Skip to main content
SpringerLink
Log in

Closed-Form Analytical Formulation for Riemann–Liouville-Based Fractional-Order Digital Differentiator Using Fractional Sample Delay Interpolation

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper intends to apply a new mathematical approach based on Riemann–Liouville fractional–differential operator to the design of fractional-order digital differentiators (FODDs). Under the research area of fractional-order calculus, Grünwald–Letnikov (GL) and Riemann–Liouville (RL) are most widely used fractional–differential operators. The GL-based methods have been extensively investigated by research community to design FODD, but there seems to be an improvement window for designing FIR filters based upon RL fractional–differential operator. Therefore, this paper establishes a generalized framework for the design of RL-based FODD (RL-FODD) and compares its performance with well-established GL-FODD designs, based upon their ability to yield ideal frequency response of FODD. Initially, closed-form analytical expression is formulated for computing FIR filter coefficients of RL-FODD. Then, the design accuracy of proposed filter in the high-frequency region is improved by incorporating non-integer sample delay into the design process. Several design examples are presented to illustrate the comparative analysis between conventional GL-FODD and proposed RL-FODD for varying fractional orders. The proposed method is evaluated, taking into account several amplitude-modulated and harmonic signals corrupted by AWGN and high-frequency chirp noise. Furthermore, the application in parameter estimation of fractional noise process is investigated. Compared with conventional GL-FODD, results of the proposed study validate its superiority and robustness based upon various experimental simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data Availability

Data sharing was not applicable to this article as no datasets were generated or analysed during the current study.

Abbreviations

\( \alpha \) :

Fractional-order

\( \varGamma \left( \cdot \right) \) :

Gamma function

\( L \) :

Number of samples

\( h \) :

Non-integer delay

D :

Delay

\( a\left( k \right) \) :

FIR filter coefficients of proposed RL-FODD

\( h_{1} \left( m \right) \) :

FIR filter coefficients of proposed L-RL-FODD

\( h_{2} \left( m \right) \) :

FIR filter coefficients of proposed RBF-RL-FODD

References

  1. D. Baleanu, J.T. Machado, A.C.J. Luo, Fractional Dynamics and Control (Springer, New York, 2011)

    Google Scholar 

  2. R.S. Barbosa, J.A. Tenreiro Machado, M.F. Silva, Time domain design of fractional differintegrators using least-squares. Signal Process. 86, 2567–2581 (2006)

    Article  Google Scholar 

  3. D.W. Brzeziński, Fractional order derivative and integral computation with a small number of discrete input values using Grünwald–Letnikov formula. Int. J. Comput. Methods 17, 1–16 (2020)

    Article  Google Scholar 

  4. M. Cai, C. Li, Numerical approaches to fractional integrals and derivatives: a review. Mathematics. 8, 1–53 (2020)

    Google Scholar 

  5. A. Charef, H.H. Sun, Y.Y. Tsao, B. Onaral, Fractal system as represented by singularity function. IEEE Trans. Automat. Contr. 37, 1465–1470 (1992)

    Article  MathSciNet  Google Scholar 

  6. D. Chen, D. Xue, Y. Chen, Digital fractional order Savitzky-Golay differentiator. IEEE Trans. Circuits Syst. II Express Briefs. 58, 293–302 (2012)

    Google Scholar 

  7. Y.Q. Chen, K.L. Moore, Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49, 363–367 (2002)

    Article  Google Scholar 

  8. M. Deriche, A.H. Tewfik, Signal modeling with filtered discrete fractional noise processes. IEEE Trans. Signal Process. 41, 2839–2849 (1993)

    Article  Google Scholar 

  9. R. Garrappa, E. Kaslik, M. Popolizio, Evaluation of fractional integrals and derivatives of elementary functions: overview and tutorial. Mathematics 7, 407–428 (2019)

    Article  Google Scholar 

  10. R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, Singapore, 2014)

    Book  Google Scholar 

  11. B.T. Krishna, Studies on fractional order differentiators and integrators: a survey. Signal Process. 91, 386–426 (2011)

    Article  Google Scholar 

  12. S. Kumar, Analysis and Design of Non Recursive Digital Differentiators in Fractional Domain for Signal Processing Applications (Thapar University, Patiala, 2014)

    Google Scholar 

  13. S. Kumar, R. Saxena, K. Singh, Fractional Fourier transform and fractional-order calculus-based image edge detection. Circuits Syst. Signal Process. 36, 1493–1513 (2017)

    Article  Google Scholar 

  14. S. Kumar, K. Singh, R. Saxena, Caputo-based fractional derivative in fractional Fourier transform domain. Circuits Syst. Signal Process. 32, 1875–1889 (2013)

    Article  Google Scholar 

  15. T.I. Laakso, V. Välimäki, M. Karjalainen, U.K. Laine, Splitting the unit: delay tools for fractional delay filter design. IEEE Signal Process. Mag. 13, 30–60 (1996)

    Article  Google Scholar 

  16. D.Y. Liu, O. Gibaru, W. Perruquetti, T.M. Laleg-Kirati, Fractional order differentiation by integration and error analysis in noisy environment. IEEE Trans. Autom. Contr. 60, 2845–2960 (2015)

    Article  MathSciNet  Google Scholar 

  17. S. Mahata, S.K. Saha, R. Kar, D. Mandal, A metaheuristic optimization approach to discretize the fractional order Laplacian operator without employing a discretization operator. Swarm Evol. Comput. 44, 534–545 (2019)

    Article  Google Scholar 

  18. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Model (World Scientific, Italy, 2010)

    Book  Google Scholar 

  19. G. Maione, On the Laguerre rational approximation to fractional discrete derivative and integral operators. IEEE Trans. Automat. Contr. 58, 1575–1585 (2013)

    Article  MathSciNet  Google Scholar 

  20. K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Academic, New York, 1974)

    MATH  Google Scholar 

  21. M.D. Ortigueira, Fractional Calculus for Scientists and Engineers (Springer, Berlin, 2011)

    Book  Google Scholar 

  22. A. Oustaloup, F. Levron, B. Mathieu, F.M. Nanot, Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 47, 25–39 (2000)

    Article  Google Scholar 

  23. I. Podlubny, I. Petráš, B.M. Vinagre, P. O’Leary, L. Dorčák, Analogue realizations of fractional-order controllers. Nonlinear Dyn. 29, 281–296 (2002)

    Article  MathSciNet  Google Scholar 

  24. Y.F. Pu, J.L. Zhou, X. Yuan, Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement. IEEE Trans. Image Process. 19, 491–511 (2010)

    Article  MathSciNet  Google Scholar 

  25. K.P.S. Rana, V. Kumar, Y. Garg, S.S. Nair, Efficient design of discrete fractional-order differentiators using Nelder-Mead simplex algorithm. Circuits Syst. Signal Process. 35, 2155–2188 (2016)

    Article  Google Scholar 

  26. S. Samadi, M.O. Ahmad, M.N.S. Swamy, Exact fractional-order differentiators for polynomial signals. IEEE Signal Process. Lett. 11, 529–532 (2004)

    Article  Google Scholar 

  27. R.W. Schafer, What is a Savitzky–Golay filter? IEEE Signal Process. Mag. 28, 111–117 (2011)

    Article  Google Scholar 

  28. H. Sheng, Y. Chen, T. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications (Springer, New York, 2011)

    MATH  Google Scholar 

  29. C.C. Tseng, Improved design of digital fractional-order differentiators using fractional sample delay. IEEE Trans. Circuits Syst. I Regul. Pap. 53, 193–203 (2006)

    Article  Google Scholar 

  30. C.C. Tseng, Design of fractional order digital FIR differentiators. IEEE Signal Process. Lett. 8, 77–79 (2001)

    Article  Google Scholar 

  31. C.C. Tseng, S.L. Lee, Closed-form design of fractional order differentiator using discrete cosine transform, in Proceedings - IEEE International Symposium on Circuits and Systems, pp. 2609–2612 (2013)

  32. C.C. Tseng, S.L. Lee, Design of fractional order digital differentiator using radial basis function. IEEE Trans. Circuits Syst. I Regul. Pap. 57, 1708–1718 (2010)

    Article  MathSciNet  Google Scholar 

  33. C.C. Tseng, S.C. Pei, S.C. Hsia, Computation of fractional derivatives using Fourier transform and digital FIR differentiator. Signal Process. 80, 151–159 (2000)

    Article  Google Scholar 

  34. J. Wang, Y. Ye, X. Pan, X. Gao, C. Zhuang, Fractional zero-phase filtering based on the Riemann-Liouville integral. Signal Process. 98, 150–157 (2014)

    Article  Google Scholar 

  35. B.J. West, Nature’s Patterns and the Fractional Calculus (Walter de Gruyter GmbH & Co KG, Berlin, 2017)

    Book  Google Scholar 

Download references

Acknowledgements

The authors thank the Editor-in-Chief, Associate Editor, and anonymous reviewers for their rigorous reviews, constructive comments, and valuable suggestions which greatly improved the quality and clarity of manuscript presentation. The work is supported by Science and Engineering Research Board (SERB) (No. SB/S3/EECE/0149/2016), Department of Science and Technology (DST), Government of India, India.

Author information

Authors and Affiliations

  1. Biomedical Signal Analysis and Interpretation Laboratory (BioSAIL), Department of Electronics and Communication Engineering, Thapar Institute of Engineering and Technology, Patiala, 147004, India

    Anmol Gupta & Sanjay Kumar

Authors
  1. Anmol Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sanjay Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sanjay Kumar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gupta, A., Kumar, S. Closed-Form Analytical Formulation for Riemann–Liouville-Based Fractional-Order Digital Differentiator Using Fractional Sample Delay Interpolation. Circuits Syst Signal Process 40, 2535–2563 (2021). https://doi.org/10.1007/s00034-020-01589-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-020-01589-2

Keywords

Search

Navigation