Circuits, Systems, and Signal Processing

, Volume 37, Issue 5, pp 2001–2020 | Cite as

Chained-Function Filter Synthesis Based on the Legendre Polynomials

  • Nikola Stojanović
  • Negovan Stamenković
  • Ivan Krstić
Article
  • 52 Downloads

Abstract

A new kind of filter called Legendre chained-function (LCF) filter with characteristic function given by the product of low-degree Legendre orthogonal polynomials (seed functions) is studied in this paper. LCF filter magnitude response exhibits unequal ripple level in the passband compared to Chebyshev chained-function filter with identical edge ripple factor at the passband edge. A proper combination of seed functions, i.e., a product of them, is used to control the maximum ripple in the passband, which affects the return loss, selectivity and a group delay characteristics of a filter. By selecting one of the possible combinations of seed functions (thus obtaining various degrees of freedom in filter design), filters with improved performances compared to traditional approximation techniques can be obtained. The degrees of freedom increase if the degree of filter increases. Compared to existing Chebyshev chained-function (CCF) filters, whose performances are also presented, and Butterworth function filters as a special case of both LCF and CCF filters, the new family of LCF filters has many advantages. A table summarizing properties of CCF and LCF filters is given for design purpose.

Keywords

Chained-function filter Legendre polynomial Chebyshev polynomial Magnitude response Group delay characteristic Monte Carlo simulation 

Notes

Acknowledgements

The authors wish to thank Professor V. S. Stojanović of the University of Niš, Niš, Serbia for his valuable comments and suggestions. The work presented here was partly supported by the Serbian Ministry of Education and Science in the frame of the Projects TR 32009.

References

  1. 1.
    M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55, 9th edn. (Dover, New York, 1972)MATHGoogle Scholar
  2. 2.
    A. Budak, P. Aronhime, Transitional Butterworth–Chebyshev filters. IEEE Trans. Circuits Theory 18(5), 413–415 (1971)CrossRefGoogle Scholar
  3. 3.
    S. Butterworth, On the theory filter amplifier. Exp. Wirel. Radio Eng. 7, 536–541 (1930)Google Scholar
  4. 4.
    C.E. Chrisostomidis, S. Lucyszyn, On the theory of chained-function filters. IEEE Trans. Microw. Theory Tech. 53(10), 3142–3151 (2005). doi: 10.1109/TMTT.2005.855358 CrossRefGoogle Scholar
  5. 5.
    M.T. Chryssomallis, J.N. Sahalos, Filter synthesis using products of Legendre polynomials. Electr. Eng. 81(6), 419–424 (1999)CrossRefGoogle Scholar
  6. 6.
    S.B. Cohn, Dissipation loss in multiple-coupled-resonator filters. Proc. IRE 47(8), 1342–1348 (1959). doi: 10.1109/JRPROC.1959.287201 CrossRefGoogle Scholar
  7. 7.
    S. Darlington, Synthesis of reactance 4-poles which produce prescribed insertion loss characteristics: including special applications to filter design. J. Math. Phys. 18(1–4), 257–353 (1939). doi: 10.1002/sapm1939181257 MathSciNetCrossRefGoogle Scholar
  8. 8.
    H.G. Dimopoulos, Analog Electronic Filters: Theory, Design and Synthesis (Springer, Dordrecht, 2012)CrossRefGoogle Scholar
  9. 9.
    M. Guglielmi, G. Connor, Chained function filters. IEEE Microw. Guided Wave Lett. 7(12), 390–392 (1997). doi: 10.1109/75.645181 CrossRefGoogle Scholar
  10. 10.
    J.S. Hong, M.J. Lancaster, Microstrip Filters for RF/Microwave Applications (Wiley, New York, 2001)CrossRefGoogle Scholar
  11. 11.
    A.B. Jayyousi, M.J. Lancaster, F. Huang, Filtering functions with reduced fabrication sensitivity. IEEE Microw. Wirel. Compon. Lett. 15(5), 360–362 (2005). doi: 10.1109/LMWC.2005.847713 CrossRefGoogle Scholar
  12. 12.
    A. Kumar, A. Verma, Design of Bessel low-pass filter using dgs for RF/microwave applications. Int. J. Electron. 103(9), 1460–1474 (2016). doi: 10.1080/00207217.2015.1126860 Google Scholar
  13. 13.
    S. Kumari, S. Gupta, N. Pandey, R. Pandey, R. Anurag, LC-ladder filter systematic implementation by OTRA. Eng. Sci. Technol. 19(4), 1808–1814 (2016). doi: 10.1016/j.jestch.2016.10.003 Google Scholar
  14. 14.
    Z.D. Milosavljević, M.V. Gmitrovic, Design of maximally selective generalized Chebyshev filters. Circuits Syst. Signal Process. 21(2), 195–205 (2002). doi: 10.1007/s00034-002-2006-8 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    J. Neirynck, L. Milić, Equal ripple tolerance characteristics. Int. J. Circuit Theory Appl. 4(1), 99–104 (1976). doi: 10.1002/cta.4490040110 CrossRefGoogle Scholar
  16. 16.
    H. Orchard, Loss sensitivities in singly and doubly terminated filters. IEEE Trans. Circuits Syst. 26(5), 293–297 (1979). doi: 10.1109/TCS.1979.1084643 CrossRefMATHGoogle Scholar
  17. 17.
    H.J. Orchard, Inductorless filters. Electron. Lett. 2(6), 224–225 (1966)CrossRefGoogle Scholar
  18. 18.
    D. Packiaraj, K. Vinoy, M. Ramesh, A. Kalghatgi, Design of compact low pass filter with wide stop band using tri-section stepped impedance resonator. AEÜ Int. J. Electron. Commun. 65(12), 1012–1014 (2011). doi: 10.1016/j.aeue.2011.03.018 CrossRefGoogle Scholar
  19. 19.
    D.M. Pozar, Microwave Engineering, 4th edn. (Wiley, New York, 2011)Google Scholar
  20. 20.
    S. Prasad, L.G. Stolarczyk, J.R. Jackson, E.W. Kang, Filter synthesis using Legendre polynomials. Proc. IEE 114(8), 1063–1064 (1967)Google Scholar
  21. 21.
    S.C.D. Roy, Modified Chebyshev lowpass filters. Int. J. Circuit Theory Appl. 38(5), 543–549 (2010). doi: 10.1002/cta.585 Google Scholar
  22. 22.
    K. Srisathit, Simple technique to extend the bandwidth of parallel-coupled microstrip bandpass filter, in 9th International Symposium on Communications and Information Technology, 2009. ISCIT 2009 (2009), pp. 901–914Google Scholar
  23. 23.
    N. Stamenković, V. Stojanović, On the design transitional Legendre–Butterworth filters. Int. J. Electron. Lett. 2(3), 186–195 (2014). doi: 10.1080/00207217.2014.894138 CrossRefGoogle Scholar
  24. 24.
    S. Winder, Analog and Digital Filter Design, 2nd edn. (Elsevier Science, Woburn, 2002)Google Scholar
  25. 25.
    Y.S. Zhu, W.K. Chen, On the design of Butterworth or Chebyshev broad-band impedance-matching ladder networks. Circuits Syst. Signal Process. 9(1), 55–73 (1990). doi: 10.1007/BF01187721 MathSciNetCrossRefGoogle Scholar
  26. 26.
    A.I. Zverev, Handbook of Filter Synthesis (Wiley, New York, 1967)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Faculty of Electronic EngineeringUniversity of NišNišSerbia
  2. 2.Faculty of Natural Science and MathematicsUniversity of PrištinaKosovska MitrovicaSerbia
  3. 3.Faculty of Technical ScienceUniversity of PrištinaKosovska MitrovicaSerbia

Personalised recommendations