Validation of Fractional-Order Lowpass Elliptic Responses of (1 + α)-Order Analog Filters

Abstract

In this paper, fractional-order transfer functions to approximate the passband and stopband ripple characteristics of a second-order elliptic lowpass filter are designed and validated. The necessary coefficients for these transfer functions are determined through the application of a least squares fitting process. These fittings are applied to symmetrical and asymmetrical frequency ranges to evaluate how the selected approximated frequency band impacts the determined coefficients using this process and the transfer function magnitude characteristics. MATLAB simulations of ( 1 + α ) order lowpass magnitude responses are given as examples with fractional steps from α = 0.1 to α = 0.9 and compared to the second-order elliptic response. Further, MATLAB simulations of the ( 1 + α ) = 1.25 and 1.75 using all sets of coefficients are given as examples to highlight their differences. Finally, the fractional-order filter responses were validated using both SPICE simulations and experimental results using two operational amplifier topologies realized with approximated fractional-order capacitors for ( 1 + α ) = 1.2 and 1.8 order filters.

Fractional-Order Chaotic Memory with Wideband Constant Phase Elements

“…The colored scale is as following: dark blue areas represent fixed point solution, green stands for limit cycle solutions, yellow and white denotes weak (LLE lower than 0.1) and strong (LLE greater than 0.1) chaos behavior. Since three-segment odd-symmetrical PWL functions∈ are considered for memory, this plot represents four-dimensional hypercube with edges g 1 inner ∈(−21, −19), g 1 outer ∈(7, 9), g 2 inner ∈(−16, −14), g 2 outer ∈ (17,19) and resolution 201 × 201 × 201 × 201 points. For this calculation, the fourth-order Runge-Kutta method in Matlab.…”

“…Some research works are focused on the general properties of this class of analog building blocks [13,14]. Other studies are aimed at specific structures such as Sallen-Key and KHN filters [15], passive and active realizations of filters having a Butterworth-type of frequency response [16], low-pass filter with transfer zeroes [17], low-pass with electronically reconfigurable parameters [18], pseudo-differential all-pass filter [19], etc. A FO filter can be constructed by using field programmable analog array as well [20,21].…”

Special Issue “Selected Papers from the 2018 41st International Conference on Telecommunications and Signal Processing (TSP)”

“…In the paper [4] by D. Kubanek et al, fractional-order transfer functions to approximate the passband and stopband ripple characteristics of a second-order elliptic lowpass filter are designed and validated. The necessary coefficients for these transfer functions are determined through the application of a least squares fitting process.…”

On the Design of Power Law Filters and Their Inverse Counterparts

“…Recent works have demonstrated the generalization of the Butterworth [23], Chebyshev [24], inverse Chebyshev [25], and elliptic filters [26] to the FO domain. Another design strategy that involves the approximation of FO filter characteristics using the integer order transfer function was also reported in the literature [27].…”