A Note on Fractional Order Derivatives on Periodic Signals According to Fourier Series Expansion

Öz

This study presents a discussion on input-output orthogonality property of derivative operators for sinusoidal functions and investigates the effects of fractional order derivative on Fourier series expansion of periodic signals. The findings of this study are useful for the interpretation of fractional order derivative operator for time periodic signals. Fourier series expansion expresses any periodical signals as the sum of sine and cosine functions. Accordingly, it is illustrated that the derivative operator takes effect on the amplitude and phase of Fourier components as follows: The first order derivative of sine and cosine functions leads to a phase shifting of the right angle and an amplitude scaling proportional to angular frequency of sinusoidal component. As a result of the right angle phase shifting of sinusoidal components, the first order derivative generates an orthogonal function for sinusoidal inputs. However, non-integer order derivatives do not conform orthogonality property for sine and cosine functions because it can lead to a phase shifting in the any fraction of right angle. It also results in an amplitude scaling proportional to -power of angular frequency of sinusoidal components. Moreover, fractional order derivative of periodic signals is expressed on the bases of Fourier series expansion and the interpretation of the operator for signals is discussed on the bases of this formula.

Anahtar Kelimeler

• : Fractional order derivative,orthogonality,Fourier series expansion

Kaynakça

1. [1]J. A. T. Machado, A probabilistic interpretation of the fractional-order differentiation Fractional Calculus and applied Analysis, vol. 6 pp.73-80,2003.
2. [2] M. Moshrefi-Torbati, J.K. Hammond, Physical and geometrical interpretation of fractional operators,Journal of the Franklin Institute, vol. 335, pp.1077-1086, 1998.
3. [3] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus & Applied Analysis, vol. 5 pp.367-386, 2002.
4. [4] F.J. Molz, G.J. Fix, S. Lu, A physical interpretation for the fractional derivative in Levy diffusion, Applied Mathematics Letters, vol.15, pp. 907-911, 2002.
5. [5] Ateş, A., Alagoz, B. B., Alisoy, G. T., Yeroğlu, C., Alisoy, H. Z ,"Fuzzy Velocity and Fuzzy Acceleration in Fractional Order Motion." Balkan Journal of Electrical and Computer Engineering vol.3,pp.98-102, 2015.
6. [6] Ortigueira, M. D., Machado, J. T., & Trujillo, J. J. (2015). Fractional derivatives and periodic functions. International Journal of Dynamics and Control, pp.1-7,2015.
7. [7] Manuel Duarte Ortigueira, Fractional Calculus for Scientists and Engineers, Springer Science & Business Media, 2011
8. [8] Bruce West, Mauro Bologna, Paolo Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer Science & Business Media, 2012.

9. [9] Tavazoei, Mohammad Saleh. A note on fractional-order derivatives of periodic functions. Automatica vol.46 no.5,pp.945-948,2010.
10. [10] Tan, N., Atherton, D.P. & Yüce, A., Computing step and impulse responses of closed loop fractional order time delay control systems using frequency response data, Int. J. Dynam. Control, pp.1-10, 2016.
11. [11] Introductory Notes on Fractional Calculus, xuru.org
12. [12] Petras I. Stability of fractional-order systems with rational orders: a survey. Fractional Calculus and Applied Analysis, vol.12 no.3,pp.269-298,2009.
13. [13] Das S. Functional Fractional Calculus (Second edition). Berlin: Springer, 2011.
14. [14] Adam Loverro, Fractional Calculus: History, Definitions and Applications for the Engineer, Lecture Notes, University of Notre Dame, Online available http://www3.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf.
15. [15] Atherton, Derek P., Nusret Tan, and Ali Yüce. Methods for computing the time response of fractional-order systems. IET Control Theory & Applications vol.9 no.6, pp817-830,2015.