Analytical impulse response of a fractional second order filter and its impulse response invariant discretization

Signal Processing

Volumn 91, Issue 3, 2011, Pages 498-507

  Li, Yan ^a,c   Sheng, Hu ^b,c   Chen, Yangquan ^c  

^a SHANDONG UNIVERSITY   (China)

^b DALIAN UNIVERSITY OF TECHNOLOGY   (China)

^c Utah State University   (United States)

Author keywords

Complex integral; Discretization method; Fractional order filter

Indexed keywords

ASYMPTOTIC PROPERTIES; COMPLEX INTEGRALS; DISCRETIZATION METHOD; DISCRETIZATIONS; FRACTIONAL ORDER; SECOND ORDER FILTERS; TIME AND FREQUENCY DOMAINS;

DISCRETE EVENT SIMULATION; VOLUME MEASUREMENT;

IMPULSE RESPONSE;

EID: 78049326583     PISSN: 01651684     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sigpro.2010.01.017     Document Type: Article

References (36)

1. K.B. Oldham, and J. Spanier The Fractional Calculus 1974 Academic Press New York
2. K.S. Miller, and B. Ross An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 Wiley New York
3. I. Podlubny Fractional Differential Equations 1999 Academic Press New York
4. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 2006 Elsevier Science Inc. New York, NY, USA
5. R.L. Bagley, and P.J. Torvik A theoretical basis for the application of fractional calculus to viscoelasticity Journal of Rheology 27 3 1983 201 210
6. A. Oustaloup, La commande CRONE, Herms, Paris, France, 1991 (in French).
7. I. Podlubny Fractional-order systems and PIλ Dμ -controllers IEEE Transactions on Automatic Control 44 1 January 1999 208 214
8. N. Laskin Fractional Schrdinger equation Physical Review E 66 5 2002 056108 (7pp.)
9. J.A. Tenreiro Machado Analysis and design of fractional-order digital control systems Systems Analysis Modelling Simulation 27 23 1997 107 122
10. B.M. Vinagre, I. Petras, P. Merchan, L. Dorcak, Two digital realization of fractional controllers: application to temperature control of a solid, in: Proceedings of the European Control Conference (ECC2001), Porto, Portugal, 2001, pp. 17641767.
11. A. Oustaloup, F. Levron, B. Mathieu, and F.M. Nanot Frequency-band complex noninteger differentiator: characterization and synthesis IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47 1 January 2000 25 39
12. Y.Q. Chen, and K.L. Moore Discretization schemes for fractional-order differentiators and integrators IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49 3 March 2002 363 367
13. Y.Q. Chen, and B.M. Vinagre A new IIR-type digital fractional order differentiator Signal Processing 83 11 2003 2359 2365
14. Ch. Lubich Discretized fractional calculus SIAM Journal on Mathematical Analysis 17 3 1986 704 719
15. Y.Q. Chen, B.M. Vinagre, and I. Podlubny Continued fraction expansion approaches to discretizing fractional order derivativesan expository review Nonlinear Dynamics 38 December 2004 155170(16)
16. B.M. Vinagre, Y.Q. Chen, and I. Petras Two direct Tustin discretization methods for fractional-order differentiator/integrator Journal of The Franklin Institute 340 August 2003 349362(14)
17. R.S. Barbosa, and J.A. Tenreiro Machado Implementation of discrete-time fractional-order controllers based on LS approximations Acta Polytechnica Hungarica 3 4 2006 5 22
18. Y. Ferdi Impulse invariance-based method for the computation of fractional integral of order 0 < α < 1 Computers and Electrical Engineering 35 5 2009 722 729
19. A. Oustaloup Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulation IEEE Transactions on Circuits and Systems 28 10 October 1981 1007 1009
20. A.G. Radwan, A.S. Elwakil, and A.M. Soliman Fractional-order sinusoidal oscillators: design procedure and practical examples IEEE Transactions on Circuits and Systems I: Regular Papers 55 7 August 2008 2051 2063
21. A.G. Radwan, A.M. Soliman, A.S. Elwakil, Design equations for fractional-order sinusoidal oscillators: practical circuit examples, in: Proceedings of the International Conference on Microelectronics, December 2007, pp. 8992.
22. S.C. Lim, and L.P. Teo The fractional oscillator process with two indices Journal of Physics A: Mathematical and Theoretical 42 2009 065208 10.1088/1751-8113/42/6/065208 (34pp.)
23. B. Davies Integral Transforms and their Applications third ed. 2002 Springer New York
24. E.I. Jury Theory and Application of the Z-Transform Method 1973 Krieger Publishing Company
25. V.E. Tarasov Fractional derivative as fractional power of derivative International Journal of Mathematics 18 3 2007 281 299
26. Y.Q. Chen, and K.L. Moore Analytical stability bound for a class of delayed fractional order dynamic systems Nonlinear Dynamics 29 2002 191 200
27. J. Sabatier, O.P. Agrawal, and J.A. Tenreiro Machado Advances in Fractional CalculusTheoretical Developments and Applications in Physics and Engineering 2007 Springer Berlin
28. J.O. Smith, Physical Audio Signal Processing. 〈 http://ccrma. stanford.edu/jos/pasp/ 〉, 2008. online book.
29. R.K. Saxena, A.M. Mathai, and H.J. Haubold On generalized fractional kinetic equations Physica A: Statistical Mechanics and its Applications 344 December 2004 657 664
30. T.W. Parks, and C.S. Burrus Digital Filter Design 1987 Wiley New York pp. 226228
31. Signal Processing Toolbox 6.12. 〈 http://www.mathworks.com/products/ signal 〉.
32. Y.Q. Chen, Low-pass IIR digital differentiator design. 〈 http://www.mathworks.com/matlabcentral/fileexchange/3517 〉, 2003.
33. Y.Q. Chen, A New IIR-type Digital Fractional order differentiator. 〈 http://www.mathworks.com/matlabcentral/fileexchange/3518 〉, 2003.
34. Y.Q. Chen, Impulse response invariant discretization of fractional order integrators/differentiators. 〈 http://www.mathworks.com/matlabcentral/ fileexchange/21342 〉, 2008.
35. Y.Q. Chen, Impulse response invariant discretization of fractional order low-pass filters. 〈 http://www.mathworks.com/matlabcentral/fileexchange/ 21365 〉, 2008.
36. H. Sheng, Impulse Response Invariant Discretization of Fractional Second Order Filter. 〈 http://www.mathworks.com/matlabcentral/fileexchange/26442 〉, 2010.