Abstract
This paper presents a realization of fractionalorder Band passfilter (FOBF) based on the concepts of fractional order inductors and fractional order capacitors. The FOBF is designed and implemented using both simulation and hardware approaches. The proposed filter order is considered up to second order or less with any real positive number. One of the cases is considered when α ≤ 1 and β ≥ 1. In the second case, the filter is designed when β ≤ 1 and α ≥ 1. In order to calculate the optimal filter parameters, the modified Particle Swarm Optimization (mPSO) algorithm has been utilized for coefficient tuning. Also, a generalized approach to design any second order FOBF is discussed in this work. The realization and performance assessment have been carried out in simulation environment as well as in lab experiment with field programmable analog array (FPAA) development board. The experimental results indicate the value of efforts to realize the fractional filter.
1 Introduction
In the last decade, Applied Mathematics with fractional calculus has become popular among researchers in different disciplines of science, technology and engineering. Fractional calculus provides an excellent instrument for the description of memory and hereditary properties of various materials and processes [1]. A fractionalorder model or transfer function has better flexibility in comparison to classical integer order model, in which system dynamics are not taken into account. The review of fractional order modeling techniques and successfully obtained results in mathematical ways was presented for quick understanding of the research topic [2].
The mathematicians, Fourier, Euler, Laplace are among the many that fiddled with fractional calculus and the mathematical consequences. Many found definitions that fit the concept of a noninteger order integral or derivative using their own notations and methodologies. Nevertheless, the most famous but, also, most complex of them owed to RiemannLiouville and GrunwaldLetnikov are given below to start the understanding of fractional derivation and its physical interpretation [1–7].
It is therefore possible to define a fractance element as one whose impedance Z is proportional to s ^{ α }; α is arbitrary. It is noted that though the fractional derivatives are being applied to the various areas in science and engineering, yet it is vital to know the key value of each definition in application fields. The fractional calculus results have been observed in various fields such as control design [8–12], electrical circuits [13], stability analysis [14], mechanics [15], electromagnetic [16, 17], and bioengineering [18, 19] and progress is ongoing to date.
While designing the analog frequency filters traditionally of integerorder, it can also be designed using fraction order. In particular, as per work presented in the paper, the purpose of fractional filter designed to estimate the ideal response behavior for any specified performance filter. The band pass filter is basically a frequency selective circuit that accepts or rejects signals of a particular frequency band [20]. The design of any analog filter needs to determine the transfer function coefficients that will yield the desired filter characteristics, i.e., magnitude, phase, or delay.
A wide range of different design procedures are used on approximating interger order filter magnitude response (Butterworth [3], Chebyshev [4], InverseChebyshev [5], Elliptic [6], arbitrary quality factor [7]) by the fractionalorder filter transfer function (FOTF). These approaches are successful and more works are largely being done on lowpass (LP) and high pass (HP) filters while little attention is given to other critical filters such as band pass, notch and comb filters. It has been found that fractional LP and HP filters of order (1 + α) have been achieved and tested practically with minimal errors in passband response, accurate –3dB frequency response and greater stability margin [8].
As per theoretical operation, the band pass filters accept and rejects signals of a particular frequency band in a basic frequency selective circuit [8, 12] to provide narrow band characteristic. The filter is required a highquality factor (Q factor). The high Q band pass filter can be used in power system engineering to reduce harmonic voltage and current distortions [13–16], the design of wireless transmitters and receivers, audio signal processing, seismology, optics, instrumentation, sonar and many more [9, 21, 22]. Literature presented using harmonic current injection, use of controllable reactors in place of fixed reactors, and direct current ripple injection to improve the performance of band pass filters [17, 18].
In this work, the desired characteristic of the parameters varies from the designed value that can be always be maintained by computing the coefficient and real value differential orders. Firstly, the modified particleswarmoptimization (mPSO) is used to optimize values of the filter in order to satisfy the constraint requirements. The other optimization procedures compared with mPSO, the advantages of the mPSO are that it is easy to implement with few parameters needing adjustment. The performance of the new fractional band pass filters has been verified. Secondly, the optimum order of approximation s ^{α} 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 waveforms from the proposed FBPF are measured with various ranges of signal input frequencies. The performance generalized approach to design fractionalorder band pass filter of orders ‘α’ and ‘β’ respectively as (α ≤ 1 and β ≥ 1) and (β ≤ 1 and α ≥ 1) has been studied and compared with corresponding integer filters through both experimentation and simulation. The obtained results confirm that the actual fractional filters behavior closely follows the theoretical approximations for all values of α.
The paper is organised as follows: Section 2 introduces fractional order BP transfer functions and finding the coefficients using modified Particle Swam Optimization (mPSO). Section 3 describes the implementation of the results using different values for α and β. Section 4 discusses the hardware implementation of fractional band pass filter using reconfigurable analog device.
2 Fractional order band pass filter transfer functions
Equation (4) shows the admittance of the circuit which will be the transfer function of the fractional BPF. The coefficients k_{1}, k_{2} and k_{3} are the parameters of the transfer function (4) and they are obtained by using the optimization algorithm technique called mPSO. The BPF transfer function with the FC and FI orders, they give attenuations in stopband of –20α dB/decade and +20β dB/decade for frequencies that are higher and lower than the center frequency. Therefore, the second order BPF transfer function for Butterworth filter provides more flexibility and a lower order than that of its integer order counterparts.
2.1 Optimization of FO BPF transfer function coefficients using mPSO
The transfer function obtained in (1) for the BPF response, its new coefficients are obtained using the mPSO optimization algorithm. Using this algorithm, the new optimized coefficients k_{1}, k_{2} and k_{3} parameters can be determined for desired filter that is required as per objective. k_{1}, k_{2} and k_{3} inputted as variables to be optimized using mPSO [8]. For the BPF, the optimization carried out for the objective function between the –3dB close to 1 rad s^{−1} and Least Square Error (LSE) is solved to vary it close to 1.
From the mPSO algorithm flow chart in Fig. 2 below, it is explained how the set of optimal coefficients are obtained for the transfer function order (α + β). Here, pbest1, pbest2 and pbest3 are said to be the best positions for particles 1, 2 and 3 revering to the objective function called in the flow diagram. Whilst for the gbest1, gbest2 and gbest3 it simply refers to global best position of particles 1, 2 and 3 respectively. H_{1+α} is the FO BPF of order (α + β) which needs to be approximated by H_{BP} and k is the coefficients (k_{1}, k_{2}, k_{3}) of FO BPF function given in (4).
Flow Chart of the bilevel PSO algorithm
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
Flow Chart of the bilevel PSO algorithm
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
Flow Chart of the bilevel PSO algorithm
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
Therefore, in this paper we have used mPSO algorithm for more computable efficiency and it is adaptable for optimization of bilevel objective functions. Thus, mPSO optimization improves the desired filter characteristics and provides a lower LSE value and high stability margin. There are generalized equations for different types of filters with stability constraints and optimization but for the one used in this work is FO band pass Butterworth filter. This paper also proposes the best designed method for the FO for BPF. The optimal new coefficient values are obtained from a bilevel optimization that approximates the passband of the Butterworth response with fractionalstep stopband attenuation in Table 1.
Values of the k coefficients
Parameter specifications  
Given coefficients  Optimized coefficients from PSO Algorithm 






3 Results and validation
3.1 Optimized values for k coefficients
The given coefficients are optimized using the mPSO algorithm for different α and β. After optimizing the new coefficients for k_{1}, k_{2} and k_{3} are obtained from curve fitting tools in MATLAB. The optimized coefficients are plotted to see the magnitude and phase response as shown in Fig. 3a and b.
(a). The magnitude plot for α
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
(a). The magnitude plot for α
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
(a). The magnitude plot for α
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
3.2 Simulation output results
The optimized coefficients are then plotted in MATLAB to see the absolute admittance response of the FO BPF for different values of α and β. There were two different cases done for the simulations. Figure 3a and 3b show the magnitude and phase plot for the case where α ≥ 1 and β ≤ 1. Figures 3 and 4 also illustrate the graph for integer (α = 1 and β = 1) and noninteger BPFs, and observed that the noninteger graph has a sharper cut off compared to the integer filter.
(a). Magnitude plot for α < 1 and β > 1. (b) phase plot of α < 1 and β > 1
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
(a). Magnitude plot for α < 1 and β > 1. (b) phase plot of α < 1 and β > 1
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
(a). Magnitude plot for α < 1 and β > 1. (b) phase plot of α < 1 and β > 1
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The second case shows the simulations for α ≤ 1 and β ≥ 1, and shows the magnitude and the phase plots as in Fig. 4, the frequency which the magnitude of admittance is maximum is not the same with the frequency at which the phase is zero. Thus for more flexibility and tuning characteristics the first case (α ≥1 and β ≤1) can be more adjusted to have sharper tuning than the second case (α ≤ 1 and β ≥1).
3.3 Quality factor
Quality factor of the optimized parameters with different values of alpha and beta
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
Quality factor of the optimized parameters with different values of alpha and beta
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
Quality factor of the optimized parameters with different values of alpha and beta
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The peak frequency response from the optimized magnitude plot
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The peak frequency response from the optimized magnitude plot
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The peak frequency response from the optimized magnitude plot
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The results in Fig. 6 show the peak frequency for different values of β values. The graph also shows that if β = 1 the peak frequencies were lower that of those β values less than 1. It is also seen that peak frequencies for β values decrease with the increase in the order of α. Thus, the advantage of FO BPF is that it has more flexibility in selection of peak frequency compared to integer order.
4 Hardware implemenation and results
The setup for the implementation of fractional band pass filter
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The setup for the implementation of fractional band pass filter
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The setup for the implementation of fractional band pass filter
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
(a). Anadigm designer
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(a). Anadigm designer
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
(a). Anadigm designer
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
Internal switched capacitor circuit to realize (a) band pass filter biquadratic CAM (b) pole and zero frequency biquadratic CAM
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
Internal switched capacitor circuit to realize (a) band pass filter biquadratic CAM (b) pole and zero frequency biquadratic CAM
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
Internal switched capacitor circuit to realize (a) band pass filter biquadratic CAM (b) pole and zero frequency biquadratic CAM
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The hardware setup
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The hardware setup
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The hardware setup
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The FO band pass output observed from spectrum analyzer
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The FO band pass output observed from spectrum analyzer
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The FO band pass output observed from spectrum analyzer
Citation: International Review of Applied Sciences and Engineering 13, 1; 10.1556/1848.2021.00308
The connection of the two CAM is shown in Fig. 8 below in the Anadigm Designer.
The fractionalorder band pass filter was realized and implemented on the FPAA development board and the setup of the hardware experiment is shown below.
5 Conclusion
The paper focused on the design and implementation of the fractionalorder band pass filter with the practical realization using reconfigurable analog device. Different fractionalorder exponents of (α, β) have been realized in this paper for 0 < α ≤ 1. The performance of design fractionalorder filter by varying the values for α and β were observed through simulation comparing with integerorder and hardware implementation on the reconfigurable analog device. The results showed that fractionalorder band pass filter gives a flexibility to design the peak frequency by varying the values of ‘α’ and ‘β’ within the fractionalorder transfer function. Also observed from the results were the output waveforms of the proposed design band pass filter measured using the oscilloscope and the spectrum analyzer for different ranges of input frequency signal. In this way, the fractionalorder filter of order (α + β) have been studied and compared with the corresponding integer order filter through simulation and experimentation.
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