Circuits, Systems, and Signal Processing https://doi.org/10.1007/s00034-019-01211-0 SHORT PAPER A Novel Approach for the Design of Optimum IIR Differentiators Using Fractional Interpolation Om Prakash Goswami 1 · Tarun K. Rawat 1 · Dharmendra K. Upadhyay 1 Received: 30 January 2019 / Revised: 8 July 2019 / Accepted: 10 July 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract 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 inter- polates 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 (dB) and − 35.04 (dB), respec- tively, 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. Keywords s-to-z transform · Al-Alaoui operator · Fractional interpolation · Optimized digital differentiators 1 Introduction Digital differentiators are considered as a fundamental building block in the diverse area of engineering related to biomedical, applied control, instrumentation, digital signal, and image processing. It derives the time derivatives of any measured and applied excitation to get useful responses according to their application needs [3,19, 29]. The digital differentiator can be classified as the finite impulse response (FIR) systems and infinite impulse response (IIR) systems. For the same specifications, IIR B Om Prakash Goswami om9837@gmail.com Tarun K. Rawat tarundsp@gmail.com Dharmendra K. Upadhyay upadhyay_d@rediffmail.com 1 Netaji Subhas Institute of Technology, Sector-3, Dwarka, New Delhi 110078, India