optimized fractional low and highpass filters of (1 + α

$: Optimized fractional low and highpass filters of (1 + α$
635Bull. Pol. Ac.: Tech. 68(3) 2020

BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 68, No. 3, 2020DOI: 10.24425/bpasts.2020.133123

Abstract. This work proposes an optimum design and implementation of fractional-order Butterworth filter of order (1 + α), with the help of analog reconfigurable field-programmable analog array (FPAA). The designed filter coefficients are obtained after dual constraint optimization to balance the tradeoffs between magnitude error and stability margin together. The resulting filter ensures better robustness with less sensitivity to parameter variation and minimum least square error (LSE) in magnitude responses, passband and stopband errors as well as a better –3 dB normalized frequency approximation at 1 rad/s and a stability margin. Finally, experimental results have shown both lowpass and highpass fractional step values. The FPAA-configured outputs represent the possibility to implement the real-time fractional filter behavior with close approximation to the theoretical design.

Key words: fractional-order filter, FPAA, realization, robustness, bilevel optimization.

Optimized fractional low and highpass filters of (1 + α) order on FPAA

N. SINGH1*, U. MEHTA2, K. KOTHARI2, and M. CIRRINCIONE2

1 School of Engineering and Physics, University of the South Pacific, Laucala, Suva, Fiji, now at SPARC Hub Headquarters, 71 Normanby Rd, Notting Hill VIC 3168, Australia

2 School of Engineering and Physics, University of the South Pacific, Laucala, Suva, Fiji

At first, fractional-order filters (FOF) were critically stud-ied in [7] and shown that fractional filters (also called frac-tional-step filters) are realizable with reasonable overshoot in the passband region. In most cases α ranges from 0.1 to 0.9. Since, as is well known, any second-order filter transfer func-tion leads to a resonant peak in the magnitude response, the magnitude response of the fractional-order Butterworth filter has been explored to address this [4, 8]. The same concept has also been used for elliptical and Chebyshev filters [9, 10]. More recently, Kubanek and Freeborn [11] have proposed a new fractional-order low-pass filter (FLPF) design focussing on the search for coefficients to approximate a second order lowpass filter transfer function with arbitrary quality factor Q. In [10] and [12], coefficients of FLPF transfer function were selected to approximate a flat passband response of a first order Butter-worth filter. Another method was proposed in [13] to approx-imate coefficients for different cases of a normalized FLPF transfer function, but this method was based on limited search for objective functions, and focused on only few parameters, such as transition bandwidth and maximum allowable peak.

Many works have recently been completed on obtaining filter parameters through optimization subroutines. Mahata et al. in [13] have used a nature-inspired optimization subrou-tine called the gravitational search algorithm (GSA) to opti-mize parameters of the (1 + α) order transfer function. In this method, authors have presented an approximation of FLO using third order transfer function instead of second order transfer function. In another recent work, a second order approxima-tion of FLO has been used to design a fractional-order low pass Butterworth filter [15]. It was developed using CMOS of the differential difference current conveyor, which was fabri-cated in a CMOS processor. The benefit of this particular filter was that it allowed low voltage operation within ±500 mV. An 

1. Introduction

Fractional calculus (FC) has been seen as more generalized than the traditional integer calculus, manifesting the potential to accomplish what integer calculus cannot [1]. Recently, it is being applied in many fields of science and technology such as engineering, biomedicine, control theory, diffusion theory, material science, robotics and signal processing [1‒3]. The FC applied to signal processing and circuit theory has shown more potential. For example, FC has been imported to electronics, making it possible to design and realize fractional order filter circuits [4]. Well known analog filters contain inductors and capacitors whose numbers determine the filter order. However, an inductor or capacitor with fractional impedance can be gen-eralized and these elements in the fractional domain are called fractance devices [5, 6]. Fractance devices are not available commercially, however it is possible to emulate them using resistor-capacitor (RC) or resistor-inductor-capacitor (RLC) trees and using platforms such as the field-programmable ana-log array (FPAA). In order to realize fractance devices physi-cally for fractional order circuits and systems, sα is used (where α 2 R). This fractional Laplacian operator (FLO) is a multi-val-ued expression with an infinite number of Riemann surfaces. When constraining sα by imposing integer-order approximations, a transfer function is described using several definitions avail-able, such as those provided by like Riemann-Liouville (R-L), Caputo, Weyl and Grunwald-Letnikov (G-L). In this work, an R-L definition is used, based on Euler՚s gamma function.

*e-mail: [email protected]

Manuscript submitted 2019-11-05, revised 2020-01-21, initially accepted for publication 2020-02-16, published in June 2020

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636

N. Singh, U. Mehta, K. Kothari, and M. Cirrincione

Bull. Pol. Ac.: Tech. 68(3) 2020

electronically reconfigurable fractional-order filter has been designed in a recent work that can be configured to work as either FLPF or FHPF [16].

In particular, [12] shows that FOF can provide precise con-trol of attenuation, i.e. –3 dB frequency and stopband atten-uation. Integer-order filters yield –20n dB/decade  stopband attenuations, where n is the integer order, however fractional order provides greater control with –20(n + α) dB/decade stop-band attenuation, where α is any real positive value less than 1 [12, 17]. Basically the (n + α) FOF can give more precise con-trol of the attenuation slope by an additional degree of freedom as compared to integer-order filters of order n. Band pass and band reject filters with asymmetric stopband characteristics [18] can be easily implemented using FOF and can be used as phase discriminators.

However, there are some challenges to obtain the optimum coefficients of the fractional-order transfer function (FOTF). It is desired to select filter coefficients which are optimum for any real differential orders while providing greater robust-ness. Indeed, variation in coefficients may affect  the –3 dB frequency, stopband attenuation and stability. In this work, the coefficient and real value differential orders are computed so that the desired characteristic can be always maintained even if the parameters vary from the designed value. Firstly, the optimized values of the filter are calculated with the help of modified particle-swarm-optimization (PSO) in order to sat-isfy the constraint requirements. Compared to other optimiza-tion routines, the advantages of the PSO are that it is easy to implement with few parameters needing adjustment. The per-formance of the new fractional-step filters has been verified. Secondly, the optimum order of approximation sα is proposed in order to implement the fractional differentiator in hardware with acceptable accuracy, and the resulting filter is implemented in the Anadigm development environment of FPAA. The wave-forms from both the proposed lowpass filter (LPF) and highpass filter (HPF) are measured with various ranges of signal input frequencies. The performance of non-integer filters of order (1 + α) has been studied and compared with corresponding integer filters through both experimentation and simulation. The obtained results confirm that the actual fractional filter՚s behavior closely follows the theoretical approximations for all values of α. Table 1 shows the comparison of previous work in terms of design approach mapped with benefits.

2. Fractional transfer functions of (1 + α) order

2.1. LPF transfer function of (1 + α) order. The FLPF trans-fer function of the (1 + α) order has previously been studied in [10‒12]. Most works focused on the design and implementation of FOTF in the following form:

H LP1 + α =

k1

s1 + α + sαk2 + k3, 0 ∙ α ∙ 1. (1)

The coefficients k2 and k3 are selected to yield a flat pass-band response while k1 is usually kept at the constant value of 1, which leads to a DC gain of 1/k3. Any filter realization is evaluated based on a flat passband with minimum error in magnitudes in the passband and stopband frequencies and with –3 dB normalized frequency almost close to 1 rad/s. This is achieved by minimizing the error objective function with the ideal normalized first order Butterworth response.

Freeborn et al. [10] have optimized the coefficients in (1) through a least squares error (LSE) approach that compared the response with a first order Butterworth response over the nor-malized frequency range of ω = 0.01 ¡ 1 rad/s. The obtained coefficients yield the minimum cumulative passband error. The numerical  search  is  limited  to 0 < k2 < 2,  0 < k3 < 1 and k1 = 1. Another attempt was presented using the MATLAB optimization tool based on a nonlinear least squares fitting [12], where the improved LSE comes at the cost of the stability mar-gin. Thus the trade-offs between LSE and stability margin are hard to be guaranteed in a uniform way against different design objectives. Because of the above difficulties, any optimization technique should be multi-objective.

In this work, modified particle swarm optimization (mPSO) has been developed to work with more than two objectives at a time. Main focus is on the design of an optimum FOTF so that the uniformity of the trade-offs between LSE and the sta-bility margin can be guaranteed. Although the computational complexity of the problem is further increased by the bilevel structure, the desired solution can be achieved in a finite time. In this paper, the novelty lies in the fact that the filter designed satisfies more than one characteristic at the same time. An objective function has been developed that gives the minimum of  the magnitude error, with  flat passband  response, –3 dB 

Table 1 Summary of design approaches and benefits

Design approach Benefits

Coefficient search to approximate TF with arbitrary Q [11] Greater stability and higher degree of freedom for coeffiecient variation

Coefficients selected to approximate flat passband response [12] Minimum passband error and higher stability

GSA used to optimize transfer function coefficients [14] Third order transfer function is used to approximate Butterworth response using FLO

Proposed design approach using PSO The proposed filter ensures better robustness with less sensitivity to parameter variation, minimum least square error (LSE) in magnitude responses, passband and stopband errors as well as a better –3 dB normalized frequency approximation at 1 rad/s, and a stability margin

637

Optimized fractional low and highpass f ilters of (1 + α) order on FPAA


frequency reached at 1 rad/s. The proposed algorithm solves a two-step problem of the so-called bilevel ptimization routine. Further, the simulated (1 + α) order LPFs with fractional steps from α = 0.01 to α = 0.99 have been developed and analyzed statistically. The results are compared with the existing FOF reported in recent literature. In the following section, the mod-ified mPSO with selected coefficients is presented. The advan-tage of global search and the optimum robust result findings feature of mPSO are investigated in this analysis.

2.2. mPSO for bilevel optimization. It is necessary to choose the best filter parameters to meet the design requirements and then these selected values can be used to implement the frac-tional filter on the reconfigurable platform. Indeed [19] shows that the basic PSO is computationally less expensive and has lower memory requirements than other optimization routines. In addition, it has a relatively small number (3 to 5) of user-defined parameters, which are not critical for the convergence and final accuracy of the algorithm. It is also suitable for solving contin-uous nonlinear optimization problems. The basic PSO version with inertia weight is described in the formula below [20].

ai Ã Ωai + R(0, ϕ1) ( pi ¡ xi) +ai + R(0, ϕ2) ( pg ¡ xi),xi Ã xi + ai

(2)

where i 2 N, Ω is an inertia weight factor which determines speed of the particle, N is the number of particles (usually N <= 40) and  is the Kronecker multiplication; xi gives the particle՚s actual location and ai defines the step velocity of the particle. The parameters ϕ1 and ϕ2 (acceleration coefficients) determine the magnitude of the random forces in the direc-tion of best particle pi and the neighborhood best pg. R(0, ϕj), ( j = 1, 2) delivers a vector of random numbers uniformly dis-tributed in

£0, ϕj¤, after each iteration and for each particle.

The inertia weight factor Ω is updated by the following law:

Ω = Ωmax ¡ (Ik ¡ 1)³Ωmax ¡ Ωmin

Im ¡ 1

´ (3)

in which Ωmax and Ωmin are maximum and minimum inertia weights, respectively, Ik is the current iteration and Im is the maximum iteration number. Table 2 shows the parameter values used in this work.

Table 2 mPSO parameters

Parameters N Ωmin Ωmax Im ϕ1 ϕ2

Value 35 0.1 0.9 100 1 2

In the implementation proposed, we have modified the boundary conditions and this constraints user settings. We concider this algorithm as mPSO. When a given boundary is violated by any of the particles, this particle i is returned to its

previous position xi. In this scenario, mPSO will reverse the step (ai) in the opposite direction (i.e. ai = –ai). In this work, k2 and k3 of (1) are fed in as the variables to be optimized. The optimization is carried out with the following bilevel objectives in order to balance the tradeoffs between LSE and –3 dB vari-ation close to the normalized frequency of 1 rad/s.

Level 1: The minimum LSE, calculated as

Ec( jω) = i =1

N

∑ jjB1( jωi)j ¡ jH LP1 + α( jωi)jj2 (4)

where Ec is the cumulative error and the magnitude responses jB1( jω)j and jH LP

1 + α( jω)j are the first order Butterworth filter and the fractional order LPF of order (1 + α) at pulsation ωi, respectively, and N is the number of samples taken between frequency 0.01‒1.5 rad/s.

Level 2: –3 dB frequency closest to 1 rad/s minimization of:

(ω3dB ¡ 1)2. (5)

The proposed mPSO with constraints in (4) and (5) and the optimum set of coefficients were obtained for all (1 + α) order transfer functions. The linear curve-fitted expressions (6, 7) were obtained in terms of parabolic function of order 3 and as a function of α as follows.

k2proposed = 0.5293α3 ¡ 0.3156α2 + 0.9672α + 0.2653 (6)

k3proposed = –0.1981α3 + 0.2471α2 + 0.2359α + 0.7233 (7)

The coefficients k2 and k3 that yielded the best performance for (1) with k1 = 1, when the order is increased from 1.01 to 1.99 in steps of 0.01, are described and compared in Fig. 1.

2.3. HPF transfer function of (1 + α) order. An FHPF can also be obtained from the FLPF transfer function by means of the transformation highlighted in [21]. There are three differ-ent transformations, each of which has its own pros and cons. In the previous section, coefficients of FLPF (1) were chosen

Fig. 1. Comparison of k2 and k3 coefficients to approximate fractional step filters: proposed, LSE [10], optimized LSE [12]

Optimized fractional low-and high-pass filters of order (1+α) on FPAA

Table 1: Summary of design approaches and benefits

Design Approach Benefits

Coefficient search to approximate TF with arbitrary Q [11] Greater stability and higher degree of freedom for coeffiecient variation.Coefficients selected to approximate flat passband response [12] Minimum passband error and higher stabilityGSA used to optimize transfer function coefficients [14] Third order transfer function is used to approximate Butterworth response using FLO

Proposed design approach using PSO

The proposed filter ensures better robustness with less sensitivity to parameter variation,minimum least square error (LSE) in magnitude responses, passband, and stopband errorsas well as a better -3dB normalized frequency approximation at 1 rad/s, and stabilitymargin.

Table 2: mPSO parametersParameters N Ωmin Ωmax Im φ1 φ2

Value 35 0.1 0.9 100 1 2

maximum iteration number. Table 2 shows the parameter val-ues used in this work.

In proposed implementation, we have modified the bound-ary conditions and this constraint sets by the user. We say thisalgorithm as mPSO. When a given boundary is violated by anyof the particles, this particle i is returned to its previous po-sition xi. In this scenario, mPSO will reverse the step (ai) inopposite direction (i.e. ai =−ai). In this work, k2 and k3 of (1)are fed in as the variables to be optimized. The optimizationis carried out with the following bilevel objectives in order tobalance the tradeoffs between LSE and -3dB variation close tothe normalized frequency of 1 rad/s.

Level 1: The minimum LSE, calculated as

Ec( jω) =N

∑i=1

| |B1( jωi)|−∣∣HLP

1+α ( jωi) |∣∣2 (4)

where Ec is the cumulative error and the magnitude responses|B1( jωi)| and

∣∣HLP1+α ( jωi)

∣∣ are first order Butterworth filterand the fractional order LPF of order (1+α) at pulsation ωi,respectively and N is the number of samples taken betweenfrequency 0.01−1.5 rad/s.

Level 2: -3dB frequency closest to 1 rad/s. minimization of:

(ω3dB −1)2 (5)

The proposed mPSO with constraints in (4) and (5) and theoptimal set of coefficients were obtained for all (1 + α) or-der transfer functions. The linear curve-fitted expressions (6-7) were obtained in terms of parabolic function of order 3 andas a function of α as follows.

k2proposed = 0.5293α3 −0.3156α2 +0.9672α +0.2653 (6)

k3proposed =−0.1981α3+0.2471α2+0.2359α+0.7233 (7)

The coefficients k2 and k3 that yielded the best performance for(1) with k1 = 1 when the order is increased from 1.01 to 1.99in steps of 0.01 are described and compared in Figure 1.

2.3. HPF transfer function of (1+α) order. An FHPF canalso be obtained from FLPF transfer function by the transfor-mation highlighted in [21]. There are three different transfor-mations each of which has its own pros and cons. In the previ-

α

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

k2,k3

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

k2 (LSE)

k3 (LSE)

k2 (Optimized LSE)

k3 (Optimized LSE)

k2 (Proposed)

k3 (Proposed)

Fig. 1: Comparison of k2 and k3 coefficients to approximatefractional step filters: Proposed, LSE [10], optimized LSE [12]

ous section, coefficients of FLPF (1) were chosen using bilevelmPSO and provided maximally flat passband response, min-imum stopband and passband errors and -3dB frequency ap-proximately or equal to 1 rad/s. Same way the transformationprovided minimum passband and stopband errors for the HPFwhen the proposed coefficients were chosen from the previ-ous subsection. It is noteworthy that this HPF transfer functionis obtained just by replacing the Laplacian operator s by 1/swhich results (1) to become:

HHP31+α (s) =

s1+α k1

s1+α k3 + sk2 +1(8)

The unity value of coefficient k1 provides a passband gain of1/k3 in (8). By Comparing (1) and (8) an interchange of de-nominator coefficients is present in (8) for the coefficient ofmiddle term k2. The term sα in (1) has been replaced by s in(8). At the frequency of 1 rad/sec, this lowpass to highpasstransformation provides the same response characteristics.

3. Numerical Comparison and Examinationwith Fixed Parameters

A detailed error analysis, stability analysis and sensitivity toparameter variation are performed in the following subsectioncomparing the performance of the proposed filter coefficientswith those presented in [12] and [10].

3.1. LPF evaluation. The FLPF offers stopband attenuationsof −20(1+α)dB per decade. The proposed filter performanceis analysed after calculating the coefficients, in terms of pass-band error, stopband error, stability, -3dB frequency and sensi-tivity to parameter variation.

Bull. Pol. Ac.: Tech. XX(Y) 2019 3

k 2, k

3

α

638



using bilevel mPSO and provided a maximally flat passband response, minimum stopband and passband errors and –3 dB frequency close to or equal to 1 rad/s. In the same manner, the transformation provided minimum passband and stopband errors for the HPF when the proposed coefficients were chosen from the previous subsection. It is noteworthy that this HPF transfer function is obtained just by replacing the Laplacian operator s by 1/s which results in (1) to become:

H HP31 + α (s) = s1 + αk1

s1 + αk3 + sk2 + 1. (8)

The unity value of coefficient k1 provides a passband gain of 1/ k3 in (8). By comparing (1) and (8), an interchange of denom-inator coefficients is present in (8) for the coefficient of middle term k2. The term sα in (1) has been replaced by s in (8). At the frequency of 1 rad/sec, this lowpass to highpass transformation provides the same response characteristics.

3. Numerical comparison and examination with fixed parameters

A detailed error analysis, stability analysis and sensitivity to parameter variation are performed in the following subsection, comparing the performance of the proposed filter coefficients with those presented in [12] and [10].

3.1. LPF evaluation. The FLPF offers stopband attenuations of –20(1 + α) dB per decade. The proposed filter performance is analysed after calculating the coefficients, in terms of passband error, stopband error, stability, –3 dB frequency and sensitivity to parameter variation.

3.1.1. Magnitude response error. It has been evaluated by using mean square error (MSE) given by (9)

MSE = i=1

M∑ jjHB1(ω i)j ¡ jH LP

1 + α(ω i)jj2

M (9)

where HB1(ω i) is the magnitude response of first order Butter-worth filter at frequency ω i for 100,000 samples taken within the frequency range from 0.001 to 100 rad/s. jH LP

1 + α(ω i)j is the magnitude response of the (1 + α) order Butterworth LPF.

The magnitude response performance of the designed filters is also compared based on two error matrices, namely passband error (PE) and stopband error (SE), defined as following.

PE = 20log10i=1

K∑ jjHB1(ω i)j ¡ jH LP

1 + α(ω i)jj2

KdB (10)

where, K = 50,000 and 0.001 ∙ ω ∙ 1.

SE = 20log10i=1

L∑ jjHB1(ω i)j ¡ jH LP

1 + α(ω i)jj2

LdB (11)

where, L = 50,000 and 1 ∙ ω ∙ 100.Both PE and SE errors of fractional (1 + α) order filters are

listed, which is calculated using the coefficients used in [10, 12] and proposed coefficients in Table 3. It is also clear from Fig. 2 that the proposed filter would mostly provide the lowest errors for almost all orders of filters, especially for PE. As for the SE values, Fig. 2b gives errors consistent but slightly lower in the range from 1 to 1.9 for (1 + α).

Fig. 2. (a) PE index values: 1. by [10], 2. by [12], 3. proposed; (b) SE index values: 1. by [10], 2. by [12], 3. proposed

N. Singh, U. Mehta, K. Konthari, and M. Cirrincione

Table 3: Comparison of PE and SE matrices for (1+α) orderfilters

ErrorMethods 1+α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

PE (dB)[10] -37.80 -38.33 -38.72 -38.90 -38.79 -38.45 -37.92 -37.27 -36.58[12] -38.07 -38.71 -39.18 -39.40 -39.31 -38.95 -38.38 -37.69 -36.95Proposed -43.03 -42.84 -42.63 -42.44 -42.29 -42.21 -42.19 -42.23 -42.30

SE (dB)[10] -47.67 -46.28 -45.31 -44.59 -44.06 -43.65 -43.34 -43.11 -42.92[12] -51.30 -49.27 -47.92 -46.97 -46.27 -45.76 -45.37 -45.07 -44.84Proposed -78.16 -70.40 -66.37 -63.85 -62.13 -60.90 -60.00 -59.33 -58.87

3.1.1. Magnitude Response Error. It has been evaluated byusing mean square error (MSE) given by (9)

MSE =

√√√√√M∑

i=1

∣∣|HB1(ωi)|−∣∣HLP

1+α(ωi)∣∣∣∣

2

M(9)

where HB1(ωi) is the magnitude response of first order Butter-worth filter at frequency, ωi for 100,000 samples taken withinthe frequency range from 0.001 to 100 rad/s.

∣∣HLP1+α(ωi)

∣∣ is themagnitude response of the (1+α) order Butterworth LPF.

The magnitude response performance of the designed filtersare also compared based on two error matrices namely pass-band error (PE) and stopband error (SE) as defined following.

PE = 20log10

√√√√√K∑

i=1


1+α(ωi)∣∣∣∣

2

K

dB (10)

where, K = 50,000 and 0.001 ≤ ω ≤ 1.

SE = 20log10

√√√√√L∑

i=1


1+α(ωi)∣∣∣∣

2

L

dB (11)

where, L = 50,000 and 1 ≤ ω ≤ 100.Both PE and SE errors of fractional (1+α) order filters are

listed which is calculated using the coefficients used in [12],[10] and proposed coefficients in Table 3. It is also clear fromFig. 2 that the proposed filter would mostly provide the lowesterrors for almost all orders of filters especially for PE. As forthe SE values, Fig. 2b gives errors consistent but slightly lowerin the range from 1 to 1.9 for (1+α).

3.1.2. Stability Region Analysis. An important criterion tobe examined with the proposed optimized coefficients is sta-bility margin, described in terms of pole angle and the regionof instability. To analyze the stability margin of FLPF, let usconvert (1) to complex W-plane [22]. The transformation ispossible for common value transfer function and it convertsthe FOTF to the W-plane by taking s = W m where α = l/m(l,m are integers and selected for the desired α value, if α is arational number).

This transformation changes (1) into

H (W ) =k1

W m+k + k2W k + k3(12)

1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Passban

dError

(dB)

-44

-42

-40

-38

-36

-34123

(a)

1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Stopban

dError

(dB)

-110

-100

-90

-80

-70

-60

-50

-40

123

(b)

Fig. 2: (a) PE index values: 1. by [10], 2. by [12] 3. proposed(b) SE index values: 1. by [10], 2. by [12] 3. proposed

The characteristic equation from (12) in W-plane should beensured that all the poles obtained with optimized coefficientsare in the stable region. It is necessary to observe further howfar the absolute pole angles, |θW |, are from the value π

2m . Ifany |θW |< π

2m then the system is unstable. The minimum rootangles have been calculated for α=0.01-0.99 with l=10 to 990in steps of 10 while m = 1000. First, by equating the denom-inator of (12) to 0 for all values of α , the absolute value ofminimum angle of the root (|θW |min) were calculated and plot-ted in Fig. 3(a). The criterion for stability is, |θW | > π

2m andaccording to chosen l and m, for stability |θW | > π

2m = 0.09.The minimum root angle using the proposed coefficients showa visibly higher margin than others. Interestingly, the opti-mized coefficients with LSE in [12] yield a lower stability thanthe proposed method even though their method gives a lowerLSE. For (1+α) between 1.2 to 1.8, the minimum root an-gles are further away from the unstable boundary as comparedwith coefficients from [12] and [10]. It can be concluded fromresults that the proposed design has obtained a better stabilitymargin with a lower value of LSE.

3.1.3. -3dB frequency analysis. It is desired that the pulsa-tion from the transfer function coefficient be approximately 1rad/s at -3dB. Thus for each order from 1.01 to 1.99, the fre-quency at which the magnitude response reaches -3dB is com-pared. According to the criterion, the fractional Butterworthfilter that reaches -3dB below its DC value at frequency 1 rad/sis the best choice for implementation. The -3dB frequenciesfor each order are given in Fig. 3(b) and numerically calculated

4 Bull. Pol. Ac.: Tech. XX(Y) 2019



ErrorMethods 1+α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

PE (dB)[10] -37.80 -38.33 -38.72 -38.90 -38.79 -38.45 -37.92 -37.27 -36.58[12] -38.07 -38.71 -39.18 -39.40 -39.31 -38.95 -38.38 -37.69 -36.95Proposed -43.03 -42.84 -42.63 -42.44 -42.29 -42.21 -42.19 -42.23 -42.30

SE (dB)[10] -47.67 -46.28 -45.31 -44.59 -44.06 -43.65 -43.34 -43.11 -42.92[12] -51.30 -49.27 -47.92 -46.97 -46.27 -45.76 -45.37 -45.07 -44.84Proposed -78.16 -70.40 -66.37 -63.85 -62.13 -60.90 -60.00 -59.33 -58.87


MSE =

√√√√√M∑

i=1


1+α(ωi)∣∣∣∣

2

M(9)


∣∣HLP1+α(ωi)



PE = 20log10

√√√√√K∑

i=1


1+α(ωi)∣∣∣∣

2

K

dB (10)

where, K = 50,000 and 0.001 ≤ ω ≤ 1.

SE = 20log10

√√√√√L∑

i=1


1+α(ωi)∣∣∣∣

2

L

dB (11)





H (W ) =k1

W m+k + k2W k + k3(12)

1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Passban

dError

(dB)

-44

-42

-40

-38

-36

-34123

(a)

1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Stopban

dError

(dB)

-110

-100

-90

-80

-70

-60

-50

-40

123

(b)









Stop

band

Err

or (d

B)

1 + α



ErrorMethods 1+α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

PE (dB)[10] -37.80 -38.33 -38.72 -38.90 -38.79 -38.45 -37.92 -37.27 -36.58[12] -38.07 -38.71 -39.18 -39.40 -39.31 -38.95 -38.38 -37.69 -36.95Proposed -43.03 -42.84 -42.63 -42.44 -42.29 -42.21 -42.19 -42.23 -42.30

SE (dB)[10] -47.67 -46.28 -45.31 -44.59 -44.06 -43.65 -43.34 -43.11 -42.92[12] -51.30 -49.27 -47.92 -46.97 -46.27 -45.76 -45.37 -45.07 -44.84Proposed -78.16 -70.40 -66.37 -63.85 -62.13 -60.90 -60.00 -59.33 -58.87


MSE =

√√√√√M∑

i=1


1+α(ωi)∣∣∣∣

2

M(9)


∣∣HLP1+α(ωi)



PE = 20log10

√√√√√K∑

i=1


1+α(ωi)∣∣∣∣

2

K

dB (10)

where, K = 50,000 and 0.001 ≤ ω ≤ 1.

SE = 20log10

√√√√√L∑

i=1


1+α(ωi)∣∣∣∣

2

L

dB (11)





H (W ) =k1

W m+k + k2W k + k3(12)

1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Passban

dError

(dB)

-44

-42

-40

-38

-36

-34123

(a)

1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Stopban

dError

(dB)

-110

-100

-90

-80

-70

-60

-50

-40

123

(b)









Pass

band

Err

or (d

B)

1 + α

(a) (b)

639



3.1.2. Stability region analysis. An important criterion to be examined with the proposed optimized coefficients is the sta-bility margin, described in terms of pole angle and the region of instability. To analyze the stability margin of FLPF, let us con-vert (1) to complex W-plane [22]. The transformation is possible for the common value transfer function and it converts the FOTF to the W-plane by taking s = W m where α = l/m (l, m are inte-gers selected for the desired α value, if α is a rational number).


H(W) = k1

W m + k + k2W k + k3. (12)

The characteristic equation from (12) in W-plane should ensure that all the poles obtained with optimized coefficients are in the stable region. It is necessary to observe further how far the absolute pole angles, jθW j, are from the value of π

2m. If any jθW j < π2m , then the system is unstable. The minimum root angles have been calculated for α = 0.01‒0.99 with  l = 10 to 990 in steps of 10 while m = 1000. First, by equating the denominator of (12) to 0 for all values of α, the absolute values of the minimum angle of the root (jθW jmin) were calculated and

plotted in Fig. 3a. The criterion for stability is jθW j < π2m and according to the chosen l and m, for stability jθW j < π2m = 0.09°. The minimum root angle using the proposed coefficients shows a visibly higher margin than others. Interestingly, the optimized coefficients with LSE in [12] yield a lower stability than the proposed method even though their method gives a lower LSE. For (1 + α) between 1.2 to 1.8, the minimum root angles are further away from the unstable boundary as compared with coefficients from [12] and [10]. It can be concluded from the results that the proposed design has obtained a better stability margin with a lower value of LSE.

3.1.3. –3 dB frequency analysis. It is desired that the pulsation from the transfer function coefficient be approximately 1 rad/s at –3 dB. Thus for each order from 1.01 to 1.99, the frequency at which the magnitude response reaches –3 dB is compared. According to the criterion, the fractional Butterworth filter that reaches –3 dB below its DC value at the frequency of  1 rad/s is the best choice for implementation. The –3 dB frequencies for each order are given in Fig. 3b and numerically calculated with different sets of coefficients from [10, 12] along with the pro-posed technique for orders from 1.01 to 1.99 in steps of 0.01. Both coefficients from [10] and [12] show similar deviations

Table 3 Comparison of PE and SE matrices for (1 + α) order filters

Error Methods1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

PE (dB)

[10] –37.80 –38.33 –38.72 –38.90 –38.79 –38.45 –37.92 –37.27 –36.58

[12] –38.07 –38.71 –39.18 –39.40 –39.31 –38.95 –38.38 –37.69 –36.95

Proposed –43.03 –42.84 –42.63 –42.44 –42.29 –42.21 –42.19 –42.23 –42.30

SE (dB)

[10] –47.67 46.28 –45.31 –44.59 –44.06 –43.65 –43.34 –43.11 –42.92

[12] –51.30 –49.27 –47.92 –46.97 –46.27 –45.76 –45.37 –45.07 –44.84

Proposed –78.16 –70.40 –66.37 –63.85 –62.13 –60.90 –60.00 –59.33 –58.87

α 1 + α

(a) Stability margin (b) –3dB frequency

Fig. 3. (a) Stability comparison: 1. by [10], 2. by [12] and 3. proposed; (b) –3 dB frequency comparison: 1. by [10], 2. by [12] and 3. proposed


α

0 0.2 0.4 0.6 0.8 1

Minim

um

phasean

gle(degree)

0.06

0.08

0.1

0.12

0.14

0.16

0.18123Limit of stability = 0.09

UNSTABLE

STABLE

(a) Stability margin

1 + α

1 1.2 1.4 1.6 1.8 2

-3dB

Frequency

(rad/s)

0.85

0.9

0.95

1

1.05

1.1

1.15123

(b) -3dB frequency

Fig. 3: (a)Stability comparison: 1. by [10], 2. by [12] and 3.proposed (b)-3dB frequency comparison: 1. by [10], 2. by [12]and 3. proposed

with different sets of coefficients from [10], [12] and proposedtechnique for orders from 1.01 to 1.99 in steps of 0.01. Bothcoefficients from [10] and [12] show similar deviations in -3dBfrequencies, at order 1.1 < (1+α) < 1.5 frequency increasesand reaches peak of 1.13 rad/s and 1.09 rad/s, respectively. Af-ter that, frequency drops gradually and at 1.8 order it crosses1 rad/sec margin in [10] and at 1.7 rad/s in [12]. However,the proposed filter coefficients show the closest agreement to1 rad/s for all orders. It can be seen that the filter has a lowerripple at -3dB frequency, fluctuating between 1.005 to 0.998rad/s. Thus, using the bilevel optimization, the desirable filtercharacteristics can be improved both in lower LSE value, andhigh stability margin and better -3dB frequency.

3.1.4. Stopband attenuation.The transfer function (1) hasdifferent roll-off characteristics with different sets of coeffi-cients. The stopband attenuation determines how the mag-nitude response changes from flat passband response to theideal stopband attenuation of −20(1+α) dB/decade. Stop-band attenuation is another characteristics of the Butterworthresponse, the slope of the roll-off characteristics determines thesuperiority of the design, that is sharper the slope, the better thedesigned filter. In order to compare the roll-off characteristicsof equation (1) from various methods, the slopes of the magni-tude of transfer functions with coefficients from [10], [12] andproposed values are given in Fig. 4. The solid green line is theideal characteristic of -20(1+α), changing from a value of -20dB/decade when (1+α) = 1 to -40 dB/decade when (1+α)=2; corresponding to the traditional integer-order attenuations

1 + α

1 1.2 1.4 1.6 1.8 2

Attenuation(dB/d

ecad

e)

-40

-35

-30

-25

-20

-15

1.46 1.48 1.5

-27

-26.5

-26

-25.5

1 [ω = 1-10]2 [ω = 1-10]3 [ω = 1-10]Ideal [ω = 1-10]1 [ω = 10-100]2 [ω = 10-100]3 [ω = 10-100]Ideal[ω = 10-100]

Fig. 4: Stopband attenuation for (1 + α) order ButterworthLPF implementation: 1. by [10], 2. by [12] 3. proposed

available for ω = [10,100] rad/s. The slope between frequen-cies ω=1 to ω=10 rad/s are shown with blue lines and ω = 10to ω = 100 rad/s with red lines for (1+α) =1.01 to 1.99 insteps of 0.01.The attenuation for all values of α using the pro-posed approximation shows the roll-off rate closest to the idealmagnitude response for ω = 1 to 10 rad/s as seen in Fig.4.

3.2. Sensitivity to Parameter Variation

3.2.1. -3dB frequency response to parameter variation.The -3dB frequency for (1+α) order transfer function (1) withvariation in coefficients by a deviation of 1% has been exploredin Fig.5. The result reveal that the best balance among the flatpassband and -3db frequency is obtained with the proposedtechnique. In Fig. 5, both k2 and k3 were varied by 1% and forall α from 0.1 to 0.9. The coefficients obtained with the pro-posed method produced the minimum percentage error whencompared to [10] and [12]. It can be concluded from the plotthat the proposed coefficients are the most suitable because thevariation in coefficients by (1%) shows the least variation of -3dB frequencies over the full range of orders.

1 + α

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

-3dB

Fre

quen

cy E

rror (

%)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6123

Fig. 5: Percentage error in -3db frequency [k2 and k3 varied] :(1) in [10], (2) in [12] and (3) in Proposed.

3.2.2. Stopband attenuation error to parameter variationThe error in percentage with respect to the ideal attenuation for(1+α) order transfer function (1) with variation in coefficientsby 1% has also been similarly explored and results are shownin Fig.6. Again the same variations in coefficients are consid-ered as previously; moreover, the effect on attenuation with



α

0 0.2 0.4 0.6 0.8 1

Minim

um

phasean

gle(degree)

0.06

0.08

0.1

0.12

0.14

0.16


UNSTABLE

STABLE


1 + α

1 1.2 1.4 1.6 1.8 2

-3dB

Frequency

(rad/s)

0.85

0.9

0.95

1

1.05

1.1

1.15123

(b) -3dB frequency




1 + α

1 1.2 1.4 1.6 1.8 2

Attenuation(dB/d

ecad

e)

-40

-35

-30

-25

-20

-15

1.46 1.48 1.5

-27

-26.5

-26

-25.5






1 + α

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

-3dB

Fre

quen

cy E

rror (

%)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6123




Min

imum

pha

se a

ngle

(deg

ree)

Min

imum

pha

se a

ngle

(deg

ree)

640



in –3 dB frequencies and at the order of  1.1 < (1 + α) < 1.5 frequency increases and reaches the peak of 1.13 rad/s and 1.09 rad/s, respectively. After that, frequency drops gradually and at the 1.8 order it crosses the 1 rad/ sec margin in [10] and at 1.7 rad/s in [12]. However, the proposed filter coefficients show the closest agreement to 1 rad/s for all orders. It can be seen that the filter has a lower ripple at –3 dB frequency, fluctuating between 1.005 to 0.998 rad/s. Thus, using bilevel optimization, the desirable filter characteristics can be improved both in the lower LSE value, and with a high stability margin and better –3 dB frequency.

3.1.4. Stopband attenuation. The transfer function (1) has dif-ferent roll-off characteristics with different sets of coefficients. Stopband attenuation determines how the magnitude response changes from flat passband response to the ideal stopband attenuation of –20(1 + α) dB/decade. Stopband attenuation is another characteristic of the Butterworth response, and the slope of the roll-off characteristics determines the superiority of the design, i.e. the sharper the slope, the better the designed filter.

In order to compare the roll-off characteristics of equation (1) from various methods, the slopes of the magnitude of transfer functions with coefficients from [10, 12] and proposed values are given in Fig. 4. The solid green line is the ideal character-istic of –20(1 + α), changing from a value of –20 dB/ decade when (1 + α) = 1 to –40 dB/decade when (1 + α) = 2; corre-sponding to the traditional integer-order attenuations available for ω = [10, 100] rad/s. The slope between frequencies ω = 1 to ω = 10 rad/s  is shown with  the blue  lines and ω = 10  to ω =  rad/s – with red lines for (1 + α) = 1.01 to 1.99 in steps of 0.01.The attenuation for all values of α using the proposed approximation shows the roll-off rate closest to the ideal mag-nitude response for ω = 1 to 10 rad/s as seen in Fig. 4.

in Fig. 5. The result reveals that the best balance among the flat passband and –3db frequency is obtained with the technique proposed. In Fig. 5, both k2 and k3 were varied by 1% and for all α varied from 0.1 to 0.9. The coefficients obtained with the proposed method produced the minimum percentage error when compared to [10] and [12]. It can be concluded from the plot that the proposed coefficients are the most suitable ones because the variation in coefficients by (1%) shows the least variation of –3 dB frequencies over the full range of orders.

Fig. 6. Percentage error in stopband attenuation [k2 and k3 varied]: 1. in [10], 2. in [12], 3. proposed


frequency ranges ω ∈ [1,10] and ω ∈ [10,100] rad/s are exam-ined. Again, Fig. 6 proves that when both k2 and k3 are variedby 1%, the proposed filter is more robust with minimum atten-uation error as compared to filters obtained with [10] and [12]methods.

1 + α

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Atte

nuat

ion

Erro

r (%

)

10-4

10-3

10-2

10-1

100

1 2 3 1 2 3

ω = 1 -10 rad/s

ω = 10 -100 rad/s

Fig. 6: Percentage error in stopband attenuation [k2 and k3 var-ied]: 1. in [10], 2. in [12] 3. proposed.

4. Real-time ImplementationIn this work, Anadigm AN231E04 FPAA kit has been config-ured to test the designed fractional filters. Anadigm FPAA ison ‘analog signal processor’ consisting of fully configurableanalog modules (CAMs) surrounded by programmable inter-connect and analogue input and output cells [23]. The CAMsin the FPAA accept transfer functions broken down into pole-zero (PZ) form. In following, it is explained briefly how frac-tional function converted into the CAMs compatible. Thereare many advantages of using FPAA that makes it suitable foranalog design applications. FPAA works totally in an analogdomain, which makes it suitable for many real-world appli-cations. The added benefits include, FPAA has low powerconsumption, high precision, drift free operation, fast process-ing speed, high level of integration, and design can be eas-ily made using PC based CAD tools. These features make iteasier to implement complex filters on FPAA. However, thereare some frequency dependent limitations of FPAA. In orderto implement filters on the FPAA, it requires large numberof linear switched capacitors which takes up lot of space inthe integrated circuit architecture. Moreover, it has operationfrequency limitation to the Nyquist rate and also it limits theachievable bandwidth and linearity.

4.1. Approximation of FLO. The approximation for the gen-eral FLO [24] can be used to realize fractional order filters.Therefore, one can convert (1) into following form,

HLP1+α(s) =

k1

s1+α + k2sα + k3

∼=k1(a2s2 +a1s+a0)

s3 + c0s2 + c1s+ c2

(13)

where a0 = α2 + 3α + 2, a1 = 8− 2α2, a2 = α2 − 3α + 2,c0 = (a1 + a0k2 + a2k3)/a0, c1 = (a1 (k2 + k3) + a2)/a0, and

c2 = (a0k3 + a2k2)/a0. The interpolated equations for coeffi-cients k2 and k3 are drawn in Fig.1 obtained from raw data withthe curve fitting function of MATLAB. With these coefficients,both fractional LPF and HPF can be implemented and realizedusing FPAA, as discussed in the following section.

4.2. (1+α) Order LPF. Previously the implemention of afilter required the determination of the values of the compo-nents to realize the transfer functions. Nowadays, however,latest features present in AnadigmDesigner 2 development en-vironment only need the transfer function in form of PZ fre-quencies and quality factor. Accordingly the transfer function(1) is decomposed into first and second order terms by usingthe bilinear and biquadratic filter CAM modules. Thus, (1) canbe written in the following form,

HLP1+α(s) = H1 (s)H2 (s) =

1s+d0

e0s2 + e1s+ e2

s2 +d1s+d2. (14)

Two CAMs are used to implement the approximated frac-tional step filters as shown in (14). The first term H1 (s) isobtained with bilinear characteristic and the other term H2 (s)with biquadratic characteristic. By equating equation (1) withequation (14) the following coefficients can be obtained:

d0 +d1 =a1 +a0k2 +a2k3

a0

d0d1 +d2 =a1 (k2 + k3)+a2

a0

d0d2 =a0k3 +a2k2

a0

e0 = k1a2

a0, e1 = k1

a1

a0, e2 = k1

(15)

CAMs have different forms of accepting variables from (14)as specified in the AN231E04 FPAA datasheet [23]. Beforetransforming (14), the following frequency transformation s =( s

ω0) = (s/2π f0) has to be performed, resulting in:

H (s) = T1 (s)T2 (s)

T1 (s) =2π f1G1

s+2π f1

T2 (s) =−s2 +

2π f2z

(Q2z)s+4π2 f2z

2

s2 +2π f2z

(Q2p)s+4π2 f2p

2

(16)

where, T1 is the transfer function of bilinear CAM, T2 is thetransfer function of biquadratic CAM, G1 is the gain of T1, f1is the pole frequency of T1, f2p,z is the PZ frequency of T2,Q2p,z is the PZ quality factor of T2 and f0 is a de-normalizedfrequency. T1 and T2 are implemented using the switched ca-pacitor technology inside the FPAA.

To implement (1+α) FLPF, the following design equationsare to be used from [10].

f1 = d0 f0, f2z = f0

√e2

e0, Q2z =

√e0e2

e1

f2p = f0√

d2, Q2p =

√d2

d1, G1 =

e0

d0

(17)


Fig. 4. Stopband attenuation for (1 + α) order Butterworth LPF implementation: 1. by [10], 2. by [12], 3. proposed

1 + α


α

0 0.2 0.4 0.6 0.8 1

Minim

um

phasean

gle(degree)

0.06

0.08

0.1

0.12

0.14

0.16


UNSTABLE

STABLE


1 + α

1 1.2 1.4 1.6 1.8 2

-3dB

Frequency

(rad/s)

0.85

0.9

0.95

1

1.05

1.1

1.15123

(b) -3dB frequency




1 + α

1 1.2 1.4 1.6 1.8 2

Attenuation(dB/d

ecad

e)

-40

-35

-30

-25

-20

-15

1.46 1.48 1.5

-27

-26.5

-26

-25.5






1 + α

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

-3dB

Fre

quen

cy E

rror (

%)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6123




1 + α

Atte

nuat

ion

(dB

/dec

ade)

Fig. 5. Percentage error in –3db frequency [k2 and k3 varied]: (1) in [10], (2) in [12] and (3) in proposed


α

0 0.2 0.4 0.6 0.8 1

Minim

um

phase

angle

(degree)

0.06

0.08

0.1

0.12

0.14

0.16


UNSTABLE

STABLE


1 + α

1 1.2 1.4 1.6 1.8 2-3dB

Frequency

(rad/s)

0.85

0.9

0.95

1

1.05

1.1

1.15123

(b) -3dB frequency




1 + α

1 1.2 1.4 1.6 1.8 2

Attenuation(dB/decade)

-40

-35

-30

-25

-20

-15

1.46 1.48 1.5

-27

-26.5

-26

-25.5






1 + α

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

-3dB

Fre

quen

cy E

rror

(%)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6123




1 + α

–3dB

Fre

quen

cy E

rror

(%)

Atte

nuat

ion

Erro

r (%

)

3.2. Sensitivity to parameter variation3.2.1. –3 dB frequency response to parameter variation. The –3 dB frequency for (1 + α) order transfer function (1) with variation in coefficients by a deviation of 1% has been explored

3.2.2. Stopband attenuation error to parameter variation. The error in percentage with respect to the ideal attenuation for (1 + α) order transfer function (1) with variation in coef-ficients by 1% has also been similarly explored and results are shown in Fig. 6. Again the same variations in coefficients are considered as previously; moreover, the effect on attenuation with frequency ranges of ω 2

£1, 10

¤ and ω 2

£10, 100

¤ rad/s

are examined. Again, Fig. 6 proves that when both k2 and k3 are varied by 1%, the proposed filter is more robust with minimum attenuation error as compared to filters obtained with [10] and [12] methods.

641



4. Real-time implementation

In this work, the Anadigm AN231E04 FPAA kit has been con-figured to test the designed fractional filters. Anadigm FPAA is an ՚analog signal processor՚ consisting of fully configurable analog modules (CAMs) surrounded by programmable inter-connect and analogue input and output cells [23]. The CAMs in the FPAA accept transfer functions broken down into pole-zero (PZ) form. In the following section, it is explained briefly how fractional function is converted into the CAMs compatible. There are many advantages of using FPAA that makes it suit-able for analog design applications. FPAA works fully in an analog domain, which makes it suitable for many real-world applications. The added benefits include FPAA's low power consumption, high precision, drift free operation, fast process-ing speed, high level of integration, and the fact that the design can be easily made using PC-based CAD tools. These features make it easier to implement complex filters on FPAA. However, there are some frequency dependent limitations of FPAA. In order to implement filters on the FPAA, a large number of linear switched capacitors is required, which takes up lot of space in the integrated circuit architecture. Moreover, it has operation frequency limitation to the Nyquist rate and also it limits the achievable bandwidth and linearity.

4.1. Approximation of FLO. Approximation for the general FLO [24] can be used to realize fractional order filters. There-fore, one can convert (1) into the following form:

H LP

1 + α(s) = k1

s1 + α + k2sα + k3 ¡¡»

H (s) ¡¡» k1(a2s2 + a1s + a0)

s3 + c0s2 + c1s + c2

(13)

where a0 = α2 + 3α + 2, a1 = 8 ¡ 2α2, a2 = α2 ¡ 3α + 2, c0 = (a1 + a0k2 + a2k3)/a0, c1 = (a1(k2 + k3) + a2)/a0, and c2 = (a0k3 + a2k2)/a0. The interpolated equations for coeffi-cients k2 and k3 are drawn in Fig. 1, obtained from raw data with the curve fitting function of MATLAB. With these coef-ficients, both fractional LPF and HPF can be implemented and realized using FPAA, as discussed in the following section.

4.2. (1 + α) order LPF. Previously the implemention of a fil-ter required the determination of the values of the components to realize the transfer functions. Nowadays, however, latest fea-tures present in AnadigmDesigner 2 development environment only need the transfer function in the form of PZ frequencies and the quality factor. Accordingly, the transfer function (1) is decomposed into first and second order terms by using the bilinear and biquadratic filter CAM modules. Thus, (1) can be written in the following form:

H LP1 + α(s) = H1(s)H2(s) = 1

s + d0

e0s2 + e1s + e2

s2 + 1d1s + d2. (14)

Two CAMs are used to implement the approximated frac-tional step filters as shown in (14). The first term H1(s) is ob-tained with the bilinear characteristic and the other term H2(s) – with the biquadratic characteristic. By equating equation (1) with equation (14) the following coefficients can be obtained:

d0 + d1 = a1 + a0k2 + a2k3

a0

d0d1 + d2 = a1(k2 + k3) + a2

a0

d0d2 = a0k3 + a2k2

a0

e0 = k1a2

a0, e1 = k1

a1

a0, e2 = k1.

(15)

CAMs have different forms of accepting variables from (14) as specif ied in the AN231E04 FPAA datasheet [23]. Before transforming (14), the following frequency transformation s = ( s

ω0) = (s/2π f0) has to be performed, resulting in:

H(s) = T1(s)T2(s)

T1(s) = 2π f1G1

s + 2π f1

T2(s) = –s2 +

2π f2z

(Q2z)s + 4π 2 f2z

2

s2 + 2π f2z

(Q2p)s + 4π 2 f2p

2

(16)

where, T1 is the transfer function of bilinear CAM, T2 is the transfer function of biquadratic CAM, G1 is the gain of T1, f1 is the pole frequency of T1, f2p, z is the PZ frequency of T2, Q2p, z is the PZ quality factor of T2 and f0 is a de-normalized frequency. T1 and T2 are implemented using the switched capacitor tech-nology inside the FPAA.

To implement (1 + α) FLPF, the following design equations are to be used from [10].

f1 = d0 f0 , f2z = f0

e2

e0, Q2z =

e0e2

e1

f2p = f0 d2 , Q2p = d2

d1, G1 = e0

d0

(17)

The FOLPF of orders (1 + α) = 1.2, 1.6, 1.9 have been realized. The approximated PZ frequencies of bilinear and biquadratic CAMs to be realized using Anadigm FPAA are shown in Table 4a when f0 = 1 kHz. Table 4b shows the val-ues of d0, 1, 2 and e0, 1, 2 from (15) used for Table 4a for the same values of α. The values for k2, 3 in Table 4b are optimized values obtained with the implementation. These values differ from the theoretical values as there are limitations on the values that can be implemented on FPAA. Indeed biquadratic and bilinear CAMs cannot realize all potential values for hardware limita-tions, since corner frequencies, quality factors and gains are

642



interrelated with the internal switched capacitor circuits of the FPAA kit. Since the manufacturers only make a finite number of capacitors, the AnadigmDesigner tool selects the best ratio of switched capacitors, matching the desired design parameters. With the coefficients proposed in this paper, the theoretical and realized values are lower than those proposed in [10] and [12] in terms of higher accuracy in passband and stopband values.

4.3. Experimental results for (1 + α) order LPF. An FLPF has been implemented on the FPAA development board: the experiment block diagram and the experimental test rig are

shown in Fig. 7. The Anadigm FPAA board was powered by 5 V DC, which drew 200 mA of current from the source. The differential-to-single and single-to-differential converters were supplied by 5 V DC supply, each separately, and drew combined current of 250 mA from the supply. The power requirements of the proposed setup ARE similar to that implemented in [10]. The cut-off frequency for the LPF has been set to 1 kHz. The (1 + α) order with α = 0.2, 0.6 and 0.9 values were imple-mented. A SSA3000X series spectrum analyser has been used to measure the magnitude response of the implemented LPF. The start frequency has been set to 100 Hz, and the stop frequency to 1 MHz. The tracking generator (TG) level was set as –10 dB. Input voltage signal was generated from the TG source port of the spectrum analyzer at 50 Ω set at 2.7 V, and connected to the input side of the implemented fractional order filter. The output was measured from the RF input port of the spectrum analyzer at 50 Ω. The filter response was recorded on the spectrum ana-lyzer screen. The measured magnitude response of FLPF of order (1 + α) = 0.2, 0.6 and 0.9 has been recorded.

The magnitude response for α = 0.2, 0.6 and 0.9 is shown in Fig. 8, which further confirms the good operation of the proposed FLPF on the FPAA. The extra degree of freedom pro-

Fig. 8. Real-time fractional (1 + α) LPF with orders 1.2, 1.4, 1.6 and 1.8


Table 5: Theoretical and realized biquad and bilinear CAMvalues for FHPF

(a)

DesignParameters

Order (1+α)1.2 1.6 1.9

Theoretical Realized Theoretical Realized Theoretical Realizedf1, kHz 29.83 29.80 21.53 21.50 13.69 13.70f2p , kHz 12.59 12.10 14.04 14.00 10.75 10.50

f2z , kHz 7.38 7.16 3.66 3.83 1.41 1.50Q2p , kHz 1.39 1.47 2.30 2.40 0.97 0.98

Q2z , kHz 0.24 0.25 0.20 0.21 0.12 0.12G1 1.00 1.00 1.00 1 1.00 1.00

(b)

value(1+α)k2

k3(1+0.2)0.46

0.78 (1+0.6)0.890.91 (1+0.9)1.29

0.99d0 2.98 2.15 1.36d1 0.90 0.60 1.09d2 1.57 1.97 1.15e0 1.00 1.00 1.00e1 3.00 1.75 1.15e2 0.54 0.13 0.02

It is to be noted that x is a dummy variable and d0 is the pos-itive real root in (18). The values of k2,3 that are used to calcu-late d0,1,2 and e0,1,2, are proposed coefficients. The values ford0,1,2 and e0,1,2 for filter orders (1+α) = 1.2, 1.6 and 1.9, werecalculated using (18) respectively and is given in Table.5(b)along with bilinear and biquadratic filter CAM parameters.

3

Frequency (Hz)

Mag

nitu

de R

espo

nse (

dB)

Fig. 8: Real-time fractional (1+α) LPF with orders 1.2, 1.4,1.6 and 1.8

There is not much difference in implementing FHPF onAN231E04 FPAA when compared to FLPF. Both the bilinearand biquadratic filter CAMs are used to implement a highpass.Only the bilinear CAM is set in highpass configuration whileis earlier set on lowpass configuration. The biquadratic filterCAM was not changed and remained same as the PZ configu-ration. For HPF f0 = 10kHz was used as a cut-off frequency.The realized and theoretical values are slightly different dueto hardware limitations to implement high decimal values, asmentioned above.

4.5. Experimental Results for (1+α) Order HPF The opti-mal transformation (8) from fractional order low-to high-passfilter is used to implement a FHPF. As discussed in an ear-lier section, the filter was implemented using the best coeffi-cients k2 and k3. The cut-off frequency for HPF was set to10kHz. Filters of (1+α) orders with α = 0.2, 0.6 and 0.9, havebeen implemented. Frequency range was setup from 1kHz to

100kHz scale. It further confirms the good operation of theproposed FLPF on the FPAA.

× 103Frequency (Hz)

0 2 4 6 8 10 12 14-100

-80

-60

-40

-20

0

α = 0.2α = 0.4α = 0.6α = 0.8

Mag

nitu

de R

espo

nse

(dB

)

Fig. 9: Real-time fractional (1+α) HPF with orders 1.2, 1.4,1.6 and 1.8

5. ConclusionsThis work has focussed on the design and the implementationof fractional order filters, by using a reconfigurable analog pro-cessor. The main advantage of this design is that it providesan extra degree of freedom in system designing, and providesprecise control of passband ripple, roll-off rate, and stopbandattenuation. This can be used in many applications such asdesigning special controllers, telecommunications and also inmodeling of various biological signals. Firstly, a new bilevelconstraint optimization was proposed to obtain the best filterparameters. The proposed filters have shown more robustnessin compared to previously proposed fractional filters. Throughanalysis, the optimal order of approximated sα was suggestedto implement fractional differentiator in hardware with accept-able accuracy. The real-time fractional filter has been imple-mentation with the analog array board. The results, obtainedwith MATLAB and in real-time, have verified the implemen-tation and operation of the fractional step filters. It is clearfrom Tables 4 and 5 that the actual fractional filter behaviorhas closely followed the theoretical one.

REFERENCES

[1] M. D. Ortigueira, J. T. M. Machado, P. O. Stalczyk, Fractionalsignals and systems, Bulletin of the Polish Academy of Sci-ences: Technical Sciences 66(4) (2018) 385–388.

[2] A. Dzielinski, D. Sierociuk, G. Sarwas, Some applications offractional order calculus, Bulletin of the Polish Academy of Sci-ences: Technical Sciences 58(4) (2010) 583–592.

[3] I. Podlubny, I. Petras, B. Vinagre, P. O’Leary, L. Dorcak, Ana-log realization of fractional - order controllers and nonlineardynamics, Nonlinear Dynamics 29(1) (2002) 281–296.

[4] A. Acharya, S. Das, I. Pan, S. Das, Extending the concept ofanalog Butterworth filter for fractional order systems, SignalProcessing 94 (2014) 409–420.

[5] A. M. Lopes, J. A. T. Machado, Fractional-order model of anon-linear inductor, Bulletin of the Polish Academy of Sci-ences: Technical Sciences 67(1) (2019) 61–67.

[6] R. Prasad, K. Kothari, U. Mehta, Flexible fractional super-capacitor model analyzed in time domain, IEEE Access 7(1)(2019) 122626–122633.


Designparameters

Order (1 + α)

1.2 1.6 1.9Theoretical Realized Theoretical Realized Theoretical Realized

f1, kHz 0.33 0.35 0.46 0.46 0.73 0.73

f2p, kHz 1.75 1.76 1.48 1.50 1.17 1.18

f2z, kHz 1.35 1.36 2.72 2.73 7.07 7.11

Q2p, kHz 0.49 0.49 0.64 0.64 0.67 0.67

Q2z, kHz 0.24 0.25 0.20 0.21 0.12 0.12

G1 1.62 1.60 0.28 0.28 0.02 0.02

value(1 + α)k2

k3

(1 + 0.2)0.460.78 (1 + 0.6)0.89

0.91 (1 + 0.9)1.290.99

d0 0.33 0.46 0.73

d1 3.55 2.29 1.73

d2 3.07 2.21 1.39

e0 0.54 0.13 0.02

e1 3.00 1.75 1.15

e2 1.00 1.00 1.00

Table 4 Theoretical and realized biquad and bilinear CAM values for FLPF

(a) (b)

Fig. 7. (a) Block diagram for hardware connection and (b) experi-mental setup


Table 4: Theoretical and realized biquad and bilinear CAMvalues for FLPF

(a)

DesignParameters

Order (1+α)1.2 1.6 1.9


f2z , kHz 1.35 1.36 2.72 2.73 7.07 7.11Q2p , kHz 0.49 0.49 0.64 0.64 0.67 0.67

Q2z , kHz 0.24 0.25 0.20 0.21 0.12 0.12G1 1.62 1.60 0.28 0.28 0.02 0.02

(b)

value(1+α)k2

k3(1+0.2)0.46

0.78 (1+0.6)0.890.91 (1+0.9)1.29

0.99d0 0.33 0.46 0.73d1 3.55 2.29 1.73d2 3.07 2.21 1.39e0 0.54 0.13 0.02e1 3.00 1.75 1.15e2 1.00 1.00 1.00

The FOLPF, of orders (1+α) = 1.2,1.6,1.9 have been re-alized. The approximated PZ frequencies of bilinear and bi-quadratic CAMs to realize using Anadigm FPAA are shownin Table 4(a) when f0 = 1 kHz. Table 4 (b) shows the val-ues of d0,1,2 and e0,1,2 from (15) used for Table 4 (a) for samevalues of α . The values for k2,3 in Table 4 (b) are optimizedvalues obtained with the implementation. These values differfrom the theoretical values as there are limitations on the val-ues that can be implemented on FPAA. Indeed biquadratic andbilinear CAMs cannot realize all possible values for hardwarelimitations, since corner frequencies, quality factors and gainsare interrelated to the internal switched capacitor circuits of theFPAA kit. Since the manufacturers only make finite number ofcapacitors, the AnadigmDesigner tool selects the best ratio ofswitched capacitors, matching the desired design parameters.With the coefficients proposed in this paper, the theoretical andrealized values is lower than those proposed in [10] and [12],in terms of higher accuracy in passband and stopband values.

4.3. Experimental Results for (1+α) Order LPF. An FLPFhas been implemented on the FPAA development board: theexperiment block diagram and the experimental test rig areshown in Fig.7. The Anadigm FPAA board was powered by5V DC which drew 200mA of current from the source. Thedifferential-to-single and single-to-differential converters weresupplied by 5V DC supply each separately and drew combinedcurrent of 250mA from the supply. The power requirements ofproposed setup is similar to that implemented in [10]. The cut-off frequency for the LPF has been set to 1kHz. The (1+α)order with α = 0.2, 0.6 and 0.9 values were implemented. ASSA3000X series spectrum analyser has been used to measurethe magnitude response of the implemented LPF. The start fre-quency has been set to 100 Hz, and the stop frequency to 1MHz. The Tracking Generator (TG) level was set as -10 dB.Input voltage signal was generated from the TG source port ofthe spectrum analyzer at 50Ω set at 2.7V and connected to theinput side of the implemented fractional order filter. The out-put was measured from RF input port of the spectrum analyzerat 50Ω. The filter response was recorded on the spectrum an-

alyzer screen. The measured magnitude response of FLPF oforder (1+α) = 0.2, 0.6 an 0.9 has been recorded.

Spectrum Analyser

Tracking Generator

(TG)

Spectrum Analyser

RF input

Level shifting Single to

Differential Converter

Differential to Single

Converter

AN231E04 FPAA

development board

Vin

I1NI1P

Vref

I1NI1P

O3N

O3P

O3N

O3P

V0

Vref

(a)

(b)

Fig. 7: (a) Block diagram for hardware connection and (b) Ex-perimental setup

The magnitude response for α = 0.2,0.6 and 0.9 is shownin Fig.8, which further confirms the good operation of the pro-posed FLPF on the FPAA. The extra degree of freedom pro-vided by the fractional order parameter allows the full manipu-lation of the filter specifications to obtain the desired responserequired by any application.

4.4. (1+α) Order HPF. An FHPF can be implemented in thesimilar manner by decomposition of the transfer function (1)into the form taken in by bi-linear and biquadratic filter CAMSgiven by (14). The bilinear CAM had inputs parameters ascorner frequency and gain, and the biquadratic filter CAM hadinput parameters as PZ configuration. The design equations ford0,1,2 and e0,1,2 slightly differ from LPF and can be calculatedusing the equation set (18).

p1 = a2k2 +a0k3 , p2 = a1k2 +a1k3 +a2

p3 = a0k2 +a2k3 +a1, p4 = a0

d0 =√[(p1)x3 − (p2)x2 +(p3)x+ p4]

d1 =a0k2 +a2k3 +a1

a0−d0

d2 =a1k2 +a1k3 +a2

a0−d0d1

e0 = k1, e1 = k1a1

a0, e2 = k1

a2

a0

(18)



Table 4: Theoretical and realized biquad and bilinear CAMvalues for FLPF

(a)

DesignParameters

Order (1+α)1.2 1.6 1.9


f2z , kHz 1.35 1.36 2.72 2.73 7.07 7.11Q2p , kHz 0.49 0.49 0.64 0.64 0.67 0.67

Q2z , kHz 0.24 0.25 0.20 0.21 0.12 0.12G1 1.62 1.60 0.28 0.28 0.02 0.02

(b)

value(1+α)k2

k3(1+0.2)0.46

0.78 (1+0.6)0.890.91 (1+0.9)1.29

0.99d0 0.33 0.46 0.73d1 3.55 2.29 1.73d2 3.07 2.21 1.39e0 0.54 0.13 0.02e1 3.00 1.75 1.15e2 1.00 1.00 1.00

The FOLPF, of orders (1+α) = 1.2,1.6,1.9 have been re-alized. The approximated PZ frequencies of bilinear and bi-quadratic CAMs to realize using Anadigm FPAA are shownin Table 4(a) when f0 = 1 kHz. Table 4 (b) shows the val-ues of d0,1,2 and e0,1,2 from (15) used for Table 4 (a) for samevalues of α . The values for k2,3 in Table 4 (b) are optimizedvalues obtained with the implementation. These values differfrom the theoretical values as there are limitations on the val-ues that can be implemented on FPAA. Indeed biquadratic andbilinear CAMs cannot realize all possible values for hardwarelimitations, since corner frequencies, quality factors and gainsare interrelated to the internal switched capacitor circuits of theFPAA kit. Since the manufacturers only make finite number ofcapacitors, the AnadigmDesigner tool selects the best ratio ofswitched capacitors, matching the desired design parameters.With the coefficients proposed in this paper, the theoretical andrealized values is lower than those proposed in [10] and [12],in terms of higher accuracy in passband and stopband values.

4.3. Experimental Results for (1+α) Order LPF. An FLPFhas been implemented on the FPAA development board: theexperiment block diagram and the experimental test rig areshown in Fig.7. The Anadigm FPAA board was powered by5V DC which drew 200mA of current from the source. Thedifferential-to-single and single-to-differential converters weresupplied by 5V DC supply each separately and drew combinedcurrent of 250mA from the supply. The power requirements ofproposed setup is similar to that implemented in [10]. The cut-off frequency for the LPF has been set to 1kHz. The (1+α)order with α = 0.2, 0.6 and 0.9 values were implemented. ASSA3000X series spectrum analyser has been used to measurethe magnitude response of the implemented LPF. The start fre-quency has been set to 100 Hz, and the stop frequency to 1MHz. The Tracking Generator (TG) level was set as -10 dB.Input voltage signal was generated from the TG source port ofthe spectrum analyzer at 50Ω set at 2.7V and connected to theinput side of the implemented fractional order filter. The out-put was measured from RF input port of the spectrum analyzerat 50Ω. The filter response was recorded on the spectrum an-

alyzer screen. The measured magnitude response of FLPF oforder (1+α) = 0.2, 0.6 an 0.9 has been recorded.

Spectrum Analyser

Tracking Generator

(TG)

Spectrum Analyser

RF input

Level shifting Single to

Differential Converter

Differential to Single

Converter

AN231E04 FPAA

development board

Vin

I1NI1P

Vref

I1NI1P

O3N

O3P

O3N

O3P

V0

Vref

(a)

(b)

Fig. 7: (a) Block diagram for hardware connection and (b) Ex-perimental setup

The magnitude response for α = 0.2,0.6 and 0.9 is shownin Fig.8, which further confirms the good operation of the pro-posed FLPF on the FPAA. The extra degree of freedom pro-vided by the fractional order parameter allows the full manipu-lation of the filter specifications to obtain the desired responserequired by any application.

4.4. (1+α) Order HPF. An FHPF can be implemented in thesimilar manner by decomposition of the transfer function (1)into the form taken in by bi-linear and biquadratic filter CAMSgiven by (14). The bilinear CAM had inputs parameters ascorner frequency and gain, and the biquadratic filter CAM hadinput parameters as PZ configuration. The design equations ford0,1,2 and e0,1,2 slightly differ from LPF and can be calculatedusing the equation set (18).

p1 = a2k2 +a0k3 , p2 = a1k2 +a1k3 +a2

p3 = a0k2 +a2k3 +a1, p4 = a0

d0 =√

[(p1)x3 − (p2)x2 +(p3)x+ p4]

d1 =a0k2 +a2k3 +a1

a0−d0

d2 =a1k2 +a1k3 +a2

a0−d0d1

e0 = k1, e1 = k1a1

a0, e2 = k1

a2

a0

(18)


a)

b)

Frequency (Hz)

Mag

nitu

de R

espo

nse

(dB

)

Differential to single converter

Level shifting single to differential converter

AN231E04 FPAA development board

Spectrum analyzer tracking generator

(TG)

Spectrum analyzer RF input

643



vided by the fractional order parameter allows the full manipu-lation of the filter specifications to obtain the desired response required by any application.

4.4. (1 + α) order HPF. An FHPF can be implemented in a similar manner by means of decomposition of the transfer function (1) into the form taken in by bi-linear and biquadratic filter CAMS given by (14). The bilinear CAM had input param-eters such as corner frequency and gain, and the biquadratic CAM filter had the input parameter in the form of PZ configu-ration. The design equations for d0, 1, 2 and e0, 1, 2 differ slightly from LPF and can be calculated using the equation set (18).

p1 = a2k2 + a0k3, p2 = a1k2 + a1k3 + a2

p3 = a0k2 + a2k3 + a1, p4 = a0

d0 = £(p1)x3 ¡ (p2)x2 + (p3)x + p4

¤

d1 = a0k2 + a2k3 + a1

a0 ¡ d0

d2 = a1k2 + a1k3 + a2

a0 ¡ d0d1

e0 = k1, e1 = k1a1

a0, e2 = k1

a2

a0.

(18)

It is to be noted that x is a dummy variable and d0 is the positive real root in (18). The values of k2, 3 that are used to calculate d0, 1, 2 and e0, 1, 2, are proposed coefficients. The values for d0, 1, 2 and e0, 1, 2 for filter orders (1 + α) = 1.2, 1.6 and 1.9, were calculated using (18), respectively, and are presented in Table 5b along with bilinear and biquadratic filter CAM param-eters.

There is not much difference in implementing FHPF on AN231E04 FPAA when compared to FLPF. Both the bilinear and biquadratic filter CAMs are used to implement a highpass. Only the bilinear CAM is set in highpass configuration while earlier it was set to lowpass configuration. The biquadratic filter CAM was not changed and remained the same as the PZ config-

uration. For HPF f0 = 10 kHz was used as a cut-off frequency. The realized and theoretical values are slightly different due to hardware limitations in implementing high decimal values, as mentioned above.

4.5. Experimental results for (1 + α) order HPF. The opti-mal transformation (8) from fractional order low to highpass filter is used to implement an FHPF. As discussed in an earlier section, the filter was implemented using the best coef-ficients k2 and k3. The cut-off frequency for HPF was set to 10 kHz. Filters of (1 + α) orders with α = 0.2, 0.6 and 0.9 have been implemented. Frequency range was set up on a 1 kHz to 100 kHz scale. This further confirms the good operation of the proposed FLPF on the FPAA.

Fig. 9. Real-time fractional (1 + α) HPF with orders 1.2, 1.4, 1.6 and 1.8


Table 5: Theoretical and realized biquad and bilinear CAMvalues for FHPF

(a)

DesignParameters

Order (1+α)1.2 1.6 1.9


f2z , kHz 7.38 7.16 3.66 3.83 1.41 1.50Q2p , kHz 1.39 1.47 2.30 2.40 0.97 0.98

Q2z , kHz 0.24 0.25 0.20 0.21 0.12 0.12G1 1.00 1.00 1.00 1 1.00 1.00

(b)

value(1+α)k2

k3(1+0.2)0.46

0.78 (1+0.6)0.890.91 (1+0.9)1.29

0.99d0 2.98 2.15 1.36d1 0.90 0.60 1.09d2 1.57 1.97 1.15e0 1.00 1.00 1.00e1 3.00 1.75 1.15e2 0.54 0.13 0.02

It is to be noted that x is a dummy variable and d0 is the pos-itive real root in (18). The values of k2,3 that are used to calcu-late d0,1,2 and e0,1,2, are proposed coefficients. The values ford0,1,2 and e0,1,2 for filter orders (1+α) = 1.2, 1.6 and 1.9, werecalculated using (18) respectively and is given in Table.5(b)along with bilinear and biquadratic filter CAM parameters.

3

Frequency (Hz)

Mag

nitu

de R

espo

nse (

dB)

Fig. 8: Real-time fractional (1+α) LPF with orders 1.2, 1.4,1.6 and 1.8

There is not much difference in implementing FHPF onAN231E04 FPAA when compared to FLPF. Both the bilinearand biquadratic filter CAMs are used to implement a highpass.Only the bilinear CAM is set in highpass configuration whileis earlier set on lowpass configuration. The biquadratic filterCAM was not changed and remained same as the PZ configu-ration. For HPF f0 = 10kHz was used as a cut-off frequency.The realized and theoretical values are slightly different dueto hardware limitations to implement high decimal values, asmentioned above.

4.5. Experimental Results for (1+α) Order HPF The opti-mal transformation (8) from fractional order low-to high-passfilter is used to implement a FHPF. As discussed in an ear-lier section, the filter was implemented using the best coeffi-cients k2 and k3. The cut-off frequency for HPF was set to10kHz. Filters of (1+α) orders with α = 0.2, 0.6 and 0.9, havebeen implemented. Frequency range was setup from 1kHz to

100kHz scale. It further confirms the good operation of theproposed FLPF on the FPAA.

× 103Frequency (Hz)

0 2 4 6 8 10 12 14-100

-80

-60

-40

-20

0

α = 0.2α = 0.4α = 0.6α = 0.8

Mag

nitu

de R

espo

nse

(dB

)

Fig. 9: Real-time fractional (1+α) HPF with orders 1.2, 1.4,1.6 and 1.8

5. ConclusionsThis work has focussed on the design and the implementationof fractional order filters, by using a reconfigurable analog pro-cessor. The main advantage of this design is that it providesan extra degree of freedom in system designing, and providesprecise control of passband ripple, roll-off rate, and stopbandattenuation. This can be used in many applications such asdesigning special controllers, telecommunications and also inmodeling of various biological signals. Firstly, a new bilevelconstraint optimization was proposed to obtain the best filterparameters. The proposed filters have shown more robustnessin compared to previously proposed fractional filters. Throughanalysis, the optimal order of approximated sα was suggestedto implement fractional differentiator in hardware with accept-able accuracy. The real-time fractional filter has been imple-mentation with the analog array board. The results, obtainedwith MATLAB and in real-time, have verified the implemen-tation and operation of the fractional step filters. It is clearfrom Tables 4 and 5 that the actual fractional filter behaviorhas closely followed the theoretical one.

REFERENCES

[1] M. D. Ortigueira, J. T. M. Machado, P. O. Stalczyk, Fractionalsignals and systems, Bulletin of the Polish Academy of Sci-ences: Technical Sciences 66(4) (2018) 385–388.

[2] A. Dzielinski, D. Sierociuk, G. Sarwas, Some applications offractional order calculus, Bulletin of the Polish Academy of Sci-ences: Technical Sciences 58(4) (2010) 583–592.

[3] I. Podlubny, I. Petras, B. Vinagre, P. O’Leary, L. Dorcak, Ana-log realization of fractional - order controllers and nonlineardynamics, Nonlinear Dynamics 29(1) (2002) 281–296.

[4] A. Acharya, S. Das, I. Pan, S. Das, Extending the concept ofanalog Butterworth filter for fractional order systems, SignalProcessing 94 (2014) 409–420.

[5] A. M. Lopes, J. A. T. Machado, Fractional-order model of anon-linear inductor, Bulletin of the Polish Academy of Sci-ences: Technical Sciences 67(1) (2019) 61–67.

[6] R. Prasad, K. Kothari, U. Mehta, Flexible fractional super-capacitor model analyzed in time domain, IEEE Access 7(1)(2019) 122626–122633.


Frequency (Hz)

Mag

nitu

de R

espo

nse

(dB

)

Designparameters

Order (1 + α)

1.2 1.6 1.9Theoretical Realized Theoretical Realized Theoretical Realized

f1, kHz 29.83 29.80 21.53 21.50 13.69 13.70

f2p, kHz 12.59 12.10 14.04 14.00 10.75 10.50

f2z, kHz 7.38 7.16 3.66 3.83 1.41 1.50

Q2p, kHz 1.39 1.47 2.30 2.40 0.97 0.98

Q2z, kHz 0.24 0.25 0.20 0.21 0.12 0.12

G1 1.00 1.00 1.00 1.00 1.00 1.00

value(1 + α)k2

k3

(1 + 0.2)0.460.78 (1 + 0.6)0.89

0.91 (1 + 0.9)1.290.99

d0 2.98 2.15 1.36

d1 0.90 0.60 1.09

d2 1.57 1.97 1.15

e0 1.00 1.00 1.00

e1 3.00 1.75 1.15

e2 0.54 0.13 0.02

Table 5 Theoretical and realized biquad and bilinear CAM values for FHPF

(a) (b)

5. Conclusions

This work has focussed on the design and implementation of fractional order filters, by using a reconfigurable analog pro-cessor. The main advantage of this design is that it provides an extra degree of freedom in system designing, and ensures control of passband ripple, roll-off rate, and stopband attenua-

644



tion. This can be used in many applications such as designing special controllers, telecommunications and also in modeling of various biological signals. Firstly, a new bilevel constraint optimization was proposed to obtain the best filter parameters. The proposed filters have shown more robustness as compared to previously proposed fractional filters. Through analysis, the optimum order of approximated sα was suggested to implement a fractional differentiator in hardware with acceptable accuracy. The real-time fractional filter has been implemented with the analog array board. The results, obtained with MATLAB and in real time, have verified the implementation and operation of the fractional step filters. It is clear from Tables 4 and 5 that actual fractional filter behavior has closely followed the theoretical one.

References [1] M.D. Ortigueira, J.T.M. Machado, and P. O. Stalczyk, “Frac-

tional signals and systems”. Bull. Pol. Ac.: Tech. 66(4), 385–388 (2018).

[2] A. Dzielinski, D. Sierociuk, and G. Sarwas, “Some applications of fractional order calculus”. Bull. Pol. Ac.: Tech. 58(4), 583–592 (2010).

[3] I. Podlubny, I. Petras, B. Vinagre, P. O’Leary, and L. Dorcak, “Analog realization of fractional – order controllers and nonlin-ear dynamics”. Nonlinear Dyn. 29(1), 281–296 (2002).

[4] A. Acharya, S. Das, I. Pan, and S. Das, “Extending the concept of analog Butterworth filter for fractional order systems”. Signal Process. 94, 409–420 (2014).

[5] A.M. Lopes and J.A.T. Machado, “Fractional-order model of a non-linear inductor”. Bull. Pol. Ac.: Tech. 67(1), 61–67 (2019).

[6] R. Prasad, K. Kothari, and U. Mehta, “Flexible fractional super- capacitor model analyzed in time domain”. IEEE Access, 7(1), 122 626–122 633 (2019).

[7] A. Radhwan, A. Elwakil, and A. Soliman, “On the generalization of second-order filters to the fractional order domain”. Journal of Circuits, Systems and Computers, 18(2), 361–386 (2009).

[8] A. Ali, A. Radwan, and A. Soliman, “Fractional order Butter-worth filter: Active and passive realizations”. IEEE J. Emerging Sel. Top. Circuits Syst. 3(3), 346–354 (2013).

[9] T. Freeborn, A. Elwakil, and B. Maundy, “Approximated frac-tional-order inverse Chebyshev lowpass filters”. Circuits, Sys-tems and Signal Process. 35(6), 1973–1982 (2015).

[10] T. Freeborn, B. Maundy, and A. Elwakil, “Field programmable analogue array implementation of fractional step filters”. IET Circuits, Devices & Systems 4(6), p. 514 (2010).

[11] D. Kubanek and T. Freeborn, “(1 + α) Fractional-order transfer functions to approximate low-pass magnitude responses with arbitrary quality factor”. Int. J. Electron. Commun. 83, 570–578 (2018).

[12] T. Freeborn, “Comparison of (1 + α) fractional-order transfer functions to approximate lowpass Butterworth magnitude re-sponses”. Circuits, Systems and Signal Process. 35(6), 1983–2002 (2016).

[13] L.A. Said, S.M. Ismail, A.G. Radwan, A.H. Madian, M.F.A. El-Ya-zeed, and A.M. Soliman, “On the optimization of fractional order low-pass filters”. Circuits, Systems, and Signal Process. 35(6), 2017–2039 (2016).

[14] S. Mahata, S.K. Saha, R. Kara, and D. Mandala, “Optimal de-sign of fractional order low pass butterworth filter with accurate magnitude response”. Digital Signal Process. 72, 96–114 (2018).

[15] F. Khateb, D. Kubanek, G. Tsirimokou, and C. Psychalinos, “Fractional-order filters based on low-voltage DDCCs”. Micro-electron. J., 50, 50–59 (2016).

[16] J. Jerabek, R. Sotner, J. Dvorak, J. Polak, D. Kubanek, N. Her-encsar, and J. Koton, “Reconfigurable fractional-order filter with electronically controllable slope of attenuation, pole frequency and type of approximation”. Journal of Circuits, Systems and Computers 26(10), 1750–1757 (2017).

[17] G. Tsirimokou, C. Psychalinos, and A. Elwakil, Design of CMOS analog integrated fractional-order circuits. Springer: Switzer-land, 2017.

[18] P. Ahmadi, B. Maundy, A.S. Elwakil, and L. Belostotski, “High-quality factor asymmetric-slope band-pass f ilters: a fractional-order capacitor approach”. IET Circuits, Devices & Systems 6(3), 187 (2012).

[19] Y. Shi and R. C. Eberhart, “Empirical study of particle swarm optimization”. in Proceedings of the 1999 Congress on Evolu-tionary Computation-CEC99, vol. 3, 1999.

[20] J. Kennedy and R. Eberhart, “Particle swarm optimization”. in IEEE International Conference on Neural Networks, Perth, WA, 1995, pp. 1942–1948.

[21] D. Kubanek, T. Freeborn, J. Koton, and N. Herencsar, “Eval-uation of (1 + α) fractional order approximated Butterworth high pass and band pass filter transfer functions”. Elektronika i Elektrotechnika 24(2), 37–41 (2018).

[22] A. Radwan, A. Soliman, A. Elwakil, and A. Sedeek, “On the sta-bility of linear systems with fractional-order elements”. Chaos, Solitons & Fractals 40(5), 2317–2328 (2009).

[23] Anadigm, “3rd generation dynamically reconfigurable dpASP”. AN231E04 Datasheet Rev 1.2, (2007).

[24] B. Krishna and K. Reddy, “Active and passive realization of fractance device of order 1/2”. Act. Passive Electron. Compon. 1, 1–5 (2008).

optimized fractional low and highpass filters of (1 + α

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