Abstract
The formula of the all-pole low-pass frequency filter transfer function of the fractional order (N + α) designated for implementation by non-cascade multiple-feedback analogue structures is presented. The aim is to determine the coefficients of this transfer function and its possible variants depending on the filter order and the distribution of the fractional-order terms in the transfer function. Optimization algorithm is used to approximate the target Butterworth low-pass magnitude response, whereas the approximation errors are evaluated. The interpolated equations for computing the transfer function coefficients are provided. An example of the transformation of the fractional-order low-pass to the high-pass filter is also presented. The results are verified by simulation of multiple-feedback filter with operational transconductance amplifiers and fractional-order element.
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K. Biswas, S. Sen, P.K. Dutta, A constant phase element sensor for monitoring microbial growxth. Sens. Actuators B Chem. 119, No 1 (2006), 186–191; DOI: 10.1016/j.snb.2005.12.011.
K. Biswas, G. Bohannan, R. Caponetto, A.M. Lopes, J.A.T. Machado, Fractional-Order Devices. Springer, Berlin/Heidelberg, Germany (2017); DOI: 10.1007/978-3-319-54460-1.
A.S. Elwakil, Fractional-order circuits and systems: An emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 10, No 4 (2010), 40–50; DOI: 10.1109/MCAS.2010.938637.
T.J. Freeborn, B. Maundy, A.S. Elwakil, Field programmable analogue array implementation of fractional step filters. IET Circuits Devices and Syst. 4, No 6 (2010), 514–524; DOI: 10.1049/iet-cds.2010.0141.
T. Freeborn, B. Maundy, A.S. Elwakil, Approximated fractional order Chebyshev lowpass filters. Math. Probl. Eng. 2015 (2015), 1–7; DOI: 10.1155/2015/832468.
T.J. Freeborn, Comparison of (1+α) fractional-order transfer functions to approximate low pass Butterworth magnitude responses. Circuits, Syst. Signal Process. 35 (2016), 1983–2002; DOI: 10.1007/s00034-015-0226-y.
T.J. Freeborn, A.S. Elwakil, B. Maundy, Approximated fractional-order inverse Chebyshev lowpass filters. Circuits, Syst. Signal Process. 35 (2016), 1973–1982; DOI: 10.1007/s00034-015-0222-2.
T.J. Freeborn, D. Kubanek, J. Koton, J. Dvorak, Validation of fractional-order lowpass elliptic responses of (1 + α)-order analog filters. Appl. Sci. 8, No 12 (2018), 1–17; DOI: 10.3390/app8122603.
D. Gupta, C.A. Lammersfeld, P.G. Vashi, J. King, S.L. Dahlk, J.F. Grutsch, C.G. Lis, Bioelectrical impedance phase angle as a prognostic indicator in breast cancer. BMC Cancer 8, No 249 (2008); DOI: 10.1186/1471-2407-8-249.
E.M. Hamed, A.M. AbdelAty, L.A. Said, A.G. Radwan, Effect of different approximation techniques on fractional-order KHN filter design. Circuits, Syst. Signal Process. 37 (2018), 5222–5252; DOI: 10.1007/s00034-018-0833-5.
T. Helie, Simulation of fractional-order low-pass filters. IEEE/ACM Trans. Audio, Speech, Language Process. 22, No 11 (2014), 1636–1647; DOI: 10.1109/TASLP.2014.2323715.
A. Kartci, A. Agambayev, M. Farhat, N. Herencsar, L. Brancik, H. Bagci, K.N. Salama, Synthesis and optimization of fractional-order elements using a genetic algorithm. IEEE Access 7 (2019), 80233–80246; DOI: 10.1109/ACCESS.2019.2923166.
D. Kubanek, T. Freeborn, (1 + α) fractional-order transfer functions to approximate low-pass magnitude responses with arbitrary quality factor. Int. J. Electron. Commun. (AEU) 83 (2018), 570–578; DOI: 10.1016/j.aeue.2017.04.031.
D. Kubanek, T. Freeborn, J. Koton, Fractional-order band-pass filter design using fractional-characteristic specimen functions. Microelectron. J. 86 (2019), 77–86; DOI: 10.1016/j.mejo.2019.02.020.
Linear Technology. LT1228 100 MHz Current Feedback Amplifier with DC Gain Control. Datasheet (2012).
S. Mahata, S.K. Saha, R. Kar, D. Mandal, Optimal design of fractional order low pass Butterworth filter with accurate magnitude response. Digit. Signal Process. 72, No C (2018), 96–114; DOI: 10.1016/j.dsp.2017.10.001.
S. Mahata, R. Kar, D. Mandal, Optimal fractional-order highpass Butterworth magnitude characteristics realization using current-mode filter. Int. J. Electron. Commun. (AEU) 102 (2019), 78–89; DOI: 10.1016/j.aeue.2019.02.014.
S. Mahata, S. Saha, R. Kar, D. Mandal, Optimal integer-order rational approximation of α and α + β fractional-order generalised analogue filters. IET Signal Process. 13, No 5 (2019), 516–27; DOI: 10.1049/iet-spr.2018.5340.
S. Mahata, S. Banerjee, R. Kar, D. Mandal, Revisiting the use of squared magnitude function for the optimal approximation of (1 + α)-order Butterworth filter. Int. J. Electron. Commun. (AEU) 110 (2019), 1–11; DOI: 10.1016/j.aeue.2019.152826.
B. Maundy, A.S. Elwakil, T.J. Freeborn, On the practical realization of higher-order filters with fractional stepping. Signal Process. 91, No 3 (2011), 484–491; DOI: 10.1016/j.sigpro.2010.06.018.
D.J. Perry, New multiple feedback active RC network. Electron. Lett. 11, No 16 (1975), 364–365; DOI: 10.1049/el:19750278.
A.G. Radwan, A.M. Soliman, A.S. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements. Chaos Solitons Fractals 40, No 5 (2009), 2317–28; DOI: 10.1016/j.chaos.2007.10.033.
Z.M. Shah, M.Y. Kathjoo, F.A. Khanday, K. Biswas, C. Psychalinos, A survey of single and multi-component Fractional-Order Elements (FOEs) and their applications. Microelectron. J. 84 (2019), 9–25; DOI: 10.1016/j.mejo.2018.12.010.
H.G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y.Q. Chen, A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64 (2018), 213–31; DOI: 10.1016/j.cnsns.2018.04.01.
A. Tepljakov, Fractional-order Modeling and Control of Dynamic Systems. Springer International Publishing, Berlin/Heidelberg, Germany (2017); DOI: 10.1007/978-3-319-52950-9.
G. Tsirimokou, C. Psychalinos, A.S. Elwakil, Design of CMOS Analog Integrated Fractional-Order Circuits: Applications in Medicine and Biology. Springer, Berlin/Heidelberg, Germany (2017); DOI: 10.1007/978-3-319-55633-8.
J. Valsa, J. Vlach, RC models of a constant phase element. Int. J. Circuit Theory Appl. 41, No 1 (2013), 59–67; DOI: 10.1002/cta.785.
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Kubanek, D., Koton, J., Jerabek, J. et al. (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response. Fract Calc Appl Anal 24, 689–714 (2021). https://doi.org/10.1515/fca-2021-0030
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DOI: https://doi.org/10.1515/fca-2021-0030