On the Design of Power Law Filters and Their Inverse Counterparts

Abstract

This paper presents the optimal modeling of Power Law Filters (PLFs) with the low-pass (LP), high-pass (HP), band-pass (BP), and band-stop (BS) responses by means of rational approximants. The optimization is performed for three different objective functions and second-order filter mother functions. The formulated design constraints help avoid placement of the zeros and poles on the right-half s-plane, thus, yielding stable PLF and inverse PLF (IPLF) models. The performances of the approximants exhibiting the fractional-step magnitude and phase responses are evaluated using various statistical indices. At the cost of higher computational complexity, the proposed approach achieved improved accuracy with guaranteed stability when compared to the published literature. The four types of optimal PLFs and IPLFs with an exponent α of 0.5 are implemented using the follow-the-leader feedback topology employing AD844AN current feedback operational amplifiers. The experimental results demonstrate that the Total Harmonic Distortion achieved for all the practical PLF and IPLF circuits was equal or lower than 0.21%, whereas the Spurious-Free Dynamic Range also exceeded 57.23 and 54.72 dBc, respectively.

Electronically Controlled Power-Law Filters Realizations

“…Thus, the fitfrd function approximation provides the smallest deviation from the ideal responses and, therefore, it will be applied in the next section for the implementation of the power-law filters. It must be mentioned at this point that optimization techniques [13] can distinctly outperform the Sanathanan-Koerner (SK) method based solution based on the magnitude and phase error metrics.…”

“…This has been followed in [7], where the Sanathanan-Koerner (SK) least-square iterative method has been employed for performing the curve-fitting approximation of both magnitude and phase frequency responses and the derived integer-order rational transfer functions have been realized by an operational amplifier (op-amp)-based Follow-the-Leader Feedback (FLF) structure. In [13], an optimization of power-law filter functions is performed for three different objective functions and 2nd-order mother filter functions. The presented implementation is based on the employment of Current-Feedback Operational Amplifiers (CFOAs) as active elements accompanied by resistors and capacitors.…”

Versatile Field-Programmable Analog Array Realizations of Power-Law Filters

“…Non-integer order signal processing has received significant research interest in the following fields [1][2][3][4][5][6]. The first field is electrical engineering, for implementing filters and oscillators [5,[7][8][9][10][11][12][13][14][15], chaotic systems [16], sensor systems [17], and control systems [2,[18][19][20][21]. This originates from the fact that both filters and oscillators offer additional degrees of freedom due to the non-integer order, which opens the door for scaling the characteristic frequencies of the filters/oscillators, as well as for precisely controlling the gradient of the transition from the pass-band to the stop-band.…”

“…In [22], voltage current conveyors (VCII) were utilized as active elements for implementing power-law filters for acoustic applications. In [12], an optimization of power-law filter functions was performed, and the presented implementation was performed using Current-Feedback Operational Amplifiers (CFOAs) as active elements The aforementioned configurations are simple solutions, perfectly working for cases of filters with pre-defined type and frequency characteristics, as no programmability or tuning of the resistors' and/or capacitors' values is provided.…”

Plant Tissue Modelling Using Power-Law Filters

“…The normalized impedance is expressed as: where , thus, according to the previous expression, the Cole model could be visualized as a fractional-order low pass filter with gain k. This filter can incorporate the power-law concept by raising the normalized impedance to a fractional-order power. It is worth noting that some approximations for modelling power-law filters were proposed in [ 35 , 36 ] to provide improved accuracy and stability but with higher computational complexity. The use of power-law filters in bio-impedance modelling has two advantages [ 34 ], namely (i) flexibility in selecting the number of parameters in the model estimated by the optimization algorithms and (ii) isolation of the mother function from the fractional-order dispersion coefficients in the model.…”