Circuits, Systems, and Signal Processing

, Volume 37, Issue 3, pp 915–938

# On the Analysis and Design of Fractional-Order Chebyshev Complex Filter

• Amr M. AbdelAty
• Ahmed Soltan
• Waleed A. Ahmed
Article

## Abstract

This paper introduces the concept of fractional-order complex Chebyshev filter. A fractional variation of Chebyshev differential equations is introduced based on Caputo fractional operator. The proposed equation is solved using fractional Taylor power series method. The condition for fractional polynomial solutions is obtained and the first four polynomials scaled using an appropriate scaling factor. The fractional-order complex Chebyshev low-pass filter based on the obtained fractional polynomials is developed. Two methods for obtaining the transfer functions of the complex filter are discussed. Circuit implementations are simulated using Advanced Design System (ADS) and compared with MATLAB numerical simulation of the obtained transfer functions to prove the validity of the two approaches.

## Keywords

Fractional differential equation Series solution Chebyshev polynomials Chebyshev filter Fractional-order filter Complex filter

## References

1. 1.
A. Acharya, S. Das, I. Pan, S. Das, Extending the concept of analog Butterworth filter for fractional order systems. Sig. Process. 94(1), 409–420 (2014)
2. 2.
A. Adhikary, S. Sen, K. Biswas, Practical realization of tunable fractional order parallel resonator and fractional order filters. IEEE Trans. Circuits Syst. I Regul. Pap. 63(8), 1142–1151 (2016)
3. 3.
M.A. Al-Bassam, On the existence of series solution of differential equations of generalized order. Port. Math. 29(1), 5–11 (1970)
4. 4.
A.S. Ali, A.G. Radwan, A.M. Soliman, Fractional order butterworth filter: active and passive realizations. IEEE J. Emerg. Sel. Top. Circuits Syst. 3(3), 346–354 (2013)
5. 5.
P.C. Biswal, Ordinary Differential Equations, 2nd edn. (PHI Learning Pvt. Ltd, Delhi, 2012)
6. 6.
J.P. Boyd, Chebyshev and Fourier Spectral Methods (Dover Publications, 2000)Google Scholar
7. 7.
M.S. Chavan, R.A. Agarwala, M.D. Uplane, Comparative study of Chebyshev I and Chebyshev II filter used for noise reduction in ECG signal. Sig. Process. 2(1), 1–17 (2008)
8. 8.
Y. Chen, B.M. Vinagre, A new IIR-type digital fractional order differentiator. Sig. Process. 83(11), 2359–2365 (2003)
9. 9.
J. Crols, M. Steyaert, Transceivers in the frequency domain. in ed by C. Wirel, Transceiver Des, pp 29–70. Springer, Boston (2003)Google Scholar
10. 10.
E.C. de Oliveira, J.A. Tenreiro Machado, A review of definitions for fractional derivatives and integral. Math. Probl. Eng. 2014(1940), 1–6 (2014)
11. 11.
B.K. Dutta, L.K. Arora, Approximate solution of inhomogeneous fractional differential equation. Adv. Nonlinear Anal. 1(4), 335–353 (2012)
12. 12.
O.S. Fard, R. BeheshtAien, A note on fuzzy best approximation using Chebyshev’s polynomials. J. King Saud Univ. Sci. 23(2), 217–221 (2011)
13. 13.
M. Faryad, Q.A. Naqvi, Fractional rectangular waveguide. Prog. Electromagn. Res. 75, 383–396 (2007)
14. 14.
G. Fedele, A. Ferrise, Periodic disturbance rejection for fractional-order dynamical systems. Fract. Calc. Appl. Anal. 18(3), 603–620 (2015)
15. 15.
T. Freeborn, B. Maundy, A.S. Elwakil, Approximated fractional order Chebyshev lowpass filters. Math. Probl. Eng. 2015(4), 1–7 (2015)
16. 16.
T.J. Freeborn, Comparison of $$(1+\alpha )$$fractional-order transfer functions to approximate lowpass butterworth magnitude responses. Circuits Syst. Signal Process. 35(6), 1983–2002 (2015)
17. 17.
T.J. Freeborn, A.S. Elwakil, B. Maundy, Approximated fractional-order inverse Chebyshev lowpass filters. Circuits Syst. Signal Process. 35(6), 1973–1982 (2016)
18. 18.
T.J. Freeborn, B. Maundy, A. Elwakil, Towards the realization of fractional step filters, in ISCAS 2010 - 2010 IEEE Int. Symp. Circuits Syst. Nano-Bio Circuit Fabr. Syst., (3):1037–1040 (2010)Google Scholar
19. 19.
C.S. Goodrich, Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23(9), 1050–1055 (2010)
20. 20.
T.T. Hartley, C.F. Lorenzo, Dynamics and control of initialized fractional-order systems. Nonlinear Dyn. 29(1/4), 201–233 (2002)
21. 21.
I.S. Jesus, J.T. Machado, Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008)
22. 22.
A. Kilbas, M. Rivero, L. Rodríguez-Germá, J. Trujillo, $$\alpha$$-Analytic solutions of some linear fractional differential equations with variable coefficients. Appl. Math. Comput. 187(1), 239–249 (2007)
23. 23.
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, vol. 204 (Elsevier, Amsterdam, 2006)
24. 24.
P. Kiss, V. Prodanov, J. Glas, Complex low-pass filters. Analog Integr. Circuits Signal Process. 35(1), 9–23 (2003)
25. 25.
Y. Ku, M. Drubin, Network synthesis using legendre and hermite polynomials. J. Franklin Inst. 273(2), 138–157 (1962)
26. 26.
C. Laoudias, C. Psychalinos, 1.5-V complex filters using current mirrors. IEEE Trans. Circuits Syst. II Express Briefs 58(9), 575–579 (2011)
27. 27.
C. Laoudias, C. Psychalinos, Complex Filters for Short Range Wireless Networks (Springer, New York, 2012)
28. 28.
E. Lenzi, M. dos Santos, M. Lenzi, D. Vieira, L. da Silva, Solutions for a fractional diffusion equation: anomalous diffusion and adsorptiondesorption processes. J. King Saud Univ. Sci. 28(1), 8–11 (2015)Google Scholar
29. 29.
H. Li, Y. Luo, Y.Q. Chen, A fractional order proportional and derivative (FOPD) motion controller: tuning rule and experiments. IEEE Trans. Control Syst. Technol. 18(2), 516–520 (2010)
30. 30.
J.S. Lim, D.C. Park, A modified chebyshev bandpass filter with attenuation poles in the stopband. IEEE Trans. Microw. Theory Tech. 45(6), 898–904 (1997)
31. 31.
R.L. Magin, Fractional calculus in bioengineering (Begell House, Connecticut, 2006)Google Scholar
32. 32.
S.A. Mahmoud, A.H. Madian, A.M. Soliman, Low-voltage CMOS current feedback operational amplifier and its application. ETRI J. 29(2), 212–218 (2007)
33. 33.
K. Moaddy, A.G. Radwan, K.N. Salama, S. Momani, I. Hashim, The fractional-order modeling and synchronization of electrically coupled neuron systems. Comput. Math. Appl. 64(10), 3329–3339 (2012)
34. 34.
C. Muto, A new extended frequency transformation for complex analog filter design. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E83–A(6), 934–940 (2000)Google Scholar
35. 35.
W. Okrasiński, Ł. Płociniczak, A note on fractional Bessel equation and its asymptotics. Fract. Calc. Appl. Anal. 16(3), 559–572 (2013)
36. 36.
K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Elsevier, Amsterdam, 1974)
37. 37.
L.D. Paarmann, Design and Analysis of Analog Filters: A Signal Processing Perspective (Springer, Berlin, 2001)Google Scholar
38. 38.
I. Petráš, Tuning and implementation methods for fractional-order controllers. Fract. Calc. Appl. Anal. 15(2), 282–303 (2012). doi:
39. 39.
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications (Academic press, Cambridge, 1998)
40. 40.
A. Radwan, A. Soliman, A. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements. Chaos Solitons Fract. 40(5), 2317–2328 (2009)
41. 41.
A.G. Radwan, Resonance and quality factor of the $$RL_{\alpha }C_{\alpha }$$ fractional circuit. IEEE J. Emerg. Sel. Top. Circuits Syst. 3(3), 377–385 (2013)
42. 42.
A.G. Radwan, A.S. Elwakil, A.M. Soliman, Fractional-order sinusoidal oscillators Design procedure and practical examples. IEEE Trans. Circuits Syst. I Regul. Pap. 55(7), 2051–2063 (2008)
43. 43.
A.G. Radwan, A.S. Elwakil, A.M. Soliman, On the generalization of second-order filters to the fractional-order domain. J. Circuits Syst. Comput. 18(02), 361–386 (2009)
44. 44.
A.G. Radwan, A.M. Soliman, A.S. Elwakil, First order filter generalized to the fractional domain. J. Circuits Syst. Comput. 17(01), 55–66 (2008)
45. 45.
J.F. Ritt, A factorization theory for functions $$\sum _{i=1}^na_ie^{\alpha _ix}$$. Trans. Am. Math. Soc. 29(3), 584–584 (1927)
46. 46.
D. Saha, D. Mondal, S. Sen, Effect of initialization on a class of fractional order systems: experimental verification and dependence on nature of past history and system parameters. Circuits Syst. Signal Process. 32(4), 1501–1522 (2012)
47. 47.
P. Samiotis, C. Psychalinos, Low-voltage complex filters using current feedback operational amplifiers. ISRN Electron. 1–7, 2013 (2013)Google Scholar
48. 48.
R. Schaumann, Design of Analog Filters (Oxford University Press, Oxford, 2001)Google Scholar
49. 49.
K. Shouno, Y. Ishibashi, Synthesis of a passive complex filter using transformers. IEEE Trans. Circuits Syst. I Regul. Pap. 55(7), 1897–1903 (2003)
50. 50.
D. Sierociuk, T. Skovranek, M. Macias, I. Podlubny, I. Petras, A. Dzielinski, P. Ziubinski, Diffusion process modeling by using fractional-order models. Appl. Math. Comput. 257, 2–11 (2015)Google Scholar
51. 51.
V.E. Tarasov, Discrete model of dislocations in fractional nonlocal elasticity. J. King Saud Univ. Sci. 28(1), 33–36 (2015)
52. 52.
C. Upathamkuekool, A. Jiraseree-amornkun, J. Mahattanakul, A low-voltage low-power complex active-RC filter employing single-stage opamp. in 2012 IEEE Int. Conf. Electron Devices Solid State Circuit, pp. 1–4. IEEE, Dec 2012Google Scholar
53. 53.
J. Valsa, P. Dvoák, M. Friedl, Network model of the CPE. Radioengineering 20(3), 619–626 (2011)Google Scholar
54. 54.
J. Vankka, Digital Synthesizers and Transmitters for Software Radio  (Springer,Verlag, 2005)Google Scholar
55. 55.
X. Zhang, X. Ni, M. Iwahashi, N. Kambayashi, Realization of universal active complex filter using CCIIs and CFCCIIs. Analog Integr. Circuits Signal Process. 20(2), 129–137 (1999)
56. 56.
X. Zhang, Y. Shinada, A second-order active bandpass filter with complex coefficients and its applications to the hilbert transform. Electron. Commun. Jpn. (Part III Fundam. Electron. Sci) 79(6), 13–22 (1996)

## Authors and Affiliations

• Amr M. AbdelAty
• 1
• Ahmed Soltan
• 2
• Waleed A. Ahmed
• 1