Bilinear Double-Order Filter Designs and Application Examples

Received 7 January 2024, accepted 17 January 2024, date of publication 22 January 2024, date of current version 30 January 2024. Digital Object Identifier 10.1109/ACCESS.2024.3357092 Bilinear Double-Order Filter Designs and Application Examples JULIA NAKO 1, (Graduate Student Member, IEEE), COSTAS PSYCHALINOS 1, (Senior Member, IEEE), FABIAN KHATEB 2,3,4, AND AHMED S. ELWAKIL 5,6,7, (Senior Member, IEEE) 1Department of Physics, Electronics Laboratory, University of Patras, Rio, 265 04 Patras, Greece 2Department of Microelectronics, Brno University of Technology, 601 90 Brno, Czech Republic 3Faculty of Biomedical Engineering, Czech Technical University in Prague, 272 01 Kladno, Czech Republic 4Department of Electrical Engineering, University of Defense, 662 10 Brno, Czech Republic 5Department of Electrical Engineering, University of Sharjah, Sharjah, United Arab Emirates 6Department of Electrical and Software Engineering, University of Calgary, Alberta, AB T2N 1N4, Canada 7Nanoelectronics Integrated Systems Center (NISC), Nile University, Giza 12677, Egypt Corresponding author: Costas Psychalinos ([email protected]) This work was supported by HEAL-Link. ABSTRACT A novel kind of non-integer order bilinear filters, named double-order bilinear filters, is introduced in this work. They are based on the employment of two non-integer orders, offering the maximum design flexibility in comparison with their fractional-order and power-law counterparts. An attractive offered benefit is that this is achieved without increasing the circuit complexity, since the proposed structure is capable of realizing all non-integer kinds of filters. Two design examples are provided, where it is shown that lead/lag compensators utilized in control applications and low/high shelving filters employed in acoustic applications are actually bilinear filters with suitably selected pole/zero frequencies. Simulation and experimental results, using the OrCAD PSpice simulator and a Field Programmable Analog Array device, respectively, support the findings of this work. INDEX TERMS Analog filters, bilinear filters, compensators, curve-fitting approximation, field programmable analog array, fractional-order filters, power-law filters, shelving filters. I. INTRODUCTION The term "bilinear filter" is used for the characterization of filters which are expressed as a ratio of two linear functions. The transfer function of a first-order bilinear filter is HIO(s)=GL·yτs+1 xτs+1,(1) with x,y>0 being dimensionless scaling factors, τbeing a time constant, and GLbeing the low-frequency gain of the filter. Employing fractional calculus, the transfer function of a fractional-order bilinear filter is HFO(s)=GL·y(τs)α+1 x(τs)α+1,(2) The associate editor coordinating the review of this manuscript and approving it for publication was Andrea De Marcellis . with 0 < α < 1 being the order of the filter. The transfer function of a power-law bilinear filter of order 0< β < 1 is given by HPL(s)=GL·yτs+y0 xτs+x0β .(3) Meanwhile, the standard first-order low-pass and high-pass filter functions are directly derived from (1), by setting y= 0 and x=y, respectively. Forms of fractional-order bilinear filter functions have been realized in [1],[2],[3], and [4], while the corresponding realization of power-law ones have been presented in [3]. Both the aforementioned kinds of filters offer improved design flexibility with regards to their integer-order counterparts, because of the variable non-integer order of the filters, which allows the adjustment of the main characteristics of their frequency behavior. 14040 2024 The Authors. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/ VOLUME 12, 2024 J. Nako et al.: Bilinear Double-Order Filter Designs and Application Examples In this work, a double-order bilinear filter function is introduced where two degrees of freedom are offered because of the employment of two orders, instead of a single order in fractional-order and power-law filters. This enables having full control of the characteristics of the filter. This work is an extension of the work presented in [5]. Two possible implementations are demonstrated, with the first one based on the employment of a single Current Feedback Operational Amplifier (CFOA) as the active element, which also offers the capability of realizing all non-integer order functions by the same RC network, and just adjusting the values of resistors and capacitors. The second implementation is based on the utilization of a Field Programmable Analog Array (FPAA) device, which offers design programmability and versatility in the sense that all kinds of (non-integer order) bilinear filters can be implemented by re-programming the characteristics of the intermediate stages. This work is organized as follows: a systematic review of the bilinear filter transfer functions, presented in the literature, is performed in Section II. The proposed generalized bilinear filter transfer function is introduced in Section II, where its possible implementations are also discussed. Two application examples are provided in Section IV, and the evaluation of the performance of the resulting schemes is performed through simulation results, obtained with the employment of the OrCAD PSpice suite and through experimental results using the FPAA AN231E04 device from Anadigm [6]. II. BILINEAR FILTER TRANSFER FUNCTIONS A. INTEGER-ORDER BILINEAR FILTERS Considering the expression in (1), the time constant is associated with a characteristic frequency ω0according to the formula: τ=1/ω0, and the pole and the zero are located in the left-half of the s-plane with their magnitudes being ωp=1 xτ=ω0 x, ωz=1 yτ=ω0 y.(4) According to (4), the pole (ωp) and zero (ωz) frequencies are not symmetrically located around the characteristic frequency having (in logarithmic scale) a distance equal to xand y, respectively. Using (4), the transfer function in (1) can be alternatively written as in (5) HIO(s)=GLωp ωzs+ωz s+ωp .(5) The gain at high frequencies (GH) tends to GH=GLωp ωz=GLy x.(6) Setting s=jωin (5), the derived gain and phase responses are given by (7a)–(7b), respectively |HIO(jω)| = GL·v u u u u t 1+ω ωz2 1+ω ωp2,(7a) HIO(jω)=tan−1(ω/ωz)−tan−1(ω/ωp).(7b) Generally, the knee frequencies of the filter are calculated from (7a) by setting the value of the gain to a desired level. The most common level is the ±3dB level and this will be employed in what follows. In order to simplify the analysis, it is assumed that the pole and zero of the filter are separated in such a way that they independently determine the behavior of the filter. The asymptotic (Bode) behavior of the frequency response is determined by the relative separation between the pole and the zero of the filter. This will be also assumed hereinafter for all the types of filters which will be considered. Type-I: ωz> ωp(x>yand GL>GH), then the gain response has a constant value equal to GLuntil the lower knee frequency ωL, which is equal to the pole frequency, and starts monotonically decreasing until the higher knee frequency ωH, which is equal to the zero frequency. After this frequency, it reaches a constant value equal to GH, due to the effect of the zero. Therefore, ωL=ωp=ω0 x, ωH=ωz=ω0 y.(8) Type-II: ωp> ωz(x<yor GL<GH), then the gain response has a constant value equal to GLuntil the lower knee frequency ωL, but now becomes equal to the zero frequency, and then it monotonically increases until reaching the higher knee frequency ωH, which is now equal to the pole frequency, reaching a constant value equal to GH. Thus, ωL=ωz=ω0 y, ωH=ωp=ω0 x.(9) Defining the geometric mean of the knee frequencies as the mean frequency (ωm) ωm≡√ωL·ωH,(10) then, using (8) or (9) and (10), it is readily obtained that the relationship between the mean frequency and the characteristic frequency ω0is ωm=ω0 √xy.(11) The gain at the mean frequency (Gm) is defined by (12) Gm≡| HIO(jωm)|= pGLGH,(12) which is equal to the geometric mean of the gains at low and high frequencies. It is now clear that the relationship between the gains at low (GL) and high frequencies (GH), and the gain at the mean frequency (Gm), is GL=Gmrx y,GH=Gm qx y .(13) Therefore, the gains of the filter GLand GH(in dBs) are equally spaced around Gm. VOLUME 12, 2024 14041 J. Nako et al.: Bilinear Double-Order Filter Designs and Application Examples FIGURE 1. Bode plots of the gain responses of the Type-I (blue) and Type-II (red) bilinear filters, with notation of their most important frequency characteristics. The phase at this frequency (Φm) reaches its minimum/maximum value given by Φm≡HIO(jωm)=sin−1 1−x y 1+x y!.(14) In order to facilitate the reader, the Bode plots associated with Type-I and II are demonstrated in Fig.1, where the aforementioned frequency characteristics are depicted. B. FRACTIONAL-ORDER BILINEAR FILTERS The pole and zero of the fractional-order filter function in (2) can be expressed as ωp=1 x1/ατ=ω0 x1/α , ωz=1 y1/ατ=ω0 y1/α ,(15) where the distance between the pole and the zero is controlled by the order of the filter. The transfer function in (2) can be alternatively written as HFO(s)=GLωp ωzα ·sα+ωα z sα+ωα p ,(16) with the gain at high frequencies being GH=GLωp ωzα =GLy x.(17) Setting sα=ωα·[cos(0.5απ)+jsin(0.5απ)]in (16), the gain and phase responses of the filter are given by |HFO(jω)| = GL·v u u u u t 1+ω ωz2α +2ω ωzαcos(0.5απ) 1+ω ωp2α +2ω ωpαcos(0.5απ) , (18a) HFO(jω)=tan−1sin(0.5απ) ωz ωα+cos(0.5απ) −tan−1sin(0.5απ) ωp ωα+cos(0.5απ).(18b) Thus, the asymptotic behavior will be as follows: Type-I: ωz> ωp(x>y,GL>GH), then the filter’s gain response has similar behavior as its integer-order counterpart. The difference is that the low and high knee frequencies are not equal to the pole/zero frequencies. They depend on the order of the filter, and they are given by ωL=ω0 x1/α ·hp1+cos2(0.5απ)−cos(0.5απ)i1/α , (19a) ωH=ω0 y1/α ·hp1+cos2(0.5απ)+cos(0.5απ)i1/α . (19b) Type-II: ωp> ωz(x<yor GL<GH), and the frequency behavior is similar to that of the Type-II integer-order filters. The knee frequencies are given by (19a)–(19b) after xand y interchanging. Using (15), it is readily obtained the relationship between the mean frequency (ωm) and the characteristic frequency ω0= 1/τ, given by (20) ωm=ω0 (√xy)1/α .(20) The relationship between the low and the high frequency gains with the gain at the mean frequency is also given by (13) making them equally spaced around the gain at the mean frequency. The phase at the mean frequency reaches its minimum/maximum value calculated by (21) 8m=tan−1sin(0.5απ) qx y+cos(0.5απ)−tan−1sin(0.5απ) qy x+cos(0.5απ) . (21) C. POWER-LAW BILINEAR FILTERS The pole and zero locations of the filter in (3) are determined by (4). Thus, the transfer function in (3) becomes HPL(s)=GLωp ωzβs+ωz s+ωpβ .(22) At high frequencies, the gain tends to the value GH=GLωp ωzβ =GLy xβ .(23) The gain and phase responses of the filter are |HPL(jω)| = GL·   1+ω ωz2 1+ω ωp2   β/2 ,(24a) HPL(jω)=β·htan−1(ω/ωz)−tan−1(ω/ωp)i.(24b) The asymptotic behavior of the filter is as follows: Type-I: ωz> ωp(x>yand GL>GH), with the gain response having the same behavior as that of its fractionalorder counterpart. Again, the knee frequencies are not equal 14042 VOLUME 12, 2024 J. Nako et al.: Bilinear Double-Order Filter Designs and Application Examples to the pole/zero frequencies and they depend on the order of the filter with their associated expressions given by ωL=ω0 x·p21/β −1, ωH=ω0 y·1 √21/β −1.(25) Type-II: ωp> ωz(x<yand GL<GH), where the knee frequencies are given by (26) ωL=ω0 y·p21/β −1, ωH=ω0 x·1 √21/β −1.(26) The expression of the mean frequency, derived using (25)–(26) is the same as that which corresponds to the integerorder case, i.e., (11). The phase, is calculated from Φm=β·sin−1 1−x y 1+x y!,(27) and this is the minimum/maximum value. III. PROPOSED GENERALIZED (DOUBLE-ORDER) BILINEAR FILTERS A. FILTERS CHARACTERISTICS The transfer functions of the integer-order, fractional-order, and power-law bilinear filters can be generalized according to the following form HDO(s)=GL·y(τs)α+1 x(τs)α+1β ,(28) with 0 < α, β ≤1 being the orders of the filter. According to (28), the integer-order, fractional-order, and power-law bilinear filters correspond to α=β=1, β=0, and α=0, respectively. The pole and zero of the filter are given by the expression in (15) and, therefore, the transfer function in (28) can be reformed as HDO(s)=GLωp ωzαβ sα+ωα z sα+ωα p!β .(29) Hence GL GH=ωz ωpαβ =x yβ .(30) The resulting gain and phase responses are described by (31a)–(31b) |HDO(jω)|=GL    v u u u u t 1+ω ωz2α +2ω ωzαcos(0.5απ) 1+ω ωp2α +2ω ωpαcos(0.5απ)     β , (31a) HDO(jω)=β·tan−1sin(0.5απ) ωz ωα+cos(0.5απ) −tan−1sin(0.5απ) ωp ωα+cos(0.5απ).(31b) TABLE 1. Frequency characteristics of double-order bilinear filters. The shape of the asymptotic behaviors of the filter is the same as that of Type-I and Type-II kinds, with the knee frequencies given by the expressions in (32a)–(32b) ωL=ω0 x1/α ·hp21/β −1+cos2(0.5απ)−cos(0.5απ)i1/α , (32a) ωH=ω0 y1/α ·hp21/β −1+cos2(0.5απ)−cos(0.5απ) i−1/α , (32b) for the Type-I and with the same expressions for Type II after interchanging xand y. The expression of the mean frequency of the filter is also given by (20), while the phase reaches its minimum/maximum value calculated by (33) 8m=β·tan−1sin(0.5απ) qx y+cos(0.5απ) −tan−1sin(0.5απ) qy x+cos(0.5απ).(33) The most important frequency characteristics of the generalized bilinear filter are summarized in Table 1. It must be mentioned at this point that the frequency characteristics of the integer-order filters (α=β=1), fractional-order (β=0), and power-law (α=0) bilinear filers could be readily obtained from this Table. In order to demonstrate the design flexibility offered by the double-order filter, for a given set of values {x,y,GL, ω0} the control of the frequency characteristics of the filter is described in Table 2. Considering the extra degrees of freedom {α, β} in the case of the double-order filter, it is evident from this Table that the five characteristics are controlled by five parameters, offering the highest possible freedom to the designer. B. REALIZATION OF THE PROPOSED GENERALIZED BILINEAR FILTER 1) MINIMUM ACTIVE COMPONENT COUNT REALIZATION Let us consider the structure in Fig.2a where a CFOA has been chosen as the active element [7]. The realized transfer VOLUME 12, 2024 14043 J. Nako et al.: Bilinear Double-Order Filter Designs and Application Examples TABLE 2. Controllability of the frequency characteristics of integer-order (I-O), fractional-order (F-O), power-law (P-L) and double-order (D-O) bilinear filters. FIGURE 2. (a) CFOA based generalized structure for implementing integer and non-integer order bilinear filter functions, (b) RC network for implementing Z2for Type-I or Z1for Type-II integer-order filters, and (c) Cauer-I RC network for implementing Z2for Type-I or Z1for Type-II generalized non-integer order bilinear filters. function is H(s)=Z2 Z1 .(34) Assuming that the impedance Z2is realized by the network in Fig.2b, its value is given by Z2(s)=R2 R1C1s+1 (R1+R2)C1s+1.(35) Using (34)–(35) and considering that Z1=R3, then the following transfer function is readily obtained HIO−I(s)=R2 R3 R1C1s+1 (R1+R2)C1s+1.(36) Comparing (1) and (36), it is derived that: GL=R2/R3, and x/y=1+R2/R1>1. Therefore, this topology implements the Type-I integer-order bilinear transfer function. The values of xand ydepend on the determination of the time constant. For example: •Assuming that τ=RC1, then x=(R1+R2)/Rand y=R1/R, with Rbeing an arbitrary value resistor. •Assuming that τ=τz=R1C1(i.e., equal to the time constant associated with the zero frequency), then x= 1+R2/R1and y=1. •Assuming that τ=√τpτz=√R1(R1+R2)C1(i.e., equal to the geometric mean of the time constants associated with the pole and zero frequencies), then x=1/y=√1+R2/R1. This choice establishes that the mean frequency will be equal to the characteristic frequency (ωm=ω0) and that the pole and zero frequencies will be symmetrically located around the mean frequency. The corresponding Type-II filters are implemented by interchanging the position of the network that implements Z2 in the previous case with the position of R3. As a result, and since (34) is still valid, the transfer function becomes HIO−II (s)=R3 R2 (R1+R2)C1s+1 R1C1s+1,(37) where it is obvious that x<y, and the aforementioned possible choices of the characteristic frequency are still valid. The realization of the corresponding fractional-order Type-I and Type-II bilinear filters could be performed by substituting the capacitor in Fig.2b by its fractionalorder counterpart. The approximation of its impedance Zα=1/Cαsαcan be performed by Foster or Cauer RC networks [8]. However, the realization of the power-law and double-order filters can not be performed by this way, due to the presence of non-integer orders that are not directly associated with Laplacian operators. Therefore, Type-I bilinear non-integer order filter (i.e., fractional-order, power-law, and double-order) will be realized by assuming that Z2(s)=RHDO(s)=RGLy(τs)α+1 x(τs)α+1β ,Z1=R,(38) while for the Type-II Z2=R,Z1(s)=R HDO(s)=R GLx(τs)α+1 y(τs)α+1β .(39) Employing a 3rd-order approximation and using the curvefitting based approximation employed also in [9], the frequency dependent impedances in (38)–(39) are approximated by the transfer function in (40) Zapprox (s)≃B3s3+B2s2+B1s+B0 s3+A2s2+A1s+A0 ,(40) with Aiand Bj(i=0,1,2,j=0,1,2,3)being positive and real coefficients. Considering, for example, the Cauer-I network demonstrated in Fig.2c, the continued fraction expansion of (40) takes the form Zapprox (s)=q0+1 q1s+1 q2+1 q3s+1 q4+1 q5s+q6 ,(41) and the design equations will be given by (42) R0,c=q0Ci,c=qiRj,c=qji=1,3,5. . . j=2,4,6 (42) where qi(j)are the coefficients of the continued fraction expansion in (41). 14044 VOLUME 12, 2024 J. Nako et al.: Bilinear Double-Order Filter Designs and Application Examples 2) PROGRAMMABLE REALIZATION The programmability of the proposed bilinear filter functions can be achieved as follows: starting from the transfer function in (28) and utilizing the curve-fitting based approximation (as in the previous Section), the resulting transfer function has the form Happrox (s)≃D3s3+D2s2+D1s+D0 s3+C2s2+C1s+C0 ,(43) with Ciand Dj(i=0,1,2,j=0,1,2,3)being also positive and real coefficients. The transfer function in (43) can be implemented by a multi-feedback structure described by the transfer function in (44) CFLF (s)= G3s3+G2 τ1s2+G1 τ1τ2s+G0 τ1τ2τ3 s3+1 τ1s2+1 τ1τ2s+1 τ1τ2τ3 .(44) The scaling factors and the time constants are calculated by equating the coefficients of (43) and (44). The transfer function in (44) can be implemented using Operational Transconductance Amplifiers (OTAs) as active elements, with their small-signal electronically controlled transconductance parameter used for implementing the scaling factors and time constants [7]. Another alternative is the utilization of an FPAA device such as the Anadigm AN231E04 device, where the programmability is achieved through the utilization of the switched-capacitor technique [10],[11]. IV. APPLICATION DESIGN EXAMPLES The transfer functions of integer-order, fractional-order, and power-law compensators are the following HIO,C(s)=GL·τs+1 xτs+1,(45) HFO,C(s)=GL·(τs)α+1 x(τs)α+1,(46) HPL,C(s)=GL·τs+1 xτs+1β .(47) Therefore, compensators are a special case of bilinear filters with y=1. In the case that x>1, this is a TypeI compensator known as lag-compensator, while for x< 1 the resulting Type-II compensator is known as leadcompensator [1],[2],[12],[13],[14],[15],[16],[17],[18], [19],[20],[21]. The corresponding expressions of shelving filters are HIO,SF (s)=HSF (s)=GL· τs x+1 xτs+1,(48) HFO,SF (s)=GL· (τs)α x+1 x(τs)α+1,(49) HPL,SF (s)=GL·τs x+1 xτs+1β .(50) Consequently, shelving filters are a special case of their corresponding bilinear filter counterparts, with x=1/y. Thus, the case of Type-I shelving filters corresponds to x>1, and these are known as low-shelving filters, while for x<1, the resulting Type-II shelving filters are known as highshelving filters [3],[22],[23],[24]. Concluding, shelving filters and compensators are different aspects of the same core, which is a bilinear filter. In other words, having available a bilinear filter structure, it can behave like a shelving filter or a compensator by choosing suitable values of the pole/zero frequencies. The only difference between shelving filters and compensators is related to the considered frequency range; shelving filters are employed for applications in the acoustic range (i.e., 20Hz–20kHz), while compensators are employed in control applications in the range of Hz. In other words, the difference is in the location of the mean frequency or, equivalently, in the location of the characteristic frequency ω0. The transfer functions of the proposed compensators and shelving filters will be HDO,C(s)=GL·(τs)α+1 x(τs)α+1β ,(51) HDO,SF (s)=GL·"(τs)α x+1 x(τs)α+1#β ,(52) respectively, offering the aforementioned benefits in terms of design flexibility and circuit complexity. A. DESIGNS OF COMPENSATORS Assuming for instance ω0=10rad/s, the range of approximation (10−2ω0,10+2ω0), and R=10k, then the values of passive elements (rounded to the E96 series defined in IEC 60063 standard) which are required for implementing the considered Type-I non-integer order compensators with GL=20dB and x=1/y=3.162, are summarized in Table 3a. The corresponding values in the case of Type-II compensators with GL=0dB and x=1/y=0.3162, are provided in Table 3b. The values of the passive elements {R1,R2,R3,C1}, which correspond to the case of integerorder compensators, are {10k, 100k, 10k, 3.01µF} and {1.1k, 10k, 10k, 28.7µF} for Types-I and II, respectively. The performance of bilinear compensators is evaluated using the OrCAD PSpice suite, with the AD844 discrete component biased at ±10V employed as CFOA. Using the component values in Table 3, the simulated responses are depicted in Fig.3, where the theoretical plots are also provided by dashes. The most important performance characteristics of the non-integer order filters are summarized in Table 4, accompanied by the theoretically predicted values given in parentheses. The corresponding results in the case of Type-I integer-order compensators are 3.13(3.19)rad/s, 27.43(31.29)rad/s, 9.3(10)rad/s, 10(10)dB, and −55.9o(−54.9o), while for the Type-II the results are 3.22(3.19)rad/s, 32.98(31.29)rad/s, 10.3(10)rad/s, 10.2(10)dB, and 54.3o(54.9o). VOLUME 12, 2024 14045 J. Nako et al.: Bilinear Double-Order Filter Designs and Application Examples TABLE 3. Values of the RC-network in Fig.1c for implementing (a) Type-I, and (b) Type-II non-integer order compensators. FIGURE 3. Simulated gain and phase responses of (a) Type-I and (b) Type-II integer-order, fractional-order, power-law, and double-order compensators implemented using the structure in Fig.2. FIGURE 4. Time-domain behavior of (a) Type-I, and (b) Type-II double-order compensators stimulated at their mean frequency by a 2Vp-p sinusoidal input. The time-domain behavior is evaluated in the case of Type-I and II double-order compensators. For this purpose, they are stimulated by a 2Vpeak-to-peak sinusoidal signal at their mean frequency. According to the waveforms in Fig.4a, the gain and the phase difference between the output and input waveforms are equal to {12.1dB, −32.9o} for the Type-I, with the associated theoretical values being {12dB, −33.09o}. The simulated values in the case of the Type-II compensator, derived from Fig.4b, are {8.04dB, 33.6o} close to the theoretically predicted ones {8dB, 33.08o}. TABLE 4. Frequency response performance characteristics of (a) Type-I, and (b) Type-II non-integer order compensators. FIGURE 5. Monte-Carlo analysis results about the mean frequency of (a) Type-I, and (b) Type-II double-order compensators. TABLE 5. Values of the coefficients and time constants in (44) for implementing (a) Type-I, and (b) Type-II non-integer order shelving filters. The sensitivity analysis is performed by employing the Monte-Carlo analysis, offered by the Advanced Analysis tool of the OrCAD PSpice. For this purpose, ±5% deviation from the nominal values of passive elements is assumed and the obtained histograms, for N=500 runs, of the mean frequency of the Type-I and II compensators are given in Fig.5. The value of the standard deviation in the case of Type-I compensator is 0.21rad/s, while for Type-II is 0.24rad/s, confirming that the presented schemes have reasonable sensitivity characteristics. 14046 VOLUME 12, 2024 J. Nako et al.: Bilinear Double-Order Filter Designs and Application Examples FIGURE 6. Implementation of the proposed generalized bilinear filter (a) Design obtained using the Anadigm Designer®2EDA software, (b) experimental setup, and (c) FPAA board with an extra board including the input and output interfaces for single-to-differential conversion and vice versa. FIGURE 7. Experimental time-domain behavior of (a) Type-I, and (b) Type-II double-order shelving filters stimulated at their mean frequency by a sinusoidal input (green:input, magenta:output). FIGURE 8. Experimental demonstration of the programmability of the proposed double-order filter for (α=0.8, β =0.7) (a) Type-I, and (b) Type-II (green:input, magenta:output). B. DESIGNS OF SHELVING FILTERS Assuming that ω0=104rad/sand a range of approximation (10−2ω0,10+2ω0), the values of the coefficients and time constants in (44) for implementing non-integer order TypesI and II shelving filters with low-frequency gains and time constant scaling factors the same as in the previous designs, are provided in Tables 5a–b, respectively. The utilized clock frequency is equal to 1MHz. Using the Anadigm Designer®2 EDA software, the resulting design is demonstrated in Fig.6a, while the experimental setup is depicted in Figs.6b-c. The gain and the input-output phase difference of the filters measured from the input-output waveforms in Fig.7, which are obtained in the cases of Type-I and Type-II double-order shelving filters stimulated by a sinusoidal signal at their mean TABLE 6. Values of the coefficients and time constants in (44) for implementing (a) Type-I, and (b) Type-II double-order shelving filters with (α=0.8, β =0.7) and (α=0.9, β =0.8). FIGURE 9. Experimental demonstration of the programmability of the proposed double-order filter for (α=0.9, β =0.8) (a) Type-I, and (b) Type-II (green:input, magenta:output). frequency, are {11.9dB, −33o} for Type-I, and {8.4dB, 29o} for Type-II. The corresponding theoretical predicted values are {12dB, −33.08o} and {8dB, 33.08o}. The programmability feature of the proposed double-order filter is demonstrated for (α=0.8, β =0.7) and (α= 0.9, β =0.8). The values of the scaling factors and VOLUME 12, 2024 14047 J. Nako et al.: Bilinear Double-Order Filter Designs and Application Examples time constants for implementing the resulting approximation transfer functions are given in Table 6. The time-domain behavior of these filters is demonstrated in Fig.8for the first case, while Fig.9corresponds to the second one. In both cases, the filters are stimulated by a sinusoidal signal at their mean frequency. The values of the gain and phase are {13.04dB, −29o} for the Type-I and {7.46dB, 26o} for the Type-II filters with (α=0.8, β =0.7), when the corresponding theoretical values are {13dB, -28.95o} and {7dB, 28.95o}. Respectively, when (α=0.9, β =0.8), the measured values of gain and phase are {12.02dB, −39o} for Type-I and {8.76dB, 32o} for Type-II double-order shelving filters, when the corresponding theoretical predicted values are equal to {12dB, −38.28o} and {8dB, 38.28o}. V. CONCLUSION Bilinear filters are applicable in control systems as lead/lagcompensators and in acoustic systems as low/high-shelving filters. The employment of non-integer orders in their transfer functions offers design flexibility but, in general, suffers from the increased circuit complexity. The proposed double-order bilinear filter function offers the most economical solution in terms of the active component count, while the presented FPAA based implementation offers programmability. All possible non-integer order versions (i.e., fractional-order, power-law, and double-order) are implementable by the same core, just by interchanging the impedances in the case of active RC implementation, or adjusting the coefficients of the approximation transfer function. Future research plans include exploring double-order PID controllers, which generalize fractional-order controllers. ACKNOWLEDGMENT The publication of the article in OA mode was financially supported by HEAL-Link. REFERENCES [1] C. Mu niz-Montero, L. A. Sánchez-Gaspariano, C. Sánchez-López, V. R. González-Díaz, and E. Tlelo-Cuautle, ‘‘On the electronic realizations of fractional-order phase-lead-lag compensators with OpAmps and FPAAs,’’ in Fractional Order Control and Synchronization of Chaotic Systems. Berlin, Germany: Springer, 2017, pp. 131–164. [2] S. Kapoulea, G. Tsirimokou, C. Psychalinos, and A. S. Elwakil, ‘‘OTA-C implementation of fractional-order lead/lag compensators,’’ in Proc. Novel Intell. Lead. Emerg. Sci. Conf. (NILES), vol. 1, Oct. 2019, pp. 38–41. [3] S. Kapoulea, A. Yesil, C. Psychalinos, S. Minaei, A. S. Elwakil, and P. Bertsias, ‘‘Fractional-order and power-law shelving filters: Analysis and design examples,’’ IEEE Access, vol. 9, pp. 145977–145987, 2021. [4] F. Sen, A. Kircay, B. Sonbas Cobb, and H. Karci, ‘‘Current-mode fractional-order shelving filters using MCFOA for acoustic applications,’’ Int. J. Electron. Commun., vol. 163, May 2023, Art. no. 154608. [5] J. Nako, C. Psychalinos, A. S. Elwakil, and S. Minaei, ‘‘Noninteger order generalized filters designs,’’ IEEE Access, vol. 11, pp. 116846–116859, 2023. [6] Anadigm. AN231E04 dpASP: The AN231E04 dpASP Dynamically Reconfigurable Analog Signal Processor. Accessed: Jan. 24, 2024. [Online]. Available: https://www.anadigm.com/an231e04.asp [7] J. Nako, C. Psychalinos, A. S. Elwakil, and D. Jurisic, ‘‘Design of higherorder fractional filters with fully controllable frequency characteristics,’’ IEEE Access, vol. 11, pp. 43205–43215, 2023. [8] G. Tsirimokou, ‘‘A systematic procedure for deriving RC networks of fractional-order elements emulators using MATLAB,’’ Int. J. Electron. Commun., vol. 78, pp. 7–14, Aug. 2017. [9] S. Kapoulea, C. Psychalinos, and A. S. Elwakil, ‘‘Power law filters: A new class of fractional-order filters without a fractional-order Laplacian operator,’’ Int. J. Electron. Commun., vol. 129, Feb. 2021, Art. no. 153537. [10] E. Tlelo-Cuautle, A. D. Pano-Azucena, O. Guillén-Fernández, and A. Silva-Juárez, ‘‘Analog implementations of fractional-order chaotic systems,’’ in Analog/Digital Implementation of Fractional Order Chaotic Circuits and Applications. Cham, Switzerland: Springer, 2020, pp. 93–114. [11] A. M. Hassanein, A. H. Madian, A. G. G. Radwan, and L. A. Said, ‘‘On the design flow of the fractional-order analog filters between FPAA implementation and circuit realization,’’ IEEE Access, vol. 11, pp. 29199–29214, 2023. [12] C. A. Monje, A. J. Calderon, B. M. Vinagre, and V. Feliu, ‘‘The fractional order lead compensator,’’ in Proc. 2nd IEEE Int. Conf. Comput. Cybern. (ICCC), Aug./Sep. 2004, pp. 347–352. [13] C. Yeroglu and N. Tan, ‘‘Classical controller design techniques for fractional order case,’’ ISA Trans., vol. 50, no. 3, pp. 461–472, Jul. 2011. [14] M. S. Tavazoei and M. Tavakoli-Kakhki, ‘‘Compensation by fractionalorder phase-lead/lag compensators,’’ IET Control Theory Appl., vol. 8, no. 5, pp. 319–329, Mar. 2014. [15] M. C. Boskovic, M. R. Rapaic, T. B. Sekara, P. D. Mandic, M. P. Lazarevic, B. Cvetkovic, B. Lutovac, and M. Dakovic, ‘‘On the rational representation of fractional order lead compensator using Padé approximation,’’ in Proc. 7th Medit. Conf. Embedded Comput. (MECO), Jun. 2018, pp. 1–4. [16] A. A. Dastjerdi, B. M. Vinagre, Y. Chen, and S. H. HosseinNia, ‘‘Linear fractional order controllers; a survey in the frequency domain,’’ Annu. Rev. Control, vol. 47, pp. 51–70, Apr. 2019. [17] G. Maione, ‘‘Design of cascaded and shifted fractional-order lead compensators for plants with monotonically increasing lags,’’ Fractal Fractional, vol. 4, no. 3, p. 37, Jul. 2020. [18] A. Tepljakov, B. B. Alagoz, C. Yeroglu, E. A. Gonzalez, S. H. Hosseinnia, E. Petlenkov, A. Ates, and M. Cech, ‘‘Towards industrialization of FOPID controllers: A survey on milestones of fractional-order control and pathways for future developments,’’ IEEE Access, vol. 9, pp. 21016–21042, 2021. [19] I. Petráš, Handbook of Fractional Calculus with Applications, no. 6. Berlin, Germany: De Gruyter, 2019. [20] P. Prommee and P. Pienpichayapong, ‘‘Reconfigurable fractional-order operator and bandwidth expansion suitable for PIαcontroller,’’ IEEE Trans. Ind. Electron., vol. 71, no. 5, pp. 5126–5136, May 2024. [21] G. Avon, R. Caponetto, E. Murgano, and M. G. Xibilia, ‘‘Implementation of a fully analog feedback loop with a carbon-black-based fractional order controller,’’ ISA Trans., vol. 135, pp. 105–114, Apr. 2023. [22] S. Sarkka and A. Huovilainen, ‘‘Accurate discretization of analog audio filters with application to parametric equalizer design,’’ IEEE Trans. Audio, Speech, Language Process., vol. 19, no. 8, pp. 2486–2493, Nov. 2011. [23] V. Välimäki and J. Reiss, ‘‘All about audio equalization: Solutions and frontiers,’’ Appl. Sci., vol. 6, no. 5, p. 129, May 2016. [24] G. A. Haidar, R. A. Z. Daou, and X. Moreau, ‘‘Synthesis of a fractional order audio boost filter,’’ in Proc. 29th Int. Conf. Microelectron. (ICM), Dec. 2017, pp. 1–5. JULIA NAKO (Graduate Student Member, IEEE) received the B.Sc. and M.Sc. degrees from the University of Patras, Greece, in 2021 and 2023, respectively, where she is currently pursuing the Ph.D. degree with the Postgraduate Studies Program ‘‘Electronics-Circuits and Systems,’’ Physics Department. She is a member of the Analog VLSI Design Team, Electronics Laboratory, working under the supervision of Prof. Costas Psychalinos. Her current research interests include the design of analog integrated circuits and systems for signal processing, including non-integer order circuits, control systems, and biomedical circuits. 14048 VOLUME 12, 2024