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

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

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