Optimized fractional-order Butterworth filter design in complex F-plane

Abstract

This paper introduces a new technique to optimally design the fractional-order Butterworth low-pass filter in the complex F-plane. Design stability is assured by incorporating the critical phase angle as an inequality constraint. The poles of the proposed approximants reside on the unit circle in the stable region of the F-plane. The improved accuracy of the suggested scheme as compared to the recently published literature is demonstrated. A mixed-integer genetic algorithm which considers the parallel combinations of resistors and capacitors for the Valsa network is used to optimize the frequency responses of the fractional-order capacitor emulators as part of the experimental verification using the Sallen–Key filter topology. The total harmonic distortion and spurious-free dynamic range of the practical 1.5th-order Butterwoth filter are measured as 0.13% and 62.18 dBc, respectively; the maximum and mean absolute relative magnitude errors are 0.03929 and 0.02051, respectively.

Deep Learning for Detecting Multi-Level Driver Fatigue Using Physiological Signals: A Comprehensive Approach

“…The 50 Hz frequency of city power was removed using a notch filter [32] in the second stage to eliminate undesired artifacts from the recorded EEG data. The data were subjected to a first-order Butterworth filter in the third phase, which ensured that only useful information in the frequency range of 0.05 to 60 Hz was included in the processing [33]. The Eye Blinking Removal plugin, which is included in the EEG Lab toolbox (ver15) [34], was used in the fourth stage to automatically eliminate the participants' unnecessary blinks throughout the experiment.…”

Design of Higher-Order Fractional Filters With Fully Controllable Frequency Characteristics

“…The first attempt for implementing fractionalorder Butterworth filters of order (n + α) > 2 was reported in [5], [6], where the (approximated) transfer function of order (1 + α) Butterworth low-pass filter is divided by an (n − 1) order Butterworth polynomial, in order to realize an (n + α) order filter. The work in [22] introduces a new technique to optimally design the fractional-order Butterworth low-pass filters in the complex F-plane using the constrained composite differential evolution (C 2 oDE) algorithm. The presented 1.5 order fractional low-pass filter realization is performed by substituting the capacitors in the well-known Sallen-Key circuit by fractional-order ones, approximated by the Valsa network [28].…”

“…More specifically, while the slope of the pass-band and stop-band gradients are fully determined by the order of the filter, being equal to ±20(1 + α) dB/dec., the cutoff frequency is not controlled without the application of an appropriate optimization algorithm. Significant research effort has been performed to overcome this obstacle, where appropriate cost functions, error minimization, metaheuristic, and genetic algorithms have been utilized [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. The design of fractional-order filters with orders greater than two has been introduced in [5], [6], [7], [22], [25], [26], [27], where the aforementioned algorithms have been employed.…”

“…Significant research effort has been performed to overcome this obstacle, where appropriate cost functions, error minimization, metaheuristic, and genetic algorithms have been utilized [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. The design of fractional-order filters with orders greater than two has been introduced in [5], [6], [7], [22], [25], [26], [27], where the aforementioned algorithms have been employed. The corresponding realizations are actually active-RC implementations of the derived rational integer-order functions and, therefore, they do not offer fully adjustable frequency characteristics of the filter structures.…”

Optimal design and analysis of fractional-order linear phase filter

“…At the beginning of this century, some scholars introduced the theory of fractional-order calculus into the field of signal processing and proposed the concept of fractional-order analog filter, and since then academic research on the fractional-order analog filter has been deepened. So far, most of the academic studies on fractional-order filters have focused on fractional-order Butterworth filters and fractional-order Chebyshev filters, mainly discussing the amplitudefrequency characteristics of fractional-order filters [7]- [11]. In recent years, several papers on fractional-order Bessel filters have discussed the phase-frequency characteristics of fractional-order filters [12]- [14], but the design method is to use the optimization algorithm to fit the amplitude-frequency characteristic curve of fractional-order filters to the amplitude-frequency characteristic curve of integer-order Bessel filters.…”