MIXED SIGNAL LETTER Fractional order inverse filters using operational amplifier D. R. Bhaskar 1 • Manoj Kumar 2 • Pragati Kumar 2 Received: 18 March 2018 / Revised: 20 July 2018 / Accepted: 6 August 2018 Ó Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract Two new op-amp-based multifunction filter structures are proposed which can realize fractional order inverse low pass, high pass and band pass filters. In the first configuration, op-amp is used in an inverting mode while in the second structure op-amp is employed in non-inverting mode. To the best knowledge of the authors, any fractional order inverse filter structure employing any active element/device has not been reported in the open-literature so far. The proposed inverse filters have been simulated in PSPICE using lA741-type op-amp as well as in MATLAB to validate the theoretical propositions. Keywords Analog signal processing  Fractional order inverse filters  Fractional calculus  Operational amplifier 1 Introduction Recently, a lot of attention has been devoted to the design of fractional order analog signal processing and signal generation circuits (see [1–21] and the references cited therein). Fractional order filters are also important from the view point of enhancing the design flexibility and controllability due to the additional degree of freedom provided by the fractional order parameters [22]. Inverse filters play an important role in some of the applications such as in communication, instrumentation and control systems where the distortion of the signal caused by transmission system or the signal processor can be elimi- nated by the inverse filters [23]. An overview of fractional order filters reported earlier in open literature is as follows. In [3], realizations of fractional order low pass, high pass, band pass and all pass filters using a single fractance device have been reported and the expressions for the pole frequencies, the quality factor, the right phase frequencies and the half power frequencies have been derived. Radwan et al. [4] have generalized the design of clas- sical second-order filters to the fractional order domain considering two capacitors of the same order. Design and simulation of Sallen-Key and Kerwin, Huelsman and Newcomb (KHN) biquad filter topologies are derived in fractional domain. First ever experimental results of KHN biquad filters using real fractional order capacitors of order 0.5 (electrolytic process) have also been reported therein. Freeborn et al. [5] have reported fractional order low pass filter of order (n ? a) where n is an integer and a, varies in steps from 0.1 to 0.9. Its characteristics have been compared with first and second order Butterworth filters. In [6], Freeborn, Maundy and Elwakil have proposed the use of field programmable analogue array hardware to implement an approximated fractional step transfer func- tion of order (n ? a), where n is an integer and 0 \ a \ 1. Experimental results of (1 ? a) and (4 ? a) order low pass and high pass filters using FPAA kit and MATLAB sim- ulation are shown. Tripathy et al. in [7] have designed the KHN biquad filter using fractional order capacitors of order a and b. The frequency responses of filters obtained experimentally are compared with simulated results of MATLAB and PSPICE. The stability and sensitivity analysis of proposed filters are also presented. In [8], Maundy, Elwakil, and Freeborn have imple- mented fractional high order (n ? a) low pass and band pass filters using Op-amp MC-1458 as active elements, where n = 5 and 0 \ a \ 1. In these realizations, compact & D. R. Bhaskar drbhaskar@dtu.ac.in 1 Department of Electronics and Communication Engineering, Delhi Technological University, New Delhi, India 2 Department of Electrical Engineering, Delhi Technological University, New Delhi, India 123 Analog Integrated Circuits and Signal Processing https://doi.org/10.1007/s10470-018-1287-1