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Complexity Reduced Design Procedure of a Fractional Order All-Pass Filter

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Abstract

In this work, the design of continuous time Fractional Order All-Pass Filter (FOAPF) is proposed. Two different design methods to realize the All-Pass Filter in the fractional domain with mathematical formulations and circuit results are investigated. The first design method is based on Fractional Order Elements (FOEs) where, FOEs of order α and β are utilized to develop a second order All-Pass Filter to the fractional domain. Whereas, in second design method, the fractional filter is approximated into higher order integer filters. The frequency response of the proposed design are validated using MATLAB (2018a) and conjointly with PSPICE (OrCAD 17.2). For first circuit design, FOEs are obtained using a single R–C parallel network. However, for the second circuit design Signal Flow Graph (SFG) approach is utilized. The evaluation of the realized FOAPF is also performed through the Lissajous pattern, AC analysis and Transient Analysis. After the simulations, the achieved results show that the second order FOAPF provides almost 180° phase shift for different values of α, β.

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References

  1. Podlubny, I. (1999). Fractional differential equations, mathematics in science and engineering. San Diego: Academic Press.

    MATH  Google Scholar 

  2. Herrmann, R. (2011). Fractional calculus: An introduction for physicists. Singapore: World Scientific.

    Book  MATH  Google Scholar 

  3. Ortigueira, M. D. (2011). Fractional calculus for scientists and engineers. Berlin: Springer.

    Book  MATH  Google Scholar 

  4. Kulish, V. V., & Lage, J. L. (2002). Application of fractional calculus to fluid mechanics. Journal of Fluids Engineering, 124(3), 803–806.

    Article  Google Scholar 

  5. Magin, R. (2004). Fractional calculus in bioengineering. Part1 Critical Reviews in Biomedical Engineering, 32(1), 1–104.

    Article  Google Scholar 

  6. Vosika, Z. B., Lazovic, G. M., Misevic, G. N., & Simic-Krstic, J. B. (2013). Fractional calculus model of electrical impedance applied to human skin. PLoS ONE, 8(4), e59483.

    Article  Google Scholar 

  7. Lazo, M. J. (2011). Gauge invariant fractional electromagnetic fields. Physics Letters A, 375(41), 3541–3546.

    Article  MathSciNet  MATH  Google Scholar 

  8. Engheta, N. (1997). On the role of fractional calculus in electromagnetic theory. Departmental Papers (ESE): 2.

  9. Bia, P., Mescia, L., & Caratelli, D. (2016) Fractional calculus-based modeling of electromagnetic field propagation in arbitrary biological tissue. Mathematical Problems in Engineering 2016.

  10. Chen, Y.Q., Xue, D., & Dou, H. (2004) Fractional calculus and biomimetic control. In 2004 IEEE International Conference on Robotics and Biomimetics, pp. 901–906. IEEE.

  11. Maundy, B., Elwakil, A., & Gift, S. (2010). On a multivibrator that employs a fractional capacitor. Analog Integrated Circuits and Signal Processing, 62(1), 99.

    Article  Google Scholar 

  12. Radwan, A. G., Elwakil, A. S., & Soliman, A. M. (2008). Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Transactions on Circuits and Systems I: Regular Papers, 55(7), 2051–2063.

    Article  MathSciNet  MATH  Google Scholar 

  13. Radwan, A. G., Soliman, A. M., & Elwakil, A. S. (2008). First-order filters generalized to the fractional domain. Journal of Circuits, Systems, and Computers, 17(01), 55–66.

    Article  Google Scholar 

  14. Radwan, A. G., Elwakil, A. S., & Soliman, A. M. (2009). On the generalization of second-order filters to the fractional-order domain. Journal of Circuits, Systems, and Computers, 18(02), 361–386.

    Article  Google Scholar 

  15. Ali, A. S., Radwan, A. G., & Soliman, A. M. (2013). Fractional order Butterworth filter: active and passive realizations. IEEE Journal on emerging and selected topics in circuits and systems, 3(3), 346–354.

    Article  Google Scholar 

  16. Kaur, G., Ansari, A. Q., & Hashmi, M. S. (2017). Fractional order multifunction filter with 3 degrees of freedom. AEU-International Journal of Electronics and Communications, 82, 127–135.

    Google Scholar 

  17. Said, L. A., Ismail, S. M., Radwan, A. G., Madian, A. H., El-Yazeed, M. F. A., & Soliman, A. M. (2016). On the optimization of fractional order low-pass filters. Circuits, Systems, and Signal Processing, 35(6), 2017–2039.

    Article  MathSciNet  Google Scholar 

  18. Oldham, K. B., & Zoski, C. G. (1983). Analogue instrumentation for processing polarographic data. Journal of electroanalytical chemistry and interfacial electrochemistry, 157(1), 27–51.

    Article  Google Scholar 

  19. Mondal, D., & Biswas, K. (2013). Packaging of single-component fractional order element. IEEE Transactions on Device and Materials Reliability, 13(1), 73–80.

    Article  Google Scholar 

  20. Adhikary, A., Khanra, M., Sen, S., & Biswas, K. (2015). Realization of a carbon nanotube based electrochemical fractor. In 2015 IEEE international symposium on circuits and systems (ISCAS), pp. 2329–2332. IEEE.

  21. Carlson, G., & Halijak, C. (1964). Approximation of fractional capacitors (1/s)^(1/n) by a regular Newton process. IEEE Transactions on Circuit Theory, 11(2), 210–213.

    Article  Google Scholar 

  22. Steiglitz, K. (1964). An RC impedance approximation to s-1/2. IEEE Transactions on Circuits Systeme., 11, 160–161.

    Google Scholar 

  23. Biswas, K., Sen, S., & Dutta, P. K. (2006). Realization of a constant phase element and its performance study in a differentiator circuit. IEEE Transactions on Circuits and Systems II: Express Briefs, 53(9), 802–806.

    Article  Google Scholar 

  24. Krishna, B.T., & Reddy, K.V.V.S. (2008). Active and passive realization of fractance device of order 1/2. Active and passive electronic components.

  25. Nakagawa, M., & Sorimachi, K. (1992). Basic characteristics of a fractance device. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 75(12), 1814–1819.

    Google Scholar 

  26. Sierociuk, D., Podlubny, I., & Petras, I. (2013). Experimental evidence of variable-order behaviour of ladders and nested ladders. IEEE Transactions on Control Systems Technology, 21(2), 459–466.

    Article  Google Scholar 

  27. Sugi, M., Hirano, Y., Miura, Y. F., & Saito, K. (2002). Frequency behaviour of self-similar ladder circuits. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 198, 683–688.

    Article  Google Scholar 

  28. Mishra, S. K., Gupta, M., & Upadhyay, D. K. (2019). Design and implementation of DDCC-based fractional-order oscillator. International Journal of Electronics, 106(4), 581–598.

    Article  Google Scholar 

  29. Langhammer, L., Dvorak, J., Jerabek, J., Koton, J., & Sotner, R. (2018). Fractional-order low-pass filter with electronic tunability of its order and pole frequency. Journal of Electrical Engineering, 69(1), 3–13.

    Article  Google Scholar 

  30. Tripathy, M. C., Biswas, K., & Sen, S. (2013). A design example of a fractional-order Kerwin–Huelsman–Newcomb biquad filter with two fractional capacitors of different order. Circuits, Systems, and Signal Processing, 32(4), 1523–1536.

    Article  MathSciNet  Google Scholar 

  31. Soltan, A., Radwan, A. G., & Soliman, A. M. (2014). CCII based fractional filters of different orders. Journal of advanced research, 5(2), 157–164.

    Article  Google Scholar 

  32. Verma, R., Pandey, N., & Pandey, R. (2017). Electronically tunable fractional order all pass filter. In IOP Conference Series: Materials Science and Engineering, vol. 225, no. 1, p. 012229. IOP Publishing.

  33. Tsirimokou, G., Psychalinos, C., & Elwakil, A. S. (2017). Fractional-order electronically controlled generalized filters. International Journal of Circuit Theory and Applications, 45(5), 595–612.

    Article  Google Scholar 

  34. Maundy, B., Elwakil, A. S., & Freeborn, T. J. (2011). On the practical realization of higher-order filters with fractional stepping. Signal Processing, 91(3), 484–491.

    Article  MATH  Google Scholar 

  35. Tsirimokou, G., Koumousi, S., & Psychalinos, C. (2016). Design of fractional-order filters using current feedback operational amplifiers. Journal of Engineering Science and Technology Review, 9(4), 71–81.

    Article  Google Scholar 

  36. Kubanek, D., & Freeborn, T. (2018). (1+ α) fractional-order transfer functions to approximate low-pass magnitude responses with arbitrary quality factor. AEU-International Journal of Electronics and Communications, 83, 570–578.

    Google Scholar 

  37. Langhammer, L., Sotner, R., Dvorak, J., Domansky, O., Jerabek, J., & Uher, J. (2017). A 1+ α low-pass fractional-order frequency filter with adjustable parameters. In 2017 40th International Conference on Telecommunications and Signal Processing (TSP), pp. 724–729. IEEE.

  38. Freeborn, T., Maundy, B., & Elwakil, A.S. (2015). Approximated fractional order Chebyshev lowpass filters. Mathematical Problems in Engineering 2015.

  39. Freeborn, T. J., Elwakil, A. S., & Maundy, B. (2016). Approximated fractional-order inverse Chebyshev lowpass filters. Circuits, Systems, and Signal Processing, 35(6), 1973–1982.

    Article  MathSciNet  MATH  Google Scholar 

  40. Jerabek, J., Sotner, R., Dvorak, J., Langhammer, L., & Koton, J. (2016). Fractional-order high-pass filter with electronically adjustable parameters." In 2016 International Conference on Applied Electronics (AE), pp. 111–116. IEEE.

  41. Khateb, F., Kubánek, D., Tsirimokou, G., & Psychalinos, C. (2016). Fractional-order filters based on low-voltage DDCCs. Microelectronics Journal, 50, 50–59.

    Article  Google Scholar 

  42. Koton, J., Kubanek, D., Sladok, O., Vrba, K., Shadrin, A., & Ushakov, P. (2017). Fractional-order low-and high-pass filters using UVCs. Journal of Circuits, Systems and Computers, 26(12), 1750192.

    Article  Google Scholar 

  43. Ghoneim, M., Hesham, R., Yassin, H., & Madian, A. (2021). α-order universal filter realization based on single input multi-output differential voltage current conveyor. Analog Integrated Circuits and Signal Processing, 107(2), 411–422.

    Article  Google Scholar 

  44. Kaur, G., Ansari, A. Q., & Hashmi, M. S. (2020). Analysis and investigation of CDBA based fractional-order filters. Analog Integrated Circuits and Signal Processing, 105(1), 111–124.

    Article  Google Scholar 

  45. Kaur, G., Ansari, A.Q., & Hashmi, M.S. (2022). Analysis and realization of fractional step filters of order (1+ α). In Fractional-Order Design, pp. 337–372. Academic Press.

  46. Qiu, X., Feng, H., & Bo, Hu. (2021). Fractional order graph filters: Design and implementation. Electronics, 10(4), 437.

    Article  Google Scholar 

  47. Varshney, G., Pandey, N., & Pandey, R. (2021). Generalization of shadow filters in fractional domain. International Journal of Circuit Theory and Applications, 49(10), 3248–3265.

    Article  Google Scholar 

  48. Hamed, E. M., Said, L. A., Madian, A. H., & Radwan, A. G. (2020). On the approximations of CFOA-based fractional-order inverse filters. Circuits, Systems, and Signal Processing, 39(1), 2–29.

    Article  Google Scholar 

  49. Kaur, G., Ansari, A. Q., & Hashmi, M. S. (2019). "Fractional order high pass filter based on operational transresistance amplifier with three fractional capacitors of different order. Advances in Electrical and Electronic Engineering, 17(2), 155–166.

    Article  Google Scholar 

  50. Radwan, A. G., Soliman, A. M., Elwakil, A. S., & Sedeek, A. (2009). On the stability of linearsystems with fractional-order elements. Chaos, Solitons & Fractals, 40(5), 2317–2328.

    Article  MATH  Google Scholar 

  51. Radwan, A. G. (2012). Stability analysis of the fractional-order RLβCα circuit. Journal of Fractional Calculus and Applications, 3(1), 1–15.

    Google Scholar 

  52. Ozoguz, S., Toker, A., & Acar, C. (1999). Current-mode continuous-time fully-integrated universal filter using CDBAs. Electronics Letters, 35(2), 97–98.

    Article  Google Scholar 

  53. Yang, X.-J., Lopes, A. M., Hristov, J. Y., Cattani, C., Baleanu, D., & Mohyud-Din, S. T. (2016). Special issue on advances in fractional dynamics in mechanical engineering. Advances in Mechanical Engineering, 8(6), 1687814016654094.

    Google Scholar 

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Funding

This work was supported by the Collaborative Research Grant (CRP) Number 021220CRP0222 at Nazarbayev University, Kazakhstan.

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Authors and Affiliations

  1. Department of Electrical Engineering, Jamia Millia Islamia, Delhi, India

    Gagandeep Kaur & A. Q. Ansari

  2. School of Engineering and Digital Sciences, Nazarbayev University, Nur Sultan, Kazakhstan

    M. S. Hashmi

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  1. Gagandeep Kaur
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Correspondence to Gagandeep Kaur.

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Kaur, G., Ansari, A.Q. & Hashmi, M.S. Complexity Reduced Design Procedure of a Fractional Order All-Pass Filter. Wireless Pers Commun 125, 2515–2535 (2022). https://doi.org/10.1007/s11277-022-09672-4

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