Exact Fractional-Order Differentiators for Polynomial Signals

A discrete-time fractional-order differentiator is modeled as a finite-impulse response (FIR) system. The system yields fractional-order derivatives of Riemann-Liouville type for a uniformly sampled polynomial signal. The computation of the output signal is based on the additive combination of the weighted outputs of N cascaded first-order digital differentiators. For differentiators of fractional order with a terminal value equal to zero, the weights are time-varying. The weights are obtained in a closed form involving the Stirling numbers of the first kind. The system tends to a time-invariant integer-order differentiator when the order of the derivative tends to an integer value. It yields exact fractional- or integer-order derivatives of a sampled polynomial signal of a certain order.