Fractional derivative of logarithmic function and its applications as multipurpose ASP circuit Shalabh Kumar Mishra 1 • Maneesha Gupta 1 • Dharmendra Kumar Upadhyay 1 Received: 13 April 2018 / Revised: 6 September 2018 / Accepted: 6 September 2018 Ó Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract This paper presents a new approach to approximate the fractional order derivative of a logarithmic function using the Caputo definition. Further, this approximated fractional derivative has been used to conceptualize and design a multi- purpose ASP (analog signal processing) circuit, by cascading the logarithmic amplifier with fractional order differentiator. The proposed multipurpose ASP circuit is capable to generate different kinds of signals (such as step signals with controlled amplitude, tangent and cotangent signals with controllable fractional power) for different inputs. Moreover, this circuit is also helpful to develop reciprocal, square root, cube root and nth-root of the input signal, up to some extent. Several types of input signals have been taken into consideration and the corresponding output responses are obtained by varying the order of differentiation in fractional sense. Numerical as-well-as circuit simulations have been done in MATLAB and PSpice environments to validate the theoretically obtained results. Further, hardware implantation has also been done to see the practical aspects of the proposed design. Keywords Logarithmic amplifier  Fractional derivative  Analog signal processing  Caputo derivative 1 Introduction Nowadays signal processing has emerged as one of the most prevailing research topics in the various areas of applied science and engineering, such as communication system, control theory, genomics, biomedical science, medical, oceanology, geophysics, astronomy and robotics [1–8]. Signal processing is broadly categorized in analog signal processing (ASP) and digital signal processing (DSP). Although DSP has its own advantages, ASP is still widely used since it has several merits over the DSP, such as lower cost, lesser power consumption, smaller device size and no frequency constraint [9]. Besides this, ASP is free from the quantization error, which is an unavoidable source of noise in DSPs. It has also been observed that the analog circuits are more efficient and capable to model the neurological systems in a more accurate manner than the digital system [10]. Moreover, it must be noted that most of the practical signals are analog in nature; therefore ASP still plays a vital role in the modern digital world. During the past decades, applications of mathematical functions have been widely increased in the ASP circuits. For example, sinusoidal and exponential functions are widely used in communication system, medical and biomedical system, while the squarer circuit is often used as a modulator, frequency multiplier, frequency divider and adaptive filter in ASP. Recently, the concept of fractional- order calculus is extensively being applied in ASP, to avail some additional advantages such as accurate system mod- eling, precise control over the system response, and more degree-of-freedom in system designing. Several analog filters, diff-integrators, and oscillators have been designed and realized using the concept of fractional calculus [11–16]. The theory of fractional calculus deals with derivatives and integrals of arbitrary order, which generalizes the integer-order differentiation and integration. Although & Maneesha Gupta maneeshapub@gmail.com Shalabh Kumar Mishra shalabhmishra14@gmail.com Dharmendra Kumar Upadhyay upadhyay_d@rediffmail.com 1 Division of ECE, Netaji Subhas Institute of Technology, Sector-3, Dwarka, Delhi 110078, India 123 Analog Integrated Circuits and Signal Processing https://doi.org/10.1007/s10470-018-1328-9