Optimal analog‐to‐digital transformation of fractional‐order Butterworth filter using binomial series expansion with Al‐Alaoui operator

Abstract

SummaryThis paper deals with the optimal analog‐to‐digital transformation of fractional‐order Butterworth filter (FOBF) in terms of infinite impulse response templates. The fractional‐order transfer function of the analog FOBF is transformed into its digital counterpart by employing the Binomial series expansion of different truncation orders, based on the Al‐Alaoui operator. This nonoptimal solution is then treated as an initial point for a local search optimizer such as the Nelder–Mead simplex (NMS) algorithm and also injected as a super‐fit individual in the initial population of a global search constrained evolutionary optimization algorithm (CEOA). Design stability and minimum‐phase response constraints are formulated for the super‐fit scheme. Both the techniques demonstrate good modeling performance; however, the super‐fit CEOA can markedly outperform the NMS method as the problem dimensionality increases.

A Review of Recent Advances in Fractional-Order Sensing and Filtering Techniques

“…The popular analog fractional-order Butterworth filter has multiple digital implementations proposed by [ 68 , 69 , 70 , 71 , 72 ]. Analog-to-digital transformations of a fractional-order Butterworth filter are used in [ 69 ] using the infinite impulse response and Al-Alaoui operator, followed by global search constrained evolutionary algorithms to determine the parameters of the filter. Infinite impulse response is also used in [ 70 ] to design direct digital fractional-order Butterworth filters through optimization routines.…”

Numerical study for the fractional RL, RC, and RLC electrical circuits using Legendre pseudo‐spectral method

“…The importance and the very wide range of applications of the fractional models (described by the fractional derivatives) directed mathematicians and physicians to study the numerical and approximate solutions for fractional differential equations (FDEs) using many approximate techniques. We have a lot of examples for the applications of such kind of equations in our life as in fluid mechanics, [1][2][3] image processing, 4 biology, [5][6][7][8][9][10][11][12] engineering, [13][14][15] physics, [16][17][18][19][20] electrical circuits and filters, [21][22][23][24][25][26][27][28][29][30][31][32] and others. [33][34][35][36] Definition 1.…”