DVCC based (2+α) order low pass Bessel filter using optimization techniques

: This paper proposes the design and analysis of (2+α) order low pass Bessel filter using different optimization techniques. The coefficients of the proposed filter are found out by minimizing the error between transfer functions of (2+α) order low pass filter and third-order Bessel approximation using simulated annealing (SA), interior search algorithm (ISA), and nonlinear least square (NLS) optimization techniques. The best optimization technique based on the error in gain, cut off frequency, roll-off, passband, stopband, and phase is chosen for designing the proposed filter. The stability analysis of the proposed filter has also been done in W-plane. The simulated responses of the best optimized proposed filter are obtained using the FOMCON toolbox of MATLAB and SPICE. The circuit realization of 2.5 order low pass Bessel filter is done using two DVCCs (differential voltage current conveyors), one generalized impedance converter (GIC) based inductor, and one fractional capacitor. The proposed filter is implemented for the cut off frequency of 10 kHz using a wideband fractional capacitor. Monte Carlo noise analyses are also performed for the proposed filter. The MATLAB and SPICE results are shown in good agreement.


Introduction
Recently, fractional order systems have shown great attraction to researchers in the field of science and engineering. These fields contain bioengineering, control systems, signal processing, nanotechnologies, biology, electrical engineering, medicine, finances, etc. The concepts of fractional calculus can be used to model various fractional order systems since it provides various novel features along with design flexibilities. The continuous progress of fractional order systems and circuits requires the study of their mathematical explanation as well as their physical implementation [1][2]. Various signal processing blocks such as a fractional order oscillator, filters, integrators, differentiators, multivibrators, etc. have been explored in the fractional order domain. Many definitions have been proposed for fractional order derivatives [3] such as Caputo definition is as follows ( is gamma function, m is an integer and α is fractional order. Initially, fractional order filters have been designed for first and second order systems [4][5]. Further, the active and passive realization of fractional Butterworth filter has been done by Ali et al. [6]. Nowadays, the performance of fractional order filters is being improved by using optimization techniques [7][8][9]. Freeborn et al. realized fractional order Butterworth, Chebyshev, and Inverse Chebyshev filters using optimization techniques [10][11][12][13][14]. In addition to these, the comparison of different optimization techniques for designing fractional filters (Butterworth Chebyshev and Bessel) has also been done [15][16][17][18]. Thus, fractional order Butterworth, Chebyshev, Inverse Chebyshev, and Bessel filters have been designed using optimization techniques in the literature. However, there is a need to design a higher order fractional filter. Here, higher order Bessel filter is designed using optimization techniques as it is not attempted previously. In the proposed work, (2+α) order low pass Bessel filter is approximated using SA, ISA, and NLS optimization techniques. The best technique out of these three is chosen and then the proposed filter is realized using DVCC based circuit. DVCC is an advanced and most effective block for realizing analog circuits. It has benefits of the differential difference amplifier and second generation current conveyor (CC-II). This paper is organized as follows: Section 2 focuses on the optimization techniques used for the proposed filter. Section 3 presents the stability analysis in W-plane. Section 4 deals with the comparison of different optimization techniques based on various performance parameters. Section 5 emphasizes the analog realization of the proposed filter. Section 6 discusses results and finally, the main facts are summarized in section 7.

Optimization techniques for coefficient selection
In the proposed work, (2+α) order low pass Bessel filter coefficients are optimized using SA, ISA, and NLS. These optimization techniques have been shown using flow charts in Figs. 1-3. To approximate the passband behavior of the proposed filter, the frequency range from ω equals 10 -5 to 1.5 rad/sec is used to reduce the error function. The transfer function of (2+α) order low pass filter is given as follows The 3 rd order Bessel transfer function with cut off frequency 1 rad/sec is given by 3 x is a vector of filter coefficients, ( , ) i Tx is magnitude response of Eq. 1 and 3 () is the third-order Bessel approximation at a frequency i  , and k is the total number of data points. SA, ISA, and NLS optimized filter coefficients 0 1 2 3 ( , , , ) a a a a are found out for α value ranging from 0.1-0.9 and summarized in Table 1.  Analyze and compare the results of default trust-region-reflective algorithm and the levenberg-marquardt algorithm.
Are the results same?
Any of the algorithm results can be chosen.
Define Lower bound and upper bound in the form of vectors or matrices of same size as . Coefficients of are to be found out using lscurvefit.

Yes
No Define observed output matrices or vectors.
Better results out of two can be taken into consideration. Table 1: Filter coefficients of (2+ α) order low pass Bessel filter using SA, ISA, and NLS

Stability Analysis
Stability analysis is an important aspect to confirm the possibility of analog realization of the proposed filter. To explore the stability of the proposed (2+α) order Bessel filter, conversion of s-plane transfer function into the W-plane transfer function is required [19][20][21][22][23][24]. This conversion is done in the following manner i) Convert s=W m and α=k/m. ii) Choose k and m for the required value of α. iii) Converted W-plane transfer function is solved for all poles. iv) Evaluate the absolute pole angles w  , if all are greater than 2m  then the system is stable otherwise not.  i) Gain error: It is calculated by comparing the maximum gain of fractional order low pass Bessel filter with the maximum gain of the ideal Bessel filter.
ii) Cut off frequency error: It is the frequency at which the magnitude of the signal falls by 3 dB of its maximum value. Cut off frequency error is measured by comparing the cut off frequency of the proposed filter with the ideal Bessel filter (1 rad/sec). iii) Roll-off error: It is the fall in dB/decade in the transition band. The roll-off error can be determined by comparing the roll-off rate of the proposed filter with the ideal Bessel filter. iv) Passband error (PE): It is the error observed in the passband (till 1 rad/sec) when compared to the ideal Bessel response. PE can be calculated as follows-  Table 3 shows the comparison of parameters for different optimization techniques (SA, ISA, and NLS). It has been observed that NLS gives the minimum error of all parameters (gain error, cut off frequency error, roll-off error, PE, SE, and phase error) as compared to SA, and ISA for α equal to 0.2, 0.5, and 0.8. Gain and roll-off of ideal third-order Bessel filter are -20.9dB and -54.6dB/decade. The frequency and phase response of SA, ISA, and NLS optimized (2+α) order Bessel filters for the orders 2.2, 2.5, and 2.8 have been plotted in Fig. 4(a)-(c). Further, the frequency and phase responses have also been plotted for B3(s) to show the deviation of fractional order filters. These responses show that the roll-off increases as the order are increasing from 2.2 to 2.8. The errors in gain, cut off frequency, roll-off, passband (PE), stopband (SE), and phase for (2+α) order Bessel filters are calculated and tabulated in Table 3.  Table 3. Simulated parameters of (2+ α) fractional-order low pass Bessel filter

Analog realization of the proposed filter
It has been discussed earlier that the NLS gives the least gain error, cut off frequency error, roll-off error, PE, SE, and phase error as compared to SA, and ISA techniques for the proposed filter. So, there is a requirement to verify the results obtained from the NLS optimization technique. Here, DVCC is chosen to design NLS optimized (2+α) order low pass Bessel filter. DVCC is defined using the following matrix: (10) Fig. 5 shows the circuit diagram of the NLS optimized (2+α) order low pass Bessel filter using 2 DVCCs, 1 GIC based inductor, and 1 fractional capacitor. The internal structure of DVCC is given in Fig. 6   GIC based inductor is used in the proposed circuit for L=1mH. In Fig.7, the desired value of the inductor (L=1mH) is achieved by choosing R=1K, R5=100Ω, and C4=0.01µF. Equivalent input impedance to ground of above ckt. (Fig. 7) is given by 1 DVCC based 2.5 order proposed filter is used C1 as a fractional capacitor, this capacitor is used for a wide frequency range. It is made up of 10 resistances and nine capacitances. Fig. 8 shows the structure of a wideband fractional capacitor and Table 4 gives the values of resistances and capacitances used in C1 [29].   Table 4. Values of resistances and capacitances used in the fractional capacitor for α=0.5 To get the overall transfer function of DVCC based proposed filter (Fig. 5) After dividing the numerator and denominator of Eq. 16 by (R2+R3), then compare this equation with Eq. 4. The outcome of comparison gives the values of R1=5163.9Ω, R2=168200 Ω, R3=19070Ω with C1=2.5nFs α-1 , C2=0.02µF, and L=1mH. These values are used to get the magnitude response of DVCC based NLS optimized (2+α) order low pass Bessel filter, magnitude is scaled by 10000 and frequency shifted to 10 kHz.

SPICE simulated magnitude response
The SPICE simulated magnitude response of the proposed 2.5 order Bessel filter is shown in Fig. 9. The MATLAB and SPICE simulated results of 2.5 order Bessel filters have been compared. The absolute error in MATLAB and SPICE simulated results of gain and cut off frequency are 3.5 dB and 0.37 rad/sec, respectively. It specifies that the results of MATLAB and SPICE are close to each other as desired for realization at the circuit level using approximated fractional order capacitors.
In addition to it, the Monte Carlo analysis of 2.5 order DVCC based NLS optimized Bessel filter for all the resistances and capacitances used in the circuit (Fig. 5) within 5% tolerance has been done for n=100 runs. The resultant plots are shown in Fig.10 (a)-(b). The maximum variation in gain, cut off frequency and roll-off rate for 2.5 order proposed filter are (-20.18 dB to -20.22 dB), (15.80 kHz to 16.33 kHz) and (-39.65 dB/decade to -41.27dB/decade) respectively. Thus, it shows the reasonable variation in the above mentioned parameters.

Noise Analysis
Noise analysis is an important aspect to see the impact of noise on the proposed circuit. There are different kinds of noise in any electronic circuit shot noise, flicker noise, and thermal noise. The collective effect of all such noises on the proposed circuit (Fig. 5) is determined in the SPICE environment. The behavior of input and output noise voltage of 2.5 order proposed filter to frequency is shown in Fig. 11. As can be seen from this figure (Fig.11) that both the input and output noise are low in the entire passband.

Conclusion
This work presents the designing of (2+α) order low pass Bessel filter using SA, ISA, and NLS techniques. These techniques are used to optimize the filter coefficients. Further, the best optimization technique based on gain error, cut off frequency error, roll-off error, PE, SE, and phase error has been chosen to design the proposed filter using DVCCs. The NLS optimized (2+α) order low pass Bessel filter gives good similarity with SPICE simulated DVCC based circuit. Therefore, MATLAB and SPICE results show a good similarity between results. This work can be further extended for other approximations of the filter.