Optimal design and comparison of wide-band digital on-line differentiators

The problem of wide-band digital on-line differentiators is of increasing importance. The frequency approach is used to design and compare different filters. First, an error criterion that represents numerical error is derived, combining amplitude and phase errors. Conditions that ensure good low frequency behavior are included. Two design procedures to obtain the filter coefficients are discussed: an indirect approach based on modeling with polynomials and modified splines and a direct approach using an optimization scheme. Finally, the frequency characteristics of the main optimal and suboptimal wide-band differentiators are compared. The full-band requirement restricts improvements to about 33 percent with respect to the first-order difference formula.