Fractional-order multi-phase oscillators design and analysis suitable for higher-order PSK applications

  • Mohammed E. Fouda
  • Ahmed Soltan
  • Ahmed G. Radwan
  • Ahmed M. Soliman


Recently, multi-phase oscillator design witnesses a lot of progress in communication especially phase shift keying based systems. Yet, there is a lack in design multi-phase oscillator with different fractional phase shifts. Thus, in this paper, a new technique to design and analyze a multi-phase oscillator is proposed. The proposed procedure is built based on the fractional-order elements or constant phase elements in order to generate equal or different phase shifts. The general characteristics equation for any oscillator is studied to derive expressions for the oscillation conditions and oscillation frequency. Also, stability analysis is introduced to guarantee the oscillation. Then, different examples of oscillators for equal and different phase shifts are introduced with their simulations.


Oscillator Multi-phase Fractional PSK Stability Oscillator analysis 


  1. 1.
    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 Part 1 Regular Papers, 53(9), 802.CrossRefGoogle Scholar
  2. 2.
    Biswas, K., Thomas, L., Chowdhury, S., Adhikari, B., & Sen, S. (2008). Impedance behaviour of a microporous pmma-film coated constant phase element based chemical sensor. International Journal of Smart Sensing and Intelligent Systems, 1(4), 922–939.Google Scholar
  3. 3.
    Caponetto, R. (2010). Fractional order systems: Modeling and control applications (72nd ed.). Singapore: World Scientific.Google Scholar
  4. 4.
    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.CrossRefGoogle Scholar
  5. 5.
    Diethelm, K., Ford, N. J., & Freed, A. D. (2002). A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1–4), 3–22.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Elshurafa, A. M., Almadhoun, M. N., Salama, K., & Alshareef, H. (2013). Microscale electrostatic fractional capacitors using reduced graphene oxide percolated polymer composites. Applied Physics Letters, 102(23), 232901.CrossRefGoogle Scholar
  7. 7.
    Elwakil, A. S. (2010). Fractional-order circuits and systems: An emerging interdisciplinary research area. IEEE Circuits and Systems Magazine, 10(4), 40–50.CrossRefGoogle Scholar
  8. 8.
    Fouda, M., Soltan, A., Radwan, A., & Soliman, A. (2014). Multi-phase oscillator for higher-order psk applications. In 21st IEEE international conference on electronics, circuits and systems (ICECS) (pp. 494–497).Google Scholar
  9. 9.
    Haba, T. C., Ablart, G., Camps, T., & Olivie, F. (2005). Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon. Chaos, Solitons & Fractals, 24(2), 479–490.CrossRefGoogle Scholar
  10. 10.
    Haykins, S. (2010). Digital communication. New Delhi: Wiley.Google Scholar
  11. 11.
    Ibrahim, G., Hafez, A., & Khalil, A. (2013). An ultra low power qpsk receiver based on super-regenerative oscillator with a novel digital phase detection technique. AEU-International Journal of Electronics and Communications, 67(11), 967–974.CrossRefGoogle Scholar
  12. 12.
    Jesus, I. S., Machado, J., Cunha, J. B., Silva, M. F. (2006). Fractional order electrical impedance of fruits and vegetables. In Proceedings of the 25th IASTED international conference on modeling, indentification, and control (pp. 489–494). ACTA Press.Google Scholar
  13. 13.
    Krishna, B., Reddy, K. (2008). Active and passive realization of fractance device of order 1/2. Active and Passive electronic components.Google Scholar
  14. 14.
    Loescharataramdee, C., Kiranon, W., Sangpisit, W., Yadum, W. (2001). Multiphase sinusoidal oscillators using translinear current conveyors and only grounded passive components. In: Proceedings of the IEEE 33rd southeastern symposium on system theory (pp. 59–63).Google Scholar
  15. 15.
    Michio, S., Hirano, Y., Miura, Y. F., & Saito, K. (1999). Simulation of fractal immittance by analog circuits: an approach to the optimized circuits. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 82(8), 1627–1635.Google Scholar
  16. 16.
    Maundy, B., Elwakil, A., & Gift, S. (2012). On the realization of multiphase oscillators using fractional-order allpass filters. Circuits, Systems, and Signal Processing, 31(1), 3–17.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Moaddy, K., Radwan, A. G., Salama, K. N., Momani, S., & Hashim, I. (2012). The fractional-order modeling and synchronization of electrically coupled neuron systems. Computers & Mathematics with Applications, 64(10), 3329–3339.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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
  19. 19.
    Ogata, K., & Yang, Y. (1970). Modern control engineering. Englewood Cliffs: Prentice-Hall.Google Scholar
  20. 20.
    O’hara, B., Petrick, A. (2005). IEEE 802.11 handbook: A designer’s companion. IEEE Standards Association.Google Scholar
  21. 21.
    Oustaloup, A., Melchior, P., Lanusse, P., Cois, O., Dancla, F. (2000). The crone toolbox for matlab. In: IEEE international symposium on computer-aided control system design. CACSD 2000, (pp. 190–195).Google Scholar
  22. 22.
    Radwan, A., Soliman, A., Elwakil, A., & Sedeek, A. (2009). On the stability of linear systems with fractional-order elements. Chaos, Solitons & Fractals, 40(5), 2317–2328.CrossRefzbMATHGoogle Scholar
  23. 23.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Radwan, A. G., Shamim, A., & Salama, K. N. (2011). Theory of fractional order elements based impedance matching networks. IEEE Microwave and Wireless Components Letters, 21(3), 120–122.CrossRefGoogle Scholar
  25. 25.
    Rankl, W., & Effing, W. (2010). Smart card handbook. New York: Wiley.CrossRefGoogle Scholar
  26. 26.
    Sabatier, J., Agrawal, O. P., & Machado, J. T. (2007). Advances in fractional calculus. New York: Springer.CrossRefzbMATHGoogle Scholar
  27. 27.
    Saito, K., & Michio, S. (1993). Simulation of power-law relaxations by analog circuits: Fractal distribution of relaxation times and non-integer exponents. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 76(2), 204–209.Google Scholar
  28. 28.
    Soliman, A. M. (2010). Transformation of oscillators using Op Amps, unity gain cells and CFOA. Analog Integrated Circuits and Signal Processing, 65(1), 105–114.CrossRefGoogle Scholar
  29. 29.
    Soltan, A., Radwan, A., & Soliman, A. (2013). CCII based KHN fractional order filter. In: IEEE 56th international midwest symposium on circuits and systems (MWSCAS), (pp. 197–200).Google Scholar
  30. 30.
    Soltan, A., Radwan, A., & Soliman, A. M. (2012). Fractional order filter with two fractional elements of dependant orders. Microelectronics Journal, 43, 818–827.CrossRefGoogle Scholar
  31. 31.
    Soltan, A., Radwan, A. G., & Soliman, A. M. (2013). CCII based fractional filters of different orders. Journal of Advanced Research, 5, 157–164.CrossRefGoogle Scholar
  32. 32.
    Soltan, A., Radwan, A. G., & Soliman, A. M. (2014). Fractional order sallen-key and khn filters: Stability and poles allocation. Circuits, Systems, and Signal Processing, 34, 1–20.Google Scholar
  33. 33.
    Soltan, A., Soliman, A. M., & Radwan, A. G. (2014). Analog circuit design in the fractional order domain. Saarbrucken: Lap Lambert Academic.Google Scholar
  34. 34.
    Valério, D., & da Costa, J. S. (2004). Ninteger: A non-integer control toolbox for matlab. In: 1st IFAC workshop on fractional differentiation and its applications, Bordeaux.Google Scholar
  35. 35.
    Valério, D., Trujillo, J. J., Rivero, M., Machado, J. T., & Baleanu, D. (2013). Fractional calculus: A survey of useful formulas. The European Physical Journal Special Topics, 222(8), 1827–1846.CrossRefGoogle Scholar
  36. 36.
    Vassis, D., Kormentzas, G., Rouskas, A., & Maglogiannis, I. (2005). The IEEE 802.11 g standard for high data rate wlans. IEEE Network, 19(3), 21–26.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Engineering Mathematics and Physics Department, Faculty of EngineeringCairo UniversityGizaEgypt
  2. 2.School of Electrical, Electronic and Computer EngineeringNewcastle UniversityNewcastle upon TyneUnited Kingdom
  3. 3.Nano-electronic Integrated Systems CenterNile UniversityGizaEgypt
  4. 4.Electronics and Communication Engineering DepartmentCairo UniversityGizaEgypt

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