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

Year 2016, Volume: 1 Issue: 1, 29 - 38, 01.12.2016

Barış Baykant Alagöz Mehmet Emin Tağluk

Abstract

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.

Keywords

: Fractional order derivative, orthogonality, Fourier series expansion

References

• [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] 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] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus & Applied Analysis, vol. 5 pp.367-386, 2002.
• [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] 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] 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] Manuel Duarte Ortigueira, Fractional Calculus for Scientists and Engineers, Springer Science & Business Media, 2011
• [8] Bruce West, Mauro Bologna, Paolo Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer Science & Business Media, 2012.
• [9] Tavazoei, Mohammad Saleh. A note on fractional-order derivatives of periodic functions. Automatica vol.46 no.5,pp.945-948,2010.
• [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] Introductory Notes on Fractional Calculus, xuru.org
• [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] Das S. Functional Fractional Calculus (Second edition). Berlin: Springer, 2011.
• [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] 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.