed using a variety of active blocks, including Operational Amplifiers (op-amps), Second-Generation Current Conveyors (CCIIs), and Current Feedback Operational Amplifiers (CFOAs). The derived implementations are active-RC structures which do not offer the electronic tunability of the characteristics of the filters. The employment of the Operational Transconductance Amplifiers (OTAs) as active elements is a solution for overcoming this obstacle due to the fact that the transconductance parameter of the OTA can be electronically adjusted by an appropriate dc current/voltage. The price paid for this achievement is the reduced linearity of the resulting filter structures, and this is caused by the small-signal nature of the transconductance [14]. Another option could be the utilization of a digitally programmable device [15], such as the FPAA AN231E04 device provided by Anadigm [16].

Denormalizing (17) at ${\omega }_h$ = 150 krad/s and taking into account that the integration constants ($M_i$) of the FPAA device are formed as $M_i={10}^{-6}/{\tau }_i$ ($i=1...4$), (having units of $\mathsf{\mu }$s${}^{-1}$), their values are summarized in

Table 5. Values of the integration constants and scaling factors of the FPAA device for approximating Butterworth low-pass filters of order 1.2, 1.4, 1.6 (${\omega }_h$ = 150 krad/s).

Coefficient		Order
	1.2	1.4	1.6
$M_1$	1.079	1.052	0.980
$M_2$	0.264	0.265	0.252
$M_3$	0.102	0.107	0.109
$M_4$	0.030	0.030	0.031
$G_0$	1.002	0.999	0.997
$G_3$	0.796	0.784	0.761
$G_2$	0.401	0.310	0.244
$G_3$	0.086	0.048	0.027
$G_4$	0.006	0.002	0.001

,

Table 6. Values of the integration constants and scaling factors of the FPAA device for approximating Butterworth high-pass filters of order 1.2, 1.4, 1.6 (${\omega }_h$ = 150 krad/s).

Coefficient		Order
	1.2	1.4	1.6
$M_1$	0.759	0.752	0.728
$M_2$	0.220	0.209	0.206
$M_3$	0.085	0.085	0.089
$M_4$	0.021	0.021	0.023
$G_0$	0.006	0.002	0.001
$G_3$	0.086	0.047	0.027
$G_2$	0.407	0.310	0.244
$G_3$	0.796	0.784	0.761
$G_4$	1.002	0.999	0.997

,

Table 7. Values of the integration constants and scaling factors of the FPAA device for approximating Butterworth band-pass filters of order 1.2, 1.4, 1.6 (${\omega }_{peak}$ = 150 krad/s, ${\omega }_{low}$ = 75 krad/s, ${\omega }_{high}$ = 300 krad/s).

Coefficient		Order
	1.2	1.4	1.6
$M_1$	0.667	0.559	0.468
$M_2$	0.228	0.239	0.252
$M_3$	0.098	0.094	0.089
$M_4$	0.034	0.040	0.048
$G_0$	0.003	0.002	0.004
$G_3$	0.221	0.169	0.131
$G_2$	0.713	0.676	0.634
$G_3$	0.223	0.169	0.131
$G_4$	0.003	0.002	0.004

and

Table 8. Values of the integration constants and scaling factors of the FPAA device for approximating Butterworth band-stop filters of order 1.2, 1.4, 1.6 (${\omega }_{peak}$ = 60 krad/s, ${\omega }_{low}$ = 12 krad/s, ${\omega }_{high}$ = 300 krad/s).

Coefficient		Order
	1.2	1.4	1.6
$M_1$	0.484	0.816	1.1362
$M_2$	0.033	0.039	0.060
$M_3$	0.207	0.092	0.060
$M_4$	0.007	0.004	0.003
$G_0$	1.271	1.815	2.373
$G_3$	0.024	0.014	0.009
$G_2$	0.562	0.405	0.2492
$G_3$	0.024	0.014	0.009
$G_4$	1.271	1.815	2.373

, where the values of the scaling factors $G_j$ (j = 0 ...4) are also given. The clock frequency is equal to $f_{clk}$ = 2 MHz. In addition, according to the specifications of the FPAA, the realizable values of the integration constants are in the range [0.02, 17.7] $\mathsf{\mu }$s${}^{-1}$, while the weight factors of the summation stages for realizing the scaling factors $G_j$ are in the range [0.01, 8.83]. Inspecting Table 5, Table 6, Table 7 and Table 8, it is derived that there are values of $M_i$ and $G_j$ that are not realizable. As the range of the realizable values of the gain stages is [0.01, 100], the problem can be solved by following the technique presented in Figure 4. According to this, the desired value of the gain scaling factor $G_j$ is implemented as $G_j=\left(-\sqrt{G_j}\right)·\left(-\sqrt{G_j}\right)$, making the values of $\sqrt{G_j}$ to be in the realizable range. In the FPAA implementation presented in the next section, this will be performed by adding an extra gain stage to each one of the inputs of the summation stages, and both of them will have a value equal to $\sqrt{G_j}$, establishing a total gain equal to $G_j$ in the path of the associated signal. A similar approach will be followed for the integration constants.

4. Experimental Results

Using the Anadigm Designer^® 2 EDA software, the resulting design is depicted in Figure 5, while the experimental setup is demonstrated in Figure 6a. The FPAA device with the accompanied printed circuit board (PCB) for implementing the required single-to-differential and differential-to-single voltage conversion is demonstrated in Figure 6b. The corresponding circuitry of the interface for performing a single-to-differential conversion of the input signal is depicted in Figure 7a, where the corresponding one for performing the differential-to-single conversion of the output signal is demonstrated in Figure 7b. Taking into account the terminal properties of the CFOA, described by the following formulas, ${\upsilon }_X={\upsilon }_Y,i_X=i_Z,{\upsilon }_Z={\upsilon }_O$, the expressions in (20) and (21) are readily obtained for the topology in Figure 7a.

$${\upsilon }_{in,p}=\frac{{\upsilon }_{in}}{2}+\frac{E}{2},$$

(20)

$${\upsilon }_{in,n}=-\frac{{\upsilon }_{in}}{2}+\frac{E}{2}.$$

(21)

The value of the dc voltage source is E = 3 V, and this is originated from the fact that the common-mode voltage of the FPAA is equal to 1.5 V.

The topology in Figure 7b implements the following input–output relationship

$${\upsilon }_{out}={\upsilon }_{out,p}-{\upsilon }_{out,n},$$

(22)

The ${\upsilon }_{out,p}$ and ${\upsilon }_{out,n}$ are given by the formulas ${\upsilon }_{out,p}={\upsilon }_{out}/2+E/2$ and ${\upsilon }_{out,n}=-{\upsilon }_{out}/2+E/2$. The AD844 discrete component is chosen for implementing the required CFOAs [17] while the value of the basic resistor is R = 10 k$\Omega$.

The frequency responses of the Butterworth low-pass and high-pass filters, which are obtained using the HP4395A Network/Impedance analyzer, are demonstrated in Figure 8a,b, respectively. The measured values of the cutoff frequencies were 151.1 rad/s for the low-pass filter and 147.8 rad/s for the high-pass filter, close to the theoretical value of 150 krad/s. The experimental results of the band-pass and band-stop filters are provided in Figure 9a,b. The measured value of the lower cutoff frequency of the band-pass filters was 74.7 krad/s, with the theoretical value being 75 krad/s. The corresponding values of the band-stop filters were 11.8 krad/s and 298.1 krad/s, with the associated theoretical values being 12 krad/s and 300 krad/s, respectively.

These slight deviations from the theory are mainly caused by the tolerances of the resistors employed in the interfacing stages as well as by the truncation of the values of the time constants and the scaling factors performed by the FPAA device.

The time-domain behavior of the filters has been evaluated by stimulating the low-pass and high-pass filters (order equal to 1.4) by a 1Vp-p sinusoidal signal with a frequency equal to their cutoff frequency. The waveforms, obtained using a DSO6034A digital oscilloscope, are demonstrated in Figure 10a,b. The values of the corresponding gains were −3.06 dB and −3.4 dB, confirming their time-domain performance. In the case of the band-pass filters, the stimulation of a filter of an order equal to 1.4 by sinusoidal inputs of the same amplitude as in the previous case and of a frequency equal to the peak frequency results in the waveforms in Figure 10c, where the expected gain of 0 dB is confirmed.

5. Conclusions

A versatile procedure for approximating various types of Butterworth filters with fully controllable frequency characteristics is introduced in this work. The algorithm is based on the built-in commands frd, fitmagfrd, and tf of the MATLAB software, leading to a rational integer-order transfer function. This can be implemented using the well-known classical filter design techniques, such as multi-feedback or cascaded structures. The choice of the technique, as well as of the utilized active element, is up to the designer, performing a trade-off between the design complexity and electronic tunability.

The experimental results derived using an FPAA device confirm the validity of the proposed concept. Future research steps include the application of the presented concept for implementing other types of filter functions, such as Chebyshev filters [18,19,20,21], Bessel filters [22,23,24], and elliptic filters [25].