Realization of optimized fractional-order symmetric-slope bandpass filter using switched-capacitors

Abstract

Here, the design, simulation, and modelling of a symmetric-slope fractional-order bandpass filter (FOBPF) using four fractional capacitors of orders $\alpha _0, \beta _0, \alpha _1,$ and $\beta _1$ has been explored. The characteristics of cascaded integer order bandpass filters (IOBPF) are well established. However, this is not so far for a FOBPF. Different issues in frequency and time response as obtained from different combinations of fractional orders with a stepping value of 0.1 have been elaborated in this work. The genetic algorithm optimization technique has been used here to determine the fractional orders of each fractional capacitors (${FC_1}$ and ${FC_2}$) providing the maximum quality factor of the filter. The proposed FOBPF has achieved a quality factor of 11.47 as comapred to the quality factor of 0.44 in case of an integer order filter. The parameters of the proposed filter were derived analytically and then compared with the experimental observations using an RC-Symmetric approximated fractional capacitor and general impedance converter (GIC). The second phase of the work deals with the application of switched-capacitors in the proposed filter circuit. It has also been observed in simualtion as well as experimental analysis that incorporation of switched-capacitor in place of resistors reduces settling time and hence increases the speed of response.

1 Introduction

Higher-order filters can easily and directly be designed by cascading one or two-pole bi-quad filter. Similarly by cascading a high-pass filter with a low-pass filter, a bandpass response can be created. The active bandpass filter is a frequency perceptive circuit used in analog electronics systems. It is used to distinguish a signal at one perfect frequency, or a range of signals that lies within a certain “band” of frequencies. Some reports have been found in literature where the design and realization of fractional-order bandpass filter (FOBPF) have been discussed [1,2,3]. The authors have carried out an analytical study to evaluate bandwidth, and quality factors from their phase and magnitude plots. Also, few experiments on FOBPF have been reported to validate the results. In [4,5,6,7], it has also been reported that the incorporation of fractional capacitor in bandpass circuits results in the stopband attenuation of any rational number which adds an additional degree of design freedom to fit a system. Apart from that different attempts were made to increase the quality factor (Q) to establish a sharper cut-off frequency [8].

As discussed in the literature, generalization of integer-order bandpass filter (IOBPF) to the fractional domain has flourished as a research area in recent times. The authors in [9] have designed and realized FOBPF using fractional capacitors of the same exponents. Similarly, a continuous-time asymmetric-slope FOBPF with high quality-factor has been presented in [10]. As reported, high-Q FOBPF is possible when one pair of poles is located very close to the stability boundary [7]. The performance analysis of FOBPF has also been studied through both experimentations and simulations in [3, 8]. The quality factor of the proposed FOBPF increases as compared to its integer-order counterpart when the exponents of the fractional-order elements change with one fractional-capacitor of order greator than one, keeping the other parameters constant. The authors in [11] implemented FOBPF using Laguerre Impulse Response approximations and tested with EEG signal.

The fractional-order bandpass filter discussed in the literature has the advantages of higher gain stability and greater flexibility in acquiring high frequency as compared to classical prototype filters [3]. It has also been observed that these realized FOBPF studied in literature have lower quality factors as compared to prototype filters [2, 3]. To achieve high Q, the resistance value of the circuit should be kept low. But due to the low resistance, very high current flows through the circuit and hence, the large voltage may cause damage to the circuit. The advantage of a fractional-order circuit is that the bandwidth can be reduced by decreasing the exponents of the fractional capacitors, keeping the values of resistance constant. On the other hand, other filter parameters like Q-factor and center frequency are also the function of ${\alpha _0}$, ${\alpha _1}$, ${\beta _0}$, and ${\beta _1}$ which adds an extra degree of design freedom [12]. Hence, a high-Q bandpass filter with flexibility in selection of filter parameters can be implemented with the help of fractional capacitors.

To have a bandpass filter with a large Q, at least one pair of poles should be located very close to the stability boundary in integer-order filter [7]. Filters with high quality factors have a better selectivity and sharper tuning characteristics. Hence, these filters find immense applications in radar, satellite, and medical accelerators applications [13]. Due to the sharp frequency response characteristics, high-Q FOBPF can be used to remove spurious signals. Similarly, FOBPF with high-Q can be used to provide an alternative impedance path for harmonic currents generated by a load. Hence, these filters find enormous applications in power system engineering for reducing current distortion and harmonic voltage[14]. Also, high-Q tunable filters are needed in wireless systems to enable efficient utilization of the available frequency spectrum [15]. Thus, it is essential to design and realize a high Q fractional-order bandpass filter. However, most studies on fractional-order bandpass filters made so far, exhibit asymmetric-slope response with a high gain drop which deteriorates the performance of the filter circuits [10]. Here, the authors try to realize a symmetric slope fractional-order band pass filter with a sharper cut-off frequency response. Also, performance parameters like settling time, Peak frequency, and selectivity are evaluated and compared with integer-order filter.

In this paper, a cascaded bandpass filter is designed using fractional capacitors and switched capacitors. To address the issues of symmetric-slope responses, a narrow bandpass filter with an optimized quality factor has been discussed in this work. Recently, a substantial amount of work on the optimization of the filter parameters has been noted in the literature. The [16], gravitational search algorithm (GSA) is used to optimize the parameters of the Butterworth fractional-order low-pass filter. Similarly, the symbiotic organism search algorithm (SOS) is used in fractional-order high-pass Butterworth filter to optimize the order of the circuit in [17]. Also, a multi-objective optimization (MOO) technique has been used to optimize the filter specifications in [18]. The design of a low-pass Bessel filter in the fractional domain has been reported in [19] to optimize the orders ${\alpha }$ and ${\beta }$ using the interior search algorithm (ISA). The design of a fractional-order high-pass Butterworth filter (FBBPF) is approximated to an integer-order (IO) transfer function using different metaheuristic optimization approaches for investigating the asymmetric roll-off characteristics [17]. The major drawback of such algorithms is that they have a slow convergence speed and easily fall into local optimum [20]. However, the Genetic algorithm (GA) is robust, stochastic, easy to understand and easily parallelized. It can operate in various representations including mixed discrete as well as continuous problems with very less information regarding the problem [21]. The capability of GA to be implemented as a Universal Optimizer can be used for optimizing any type of problem belonging to different fields. It has simplicity, ease of implementation, proper balance between exploration and exploitation and can be achieved by setting parameters appropriately. Also, GA provides logical reasoning behind the use of operators like selection, crossover and mutation. Hence, GA is chosen here to optimize the Q-factor of the proposed FOBPF.

It is well known from the literature that the incorporation of switched capacitors in place of resistors reduces the settling time and increases the stability of the circuits [22,23,24]. Also, prototype filters employing the RC time constant can cause bandwidth variation due to the tolerance factor of resistance. Hence, the use of switched capacitor in place of resistors enhances the frequency stability and also improves the performance factor. The filters replacing the resistor with a switched capacitor eliminates the probable power losses due to resistors. Due to such significant advantages, the authors attempt to generalize the design of a FOBPF circuit using switched capacitors. Extensive simulation studies on settling time, maximum peak, and rise time of FOBPF using switched capacitors have also been reported.

The proposed filter possesses order, $\alpha _0+\beta _0+\alpha _1+\beta _1$ which performs close to a symmetrical frequency response with a higher cut-off frequency. Fractional order filters are signified by general fractional-order differential equations, and well-thought-out as the comprehensive case of the integer-order [4, 5, 25]. Here in fractional-order filters, the passive elements like capacitors are replaced by its fractional counterpart whose transfer functions are, $FC_1=1/(C_1 s^{\alpha _0})$, $FC_2=1/(C_2 s^{\beta _0})$, $FC_3=1/(C_3 s^{\alpha _1})$ and $FC_4=1/(C_4 s^ {\beta _1})$. Thereafter, on evaluating the transfer function of the filter in the s-domain it yields the fractional-order transfer function. The fractional order (FO) circuits are best modeled by fractional calculus in the form of fractional derivatives and integrations [26, 27]. Irrespective of electrical circuits and systems fractional-order has also provided promising results in biomedical engineering where the applications of low-frequency filters are eminent [28]. So, the fractional integration of $\alpha ^{th}$ order of the function f(t) is defined as [5],

$$\begin{aligned} J^{\alpha }f(t)= \frac{1}{\Gamma {(\alpha )}}\int _0^t{(t-\tau )}^{\alpha -1}{f(\tau )}{d\tau } \end{aligned}$$

(1)

Here, [29] represents the Riemann-Liouville fractional integral, where $\alpha$ is the fractional-order varying as $0< \alpha <1$. The Laplace Transform of the fractional Integral is,

$$\begin{aligned} L[{J^{\alpha }{f(t)}}]= {s^{-\alpha }{F(s)}} \end{aligned}$$

(2)

For Zero initial conditions. Here F(s) is the Laplace transform of f(t).

The rest of the paper is organized as follows; Sect. 2 provides the modelling and simulation of the cascaded bandpass filter in its integer as well as fractional counterpart and optimization of the orders using genetic algorithm. In sect. 3, the proposed fractional-order filter is designed using switched-capacitor. In sect. 4 the experimental realization of the proposed fractional-order filter as well as proposed fractional-order filter using switched-capacitors has been carried out, whereas and finally, the paper is concluded in section 5.

2 Cascaded fractional order bandpass filter

Here, a cascaded bandpass filter is being considered for design, where the integer-order capacitors are replaced with its fractional-order counterpart [30, 31]. Thereafter, the resistors in this fractional-order bandpass filter are replaced by switched-capacitors [24]. Here the fractional capacitors are $FC_1$, $FC_2$, $FC_3$ and $FC_4$ bearing orders ${\alpha _0}$, ${\beta _0}$, ${\alpha _1}$, and ${\beta _1}$ respectively. In figure 1 the cascaded fractional order bandpass filter is seen, though in literature the integer counterpart of the filter is being studied earlier, in this paper the capacitors are replaced with fractional capacitors for performance improvements. Considering the fractional capacitors and resistors in figure 1 with the values as depicted in table 1, the transfer function equation of the cascaded fractional order bandpass filter is conveyed as,

Table 1 Filter elements values.

Figure 1

Multistage fractional-order bandpass filter using four fractional capacitors.

$$\begin{aligned} {H(s)=G_{stage 1}(s). G_{stage 1}(s)} \end{aligned}$$

(3)

where,

$$\begin{aligned} G_{stage 1}(s)= & {} \frac{R_f FC_1 {s}^{\alpha _0}}{(R_i FC_1 s^{\alpha _0}+1)(R_f FC_2 s^{\beta _0}+1)}, \\ G_{stage 2}(s)= & {} \frac{R_f FC_3 {s}^{\alpha _1}}{(R_i FC_3 s^{\alpha _1}+1)(R_f FC_4 s^{\beta _1}+1)} \end{aligned}$$

Now substituting, ${s=j.\omega }$ in trasnsfer function (3),

$$\begin{aligned} H({j.\omega })=\frac{{R_f}^{2} FC_1 FC_3 {{(j.\omega )}}^{{\alpha _0}+{\alpha _1}}}{a\times b \times c \times d} \end{aligned}$$

(4)

Where, ${a=(R_i FC_1 {(j.\omega )}^{\alpha _0}+1)}$, ${b=(R_f FC_2 {(j.\omega )}^{\beta _0}+1)}$,

${c=(R_i FC_3 {(j.\omega )}^{\alpha _1}+1)}$, and ${d=(R_f FC_4 {(j.\omega )}^{\beta _1}+1)}$

The maximum of function (in magnitude) ${(H(j.\omega ))}$ can be obtained by applying the condition ${\frac{d}{d\omega }|H(j.\omega )|_{\omega =f_0}=0}$. Therefore, for stage 1, maximum of function (in magnitude) is obtained by solving ${\frac{d}{d\omega }(|G_{stage 1}(j.\omega )|)=0}$ i.e., given by,

$$\begin{aligned} \begin{aligned}&A_0\omega ^{{\beta _0}-1}+B_0\omega ^{2({\beta _0}-{\alpha _0})-1}-C_0\omega ^{-2{\alpha _0}-1}+D_0\omega ^{2{\beta _0}-{\alpha _0}-1}\\&\quad +E_0\omega ^{{\beta _0}-{\alpha _0}-1}+F_0\omega ^{{\beta _0}-1}+G_0\omega ^{{\beta _0}-2{\alpha _0}-1}-H_0\omega ^{-{\alpha _0}-1}=0 \end{aligned} \end{aligned}$$

(5)

Where,

$$\begin{aligned} A_0= & {} 2{\beta _0}{R_i}^2 {FC_1}^2, B_0=2({\beta _0}-{\alpha _0}){\frac{FC_2}{FC_1}}^2,\\ C_0= & {} 2{\alpha _0}{\frac{R_i}{R_f}}^2 cos{\alpha _0}\pi , D_0=2(2{\beta _0}-{\alpha _0})R_i FC_2 cos\frac{{\alpha _0}{\pi }}{2},\\ E_0= & {} ({\beta _0-\alpha _0})\frac{R_i}{R_f}\left[ \frac{FC_2}{FC_1} cos\frac{({\beta _0-\alpha _0})\pi }{2}+ cos\frac{({\alpha _0+\beta _0})\pi }{2}\right] ,\\ F_0= & {} \beta _0\frac{{R_i}^2}{R_f FC_1} cos\frac{{\beta _0}{\pi }}{2},\\ G_0= & {} (\beta _0-2\alpha _0)\frac{FC_2}{R_f {FC_1}^2}cos\frac{({\alpha _0+2\beta _0})\pi }{2}, \\ H_0= & {} -{\alpha _0}\frac{R_i}{{FC_1}{R_f}^2} cos\frac{{\alpha _0}\pi }{2}. \end{aligned}$$

Similarly, the maximum of function (in magnitude) of stage 2 is obtained by solving ${\frac{d}{d\omega }(|G_{stage 2}(j.\omega )|)=0}$ i.e., given by,

$$\begin{aligned} \begin{aligned}&A_1\omega ^{{\beta _1}-1}+B_1\omega ^{2({\beta _1}-{\alpha _1})-1}-C_1\omega ^{-2{\alpha _1}-1}+D_1\omega ^{2{\beta _1}-{\alpha _1}-1}\\&\quad +E_1\omega ^{{\beta _1}-{\alpha _1}-1}+F_1\omega ^{{\beta _1}-1}+G_1\omega ^{{\beta _1}-2{\alpha _1}-1}-H_1\omega ^{-{\alpha _1}-1}=0 \end{aligned} \end{aligned}$$

(6)

Where,

$$\begin{aligned} A_1= & {} 2{\beta _1}{R_i}^2 {FC_3}^2, B_1=2({\beta _1}-{\alpha _1}){\frac{FC_4}{FC_3}}^2,\\ C_1= & {} 2{\alpha _1}{\frac{R_i}{R_f}}^2 cos{\alpha _1}\pi , D_1=2(2{\beta _1}-{\alpha _1})R_i FC_4 cos\frac{{\alpha _1}{\pi }}{2},\\ E_1= & {} ({\beta _1-\alpha _1})\frac{R_i}{R_f}\left[ \frac{FC_4}{FC_3} cos\frac{({\beta _1-\alpha _1})\pi }{2}+ cos\frac{({\alpha _1+\beta _1})\pi }{2}\right] ,\\ F_1= & {} \beta _1\frac{{R_i}^2}{R_f FC_3} cos\frac{{\beta _1}{\pi }}{2},\\ G_1= & {} (\beta _1-2\alpha _1)\frac{FC_4}{R_f {FC_3}^2}cos\frac{({\alpha _1+2\beta _1})\pi }{2}, \\ H_1= & {} -{\alpha _1}\frac{R_i}{{FC_3}{R_f}^2} cos\frac{{\alpha _1}\pi }{2}. \end{aligned}$$

Now, maximum of function (in magnitude) of cascaded FOBPF (considering both stage 1 and stage 2) is given by,

$$\begin{aligned}{} & {} |G_{stage 1}(j.\omega )| \frac{d}{d\omega }\left( |G_{stage 2}(j.\omega )|\right) \nonumber \\{} & {} \qquad +|G_{stage 2}(j.\omega )| \frac{d}{d\omega }\left( |G_{stage 1}(j.\omega )|\right) =0 \end{aligned}$$

(7)

The maximum of function (in magnitude) depends on the exponents, ${\alpha _0}$, ${\beta _0}$, ${\alpha _1}$, and ${\beta _1}$. The salient features of proposed FOBPF has been studies considering stage 1 and combined stage 1 & stage 2 in the following subsections.

2.1 Considering stage 1 only

Here, the proposed fractional order bandpass filter is demostrated in stage 1 only using two resistors ${R_i}$ and ${R_f}$ as well as two fractional capacitors ${FC_1}$ and ${FC_2}$. This stage 1 of the fractional-order bandpass filter as shown in figure 1 is simulated in MATLAB and the frequency domain analysis is obtained in figure 2(a) and (b) as well as the frequency-parameter values are represented in tables 2 and 3. Here, in case of stage 1 the values of fractional capacitors are choosen as ${{FC_1}=1~\mu Fs.^{1-\alpha _0}}$ and ${{FC_2}=10~\mu Fs.^{1-\beta _0}}$. From the frequency domain plot in figure 2(a) and (b) it is seen that with a decrease in ${\alpha _0}$ value from 1.0 to 0.1 by keeping ${\beta _0}=1$(i.e., constant) the magnitude curve slowly converges down lowering the values of center (peak) frequency. In the magnitude plot it is also observed that, initially the slope of the curve slowly decreases and then abruptly increases/decreases as per the fractional orders. Again, from the phase plot it is observed that for some fractional values of ${\alpha _0}$, the phase converges below the integer values and for values i.e., ${\alpha _0}= 0.3$ and 0.8 it converges after the integer value.

Figure 2

Frequency response of (a) fractional-order bandpass filter with ${\alpha _0}$ varying and ${\beta _0 = 1}$ considering stage 1 only (b) fractional-order bandpass filter with ${\beta _0}$ varying and ${\alpha _0 = 1}$ considering stage 1 only.

Table 2 Parameters of the FOBPF by varying ${\alpha _0}$ at ${\beta _0 = 1}$ (considering stage 1 only).

Table 3 Parameters of the FOBPF by varying ${\beta _0}$ with ${\alpha _0 = 1}$ (considering stage 1 only).

It can be pointed out that both the bandwidth and the center frequency decrease with a decrease in fractional order ${\alpha _0}$. Now, from the plot shown in figure 2(b) it is observed that, on decreasing the fractional value of ${\beta _0}$ from 1.0 to 0.1, keeping ${\alpha _0=1}$ (i.e., constant) both magnitude and the phase plot varies. In the magnitude plot, the decrease in ${\beta _0}$ values raises the curve upwards from -140 dB to more than -100 dB, whereas, in the case of the phase plot, the curve shifts in frequency axis i.e., a decrease in ${\beta _0}$ results in an increase in phase sweep. Now, the genetic algorithm technique is used to determine the orders ${\alpha _0}$ and ${\beta _0}$ at which maximum quality factor can be obtained.

Figure 3

Proposed Quality factor determination methodology of a bandpass filter.

Figure 4

Flowchart of genetic algorithm optimization.

Figure 5

(a) 3-D plot of quality factor varying with ${\alpha _0}$ and ${\beta _0}$ in single stage FOBPF, and (b) Contour plot of quality factor with varying with ${\alpha _0}$ and ${\beta _0}$ in single stage FOBPF.

Figure 6

Magnitude response of optimized stage 1 FOBPF at ${\alpha _0}=0.3$ and ${\beta _0}=0.7$.

Genetic algorithms are computer-based optimization methods that use the Darwinian evolution of nature as a model [20]. The solution-base of the problem is encoded as being similar to individual chromosomes consisting of several genes. In general, genetic algorithms tend to work better than traditional optimization algorithms because they are less likely to be off track by local optima [21]. Here, the genetic algorithm (GA) is used to determine the maximum value of the quality-factor for stage 1 of FOBPF while varying the fractional exponents of the capacitors ${FC_1}$ and ${FC_2}$ (i.e., ${\alpha _0}$, and ${\beta _0}$). Here, the problem is finding the value of fractional orders up to one decimal place for which the quality factor of cascaded fractional order bandpass filter is maximum, i.e., ${Q = f (\alpha _0, \beta _0)}$ where ${\alpha _0}$, ${\beta _0}$ are the orders of two fractional capacitors in stage 1 of proposed FOBPF. Now, from figure 3, we could examine the frequency parameters in terms of the variables embedded. ${x_{identity} (y_{max})}$ is the Centre frequency ${(f_0), x_{identity} (y_{max} -3dB)}$ gives two values of frequency i.e., lower cut-off frequency and the higher cut-off frequency, whose difference provides us the bandwidth. Hence, quality factor is given by ${Q=\frac{f_0}{bandwidth}}$, it can be formulated as shown in (8).

$$\begin{aligned} Q=\frac{x_{identity}(y_{max})}{\Delta x_{identity}(y_{max}-3dB)} \end{aligned}$$

(8)

Here, $y_{max}$ is the maximum peak magnitude and $x_{identity}$ is the function which gives the value of x-axis corresponding to the result in y-axis. The algorithm represented in figure 4, when initialized in Python coding evaluates the quality factors at different values of ${\alpha _0}$ and ${\beta _0}$ for stage 1. The initialization deals with the quality factor implemented from (8). The magnitude of the (3) is obtained, then obtaining $x_{identity}(y_{max})$ gives the Centre frequency ${(f_0)}$. Thereafter, the change in ${ x_{identity} (y_{max} -3dB)}$ gives the bandwidth where Quality factor, ${Q=\frac{f_0}{bandwidth}}$. The pose is then initialized of both the fractional orders ${\alpha _0}$ and ${\beta _0}$ at 0.1 to 1.0.

Now, in this analysis 1000 frequency points were initiated for the function where the population size of parameters ${\alpha _0}$, ${\beta _0}$ are 100. In selection procedure, the quality factors as selected among all possible outcomes from all fractional orders in the specified range (i.e., 0.1 to 1.0 for both the variables). The termination condition is to reach the maximum quality factor which is set, when the outcome of each conbination of fractional-orders are stored and compared with eachother (figures 5, 6).

2.2 Considering stage 1 and stage 2

This subsection deals with simulation results and theoretical background of the proposed circuit considering both stage 1 and stage 2 as shown in figure 1. The optimized orders ${\alpha _0}$ and ${\beta _0}$ are considered here for evaluating the quality factor and symmetricity characteristics for the cascaded FOBPF circuit. It is observed that for fixed values of ${\alpha _0}$ and ${\beta _0}$, higher quality factor can be achieved when ${\alpha _1}<1$ and ${\beta _1}>1$. The various frequency-domain parameters are determined from figure 7(a) and figure 7(b) are noted in tables 4 and 5 respectively. In tables 4 and 5, it can be inferred that maximum quality factor of 11.45 can be obtained with ${\alpha _1=0.1}$, ${\beta _1=1.9}$. Similiarly, symmetricity characteristics has been obtained at ${\alpha _1=0.7}$, ${\beta _1=1.9}$ as shown in figures 8, 9 and table 6. Here, in figure 10 the stability boundary has been determined from [32] using the orders of the fractional capacitors of the proposed FOBPF. The angle of the stability region is given by, ${\theta }=max({\alpha _0},{\beta _0},{\alpha _1},{\beta _1})\frac{\pi }{2}$. Then, the pole locations of the proposed FOBPF at ${\alpha _0}=0.3$, ${\beta _0}=0.7$, ${\alpha _1}=0.7$ and ${\beta _1}=1.9$ is determined in MATLAB uisng the FOMCON toolbox as in figure 11. It is seen that two poles ${P_1}$ and ${P_2}$ appear very close to the stability boundary. As expected, all the poles lies inside the stable region of s-plane, where ${P_1=-7.78\times 10^{3}+4.481i}$ and ${P_2=-7.78\times 10^{3}-4.481i}$.

Figure 7

Frequency response of proposed fractional-order bandpass filter at ${\alpha _0 = 0.3}$, ${\beta _0 = 0.7}$ (a) with increasing ${\alpha _1}$ and decreasing ${\beta _1}$, and (b) with deccreasing ${\alpha _1}$ and increasing ${\beta _1}$ at condition ${\alpha _1}+{\beta _1}=2$.

Table 4 Parameters of the proposed fractional-order bandpass filter with varying ${\alpha _1}$ and ${\beta _1}$ (as ${\alpha _1}<{\beta _1}$) at ${\alpha _0 = 0.3}$, ${\beta _0 = 0.7}$ (considering both stage 1 and stage 2).

Table 5 Parameters of the proposed fractional-order bandpass filter with varying ${\alpha _1}$ and ${\beta _1}$ (as ${\alpha _1}>{\beta _1}$) at ${\alpha _0 = 0.3}$, ${\beta _0 = 0.7}$ (considering both stage 1 and stage 2).

Figure 8

Frequency response of cascaded fractional-order bandpass filter with at${\alpha _0=0.3}$, ${\beta _0=0.7}$, ${\beta _1}=1.9$ and varying ${\alpha _1}$.

Figure 9

Symmetricity analysis of proposed FOBPF at the three high-Q conditions.

Figure 10

Stability boundary of the proposed fractional-order bandpass filter.

Figure 11

Pole locations of the proposed fractional-order bandpass filter.

Table 6 Parameters of proposed fractional-order bandpass filter with varying ${\beta _1}$ at ${\alpha _0 = 0.3}$, ${\beta _0 = 0.7}$, and ${\alpha _1 = 1}$ (considering both stage 1 and stage 2).

3 Proposed cascaded fractional-order filter with switched-capacitors

In this section, a multistage fractional order bandpass filter (FOBPF) is considered for design and realization purposes, where all the four resistors are replaced with switched-capacitors as shown in figure 12.

Figure 12

Multistage fractional-order bandpass filter using fractional capacitors and switched-capacitors.

Figure 13

Time-domain Response of Optimized FOBPF at ${\alpha _0=0.3}$, ${\beta _0=0.7}$, ${\alpha _1=0.8}$, and ${\beta _1=0.8}$ (a) using equivalent resistance corresponding to switching fequencies (b) using switched-capacitors.

In a switched-capacitor, the resistance value can be changed by adjusting the switching speed. However, prototype filters employing RC time constant can cause bandwidth variation due to the tolerance factor of resistance. Hence, the use of switched capacitors in place of resistors enhances the frequency stability and improves the performance factor i.e., decreases settling-time, peak-time as well as, the maximum peak. In this switched-capacitor two non-overlapping clock frequencies pulse is given [22], by virtue of which the generalized pulse is the delay percentage of the duty cycle in convergence [33].

Table 7 Comparison of FOBPF parameters using switched-capacitors and equivalent Resistances.

Figure 14

Comparative analysis of time-domain response of fractional-order bandpass filter with four switched capacitors of switching frequency ${(f_c)}$.

Table 8 Time-domain parameters comparison of fractional order bandpass filter with varying switching frequency ${(f_c)}$.

Here, by making use of the switched capacitors, the circuit is provided with equivalent resistances which could be realized easily on the circuit with less power-loss [23]. A paracytic capacitor of ${0.1~\mu F}$ is used as the key component of the switched capacitor which charges and discharges simultaneously as per the clock pulse. The circuit diagram of the cascaded switched capacitors based cascaded fractional-order bandpass filter is shown in figure 12 with four fractional capacitors and four switched capacitors. Now, observing the switched capacitors independently, they are initialized with optimized fractional capacitors constituting a switched-capacitors based fractional order bandpass filter. The time-domain analysis is being simulated of different values of switching frequency ${(f_c)}$ ranging from 200 Hz to 1000 Hz in figure 13 providing a step input to the circuit. It is seen that on increasing the switching frequency the peak value decreases but the settling time increases. From figure 13 and table 7 it is clear that replacing resistors with switched capacitors of equivalent resistances decreases the settling time. Here, the fractional order bandpass filter is modeled by using its fractional order transfer function, where each Laplace terms (s) is modelled separately as a subsystem to cascade it with each other to form the entire filter. Here, the fractional exponent of ${\frac{1}{s}}$ is taken as the unit fractional capacitor in Laplace domain, since the fractional capacitor’s model in Laplace domain is ${\frac{1}{s^\alpha }}$. Where, again the value of capacitance is initiated as product of ${\frac{1}{s^\alpha }}$ and 1/C. i.e.${\frac{1}{C \dot{s}^\alpha }}$.

Now, by simulating the proposed filter in LabVIEW, the results were obtained, and compared with the MATLAB results in figure 14, with ${f_c}$ varying from 200 Hz to 1000 Hz. Consequently, from time domain analysis it is stated that keeping the switching frequency of the switched capacitors from 200 Hz to 1000 Hz, the system provides comparatively faster step response as compared to the use of equivalent resistances. From table 8, it is seen that the step response obtained of the proposed filter with varying switching frequency in MATLAB and LabVIEW are nearly similar. Further investigations are needed for practical implementations of FOBPF using switched capacitors.

4 Realiztaion and performance study of cascaded fractional-order bandpass filter

Different issues are well explained experimentally by realizing two terminal fractional capacitors in [34,35,36,37]. Subsequently it has also been mentioned in [1] that the lifetime of such two-terminal fractional capacitors is on an average of three and half months. RC symmetric circuit [38] has been used here for realization of fractional capacitors.

The Foster representation of approximated fractional order capacitor is presented in the circuit in figure 15. Here, the proposed cascaded fractional order bandpass filter has been realized using four fractional capacitors, where the fractional capacitors are realized with newly developed RC Symmetric technique [38]. As reported in [3], the fractional capacitor with order greater than one can be realized using a general impedance converter (GIC) as shown in figure 14. Here, the impedances, ${Z_1}=100 ~k\Omega$, ${Z_2}=10 ~\mu F$, ${Z_3}=0.1 ~\mu F/s^{1-0.9}$, ${Z_4}=1~ \Omega$, and ${Z_5}=1 ~\Omega$ are used to realize the fractional-capacitor of order 1.9 (figure 16).

Figure 15

FOSTER I representation of RC Symmetric Approximated fractional capacitor.

Figure 16

General impedance converter for realization of fractional-capacitor of order greator than 1.

Here the structure contains N+2 numbers of parallel RC blocks and a single resistor connected in series and indexed as 0,1, 2, ...... N+2. Using, the Foster-1 representation as shown in figure 15, the values of resistors and capacitors were obtained by and tabulated in table 9.

Table 9 RC values at fractional-orders 0.3, 0.7 and 0.9.

Figure 17

(a) Experimental Implementation of the fractional order bandpass filter using fractional capacitors of optimized orders, (b) Magnitude plot comparison of simulated fractional order, and experimental fractional order proposed bandpass filter.

The three fractional-capacitors (with orders ${\alpha _0}$=0.3, ${\beta _0}$ =0.7 and ${\alpha _1}$=0.7) are realized using RC symmetric approximation. Similarly, fractional-capacitor (with order ${\beta _1 }$ =1.9) is realized replacing ${Z_3}$ of GIC circuit with RC-symmetric approximated fractional-capacitor of order 0.9. Here, in figure 17(a), the experimental setup of the proposed fractional order filter is designed. The frequency response of the bandpass filter has been observed by varying the input frequency of the signal from 1 Hz to 1000 Hz.

$$\begin{aligned} Magnitude, H|(j.\omega )|=20log{\left( \frac{V_{0}}{V_{i}}\right) } \end{aligned}$$

(9)

Here, two different FOBPFs are realized in hardware by using resistors (two numbers of Ri and Rf) first and then replacing these resistors with switched-capacitors in figure 1. It may be noted that, simulation results are closer to the experiental results with a maginal error of 4.45% when FOBPFs are realized using resistors.

Figure 18

Comparison of simulated and experimental switched-capacitor based fractional-order bandpass filter of order ${\alpha _0=0.3}$, ${\beta _0=0.7}$, ${\alpha _1=0.7}$ and ${\beta _1=1.9}$ and switching frequency of 1000 Hz.

Table 10 Comparison of simulated and experimental results.

Then, the FOBPF are realized using switched-capacitors in place of resistors to study its performance. As evident in figure 17(b), the experimental plots are closer to the simulated results with a marginal error of 18.18%. Then the performance parameters are evaluated both in simulation platform and through experimentations in table 10. These marginal errors are due to the wire resistances, circuit complexibility, experimental errors and non-idealities of the approximated fractional capacitors and other components used in circuit design (figure 18).

5 Conclusions

In this work a symmetric slope fractional order bandpass filter (FOBPF) has been explored using optimized orders of fractional capacitors. It is envisaged that the simulation studies carried out in this paper on optimized FOBPF based on switched-capacitors will open newer applications of fractional order circuits and filters. Here, Genetic algorithm was carried out in Python using SPYDER IDLE to obtain the optimized orders of the two capacitors i.e., ${\alpha _0=0.3}$, ${\beta _0=0.7}$ of ${FC_1}$ and ${FC_2}$. It is to be noted that the quality factor of 11.47 is obtained using ${\alpha _0=0.3}$, ${\beta _0=0.7}$, ${\alpha _1=0.7}$ and ${\beta _1=1.9}$. Like conventional filters the proposed fractional-order filte results in a nearly symmetric slope frequency responses with a roll-off-rate of +21.9 dB/decade and -21.3 dB/decade. It is clear in table 7 that the performance parameters of the proposed FOBPF improve when switched capacitor replaces the resistances. One of the major concerns of the present studies is, to study the different parameters i.e., settling time, peak overshoot, and rise time of the fractional-order filter where it would give better performance if in Lab-VIEW platform as compared to MATLAB Simulink platform. To validate the performance parameters of the proposed symmetric-slope high-Q FOBPF, experimentations are carried out using both resistors and switched-capacitors has also been carried out. The experimental results clearly establish that the performance parameters of FOBPF improve when the above stated orders of fractional capacitors are utilized.