Digital Differentiators Based on Taylor Series

Much research has been performed on the efficient design and implementation of finite impulse response (FIR) digital differentiators. Some of the commonly used design procedures include windows based designs [1], minimax designs [2], eigenfilters [3], [4], optimization techniques [5], the weighted least squares technique [6], and a recent design with optimal noise attenuation [7]. Another important class of FIR differentiators is maximally linear differentiators [8]–[10] which are highly accurate at low frequencies. In this paper, we demonstrate that the coefficients of the maximally linear digital differentiator of the order of 2N + 1 are the same as the coefficients of a central difference approximation of the order of N , to the first-order derivative based on the Taylor series expansion. We suggest a modification to the formula of the coefficients, in which within a certain maximum allowable ripple on amplitude response of the resultant differentiator, its passband can be dramatically increased, although it has no more maximal linearity.