Design of Second Order Digital FIR Full-Band Differentiators Using Weighting Coefficients

Most methods for the FIR differentiator design, published so far, have rather complex algorithms, requiring significant computing time. In addition, their design procedures and equations differ for the cases of even and odd differentiator length. FIR differentiators (filters) designed by these methods mostly have the passband group delay level corresponding to that of the linear phase FIR filters. In that sense, the design procedure of the first and higher order FIR full-band differentiators, proposed in this paper, is not only more efficient, but general too. Its efficiency consists in the direct (by this means-faster) determination of numerical values of their impulse response coefficients. Its generality consists in the fact that this method uses the same procedure and formulas for the design of differentiators with both even and odd length (without any mutual difference). (In order to avoid confusion, in the following text, the order of differentiation will be denoted by k, while the length of the FIR structure will be denoted by N). In-addition, FIR differentiators designed by this method have approximately constant passband group delay level, which is lower than that of the corresponding linear phase FIR differentiators. These characteristics are enabled due to the fact that full-band differentiators, designed by the presented method, possess neither the (anti) symmetric feature of their impulse response coefficients, nor the strictly linear phase. The proposed method is based on the simultaneous approximation of the prescribed magnitude and group delay responses. The method presents an approach for the FIR differentiator frequency response approximation, directly in the complex, and not in the real domain. Two weighting coefficients, alpha(R) and alpha(I), for the approximation of the real and imaginary part of the frequency response, respectively, are introduced. In the convenient manner, they are incorporated into properly and specifically defined quadratic measure-error of the FIR differentiator's frequency response approximation. This manner enables adjusting (decreasing) contributions of the frequency response real and imaginary part approximation errors to the total approximation error. Performing a proper analysis of the real and imaginary part approximation errors (of designed FIR differentiator-frequency responses), values of these two parameters, giving a small variation of the group delay response, are-obtained simultaneously with very small magnitude response error in the passband. In order to illustrate the-proposed method effectiveness, numerical design examples of the second order FIR full-band differentiators are given.