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Using New Hybrid Integrators to Design IIR Fractional-Order Integrators via Impulse Invariance Method

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Abstract

This paper presents a method for designing fullband and non-fullband IIR digital integrators with linear phase that has been used to approximate fractional-order integrators with IIR filters. Several numerical integration rules have been considered in this study, namely, Euler, Simpson, Schneider, bilinear, and F012. The main idea in our design is based on interpolating the well-known integration rules (Euler, Bilinear and Simpson) proposed by Al-Alaoui combined with deterministic signal modeling techniques. Numerical examples are presented to illustrate the performance of the proposed integrators. It was found that the Euler–Simpson integrator gives better approximation accuracy than existing integrators.

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References

  1. M.A. Al-Alaoui, A class of second-order integrators and low-pass differentiators. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 42(4), 220–223 (1995)

    Article  Google Scholar 

  2. M.A. Al-Alaoui, Low frequency differentiators and integrators for biomedical and seismic signals. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48(8), 1006–1011 (2001)

    Article  Google Scholar 

  3. M.A. Al-Alaoui, Al-Alaoui operator and the new transformation polynomials for discretization of analogue system. Electr. Eng. 90(6), 455–467 (2008)

    Article  Google Scholar 

  4. Y. Ferdi, Computation of fractional order derivatives and integral via power series expansion and signal modeling. Nonlinear Dyn. 46(1–2), 1–15 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Y. Ferdi, Impulse invariance-based method for the computation of fractional integral of order 0<α<1. Comput. Electr. Eng. 35(5), 722–729 (2009)

    Article  MATH  Google Scholar 

  6. Y. Ferdi, B. Boucheham, Recursive filter approximation of digital fractional differentiator and integrator based on Prony’s method, in Proceedings of the 1st IFAC Workshop on Fractional Differentiation and Its Applications, ENSEIRB, Bordeaux, France, 19–21 July, 2004, p. 428 (2004)

    Google Scholar 

  7. G.F. Franklin, J.D. Powell, M.L. Workman, Digital Control of Dynamics Systems (Addison-Wesley, Reading, 1990)

    Google Scholar 

  8. E. Gurova, Modified stability approach for higher-order s-to-z mapping functions. Electron. Lett. 32, 1182–1183 (1996)

    Article  MathSciNet  Google Scholar 

  9. R.W. Hamming, Digital Filters (Prentice-Hall, Englewood Cliffs, 1989)

    Google Scholar 

  10. C.-W. Hsue, L.-C. Tsai, Y.-H. Tsai, Time constant control of microwave integrators using transmission lines. IEEE Trans. Microw. Theory Tech. 54(3), 1043–1047 (2006)

    Article  Google Scholar 

  11. F.A. Khanday, C. Psychalinos, N.A. Shah, Square-root-domain realization of single-cell architecture of complex TDCNN. Circuits Syst. Signal Process. (2012). doi:10.1007/s00034-012-9503-1

    Google Scholar 

  12. B. Kumar, A. Kumar, FIR linear phase approximations of frequency response 1/(jω) for maximal flatness at an arbitrary frequency ω 0 (0<ω 0<π). IEEE Trans. Signal Process. 47(6), 1772–1775 (1999)

    Article  Google Scholar 

  13. B. Kumar, D.R. Choudhury, A. Kumar, On the design of linear phase FIR integrators for midband frequencies. IEEE Trans. Signal Process. 44(10), 345–353 (1996)

    Google Scholar 

  14. J. Le Bihan, Novel class of digital integrators and differentiators. Electron. Lett. 29(11), 971–973 (1993)

    Article  Google Scholar 

  15. S.K. Mitra, Digital Signal Processing (McGraw-Hill, New York, 2006)

    Google Scholar 

  16. R. Pintelon, J. Schoukens, Real time integration and differentiation of analog signals by means of digital filtering. IEEE Trans. Instrum. Meas. 39(6), 923–927 (1990)

    Article  Google Scholar 

  17. L.R. Rabiner, K. Steiglitz, The design of wide-band recursive and nonrecursive digital differentiators. IEEE Trans. Audio Electroacoust. 18(2), 204–209 (1970)

    Article  Google Scholar 

  18. A.M. Schneider, J.T. Kaneshige, F.D. Groutage, Higher order s to z mapping functions and their application in digitizing continuous time filters. Proc. IEEE 79(11), 1661–1674 (1991)

    Article  Google Scholar 

  19. K. Steiglitz, L.E. Mc-Bride, A technique for the identification of linear systems. IEEE Trans. Autom. Control AC-10(4), 461–464 (1965)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank anonymous reviewers for their helpful suggestions and comments during the review process.

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Authors and Affiliations

  1. LRES Laboratory, Electrical Engineering Department, Skikda University, BP. 26, Route d’El-Hadaîek, 21000, Skikda, Algeria

    Chafia Mekhnache & Youcef Ferdi

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  1. Chafia Mekhnache
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  2. Youcef Ferdi
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Correspondence to Chafia Mekhnache.

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Mekhnache, C., Ferdi, Y. Using New Hybrid Integrators to Design IIR Fractional-Order Integrators via Impulse Invariance Method. Circuits Syst Signal Process 33, 643–653 (2014). https://doi.org/10.1007/s00034-013-9646-8

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  • DOI: https://doi.org/10.1007/s00034-013-9646-8

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