Higher degree FIR digital differentiators based on Taylor series

Simplified designs having closed form expressions for impulse response coefficients of maximally linear FIR digital differentiators are already available. We extend these designs to higher degree differentiators and present closed form expressions for their coefficients. The designs are in fact solutions of systems of linear approximate equations based on the Taylor series, formally known as central difference approximations. We avoid this computationally expensive step and present closed form expressions for the solution, which can be implemented as digital differentiators of arbitrary order and degree, and have maximal accuracy at ω = 0. We also present as new class of central difference approximations, and find explicit formulas for their coefficients. Differentiators of odd degree based on these new approximations are more accurate in the upper frequency bands.