A Novel Approach for the Design of Optimum IIR Differentiators Using Fractional Interpolation

In this paper, a novel method for designing an optimum infinite impulse response digital differentiator of the first and second orders is presented. The proposed method interpolates bilinear transform and rectangular transform fractionally, and then, unknown variables of the generalized equation are optimized using the genetic algorithm. The results obtained by the proposed designs are superior to all state-of-the-art designs in terms of magnitude responses. The first-order and second-order differentiator attains mean relative magnitude error as low as -27.702\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\,27.702$$\end{document} (dB) and -35.04\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\,35.04$$\end{document} (dB), respectively, in the complete Nyquist range. Besides, suggested low-order, differentiator design equations can also be optimized of any desired Nyquist frequency range, which makes it suitable for real-time applications.