638 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 24, NO. 2, FEBRUARY 2016 OTA-Based Logarithmic Circuit for Arbitrary Input Signal and Its Application Mousumi Bhanja, Student Member, IEEE, and Baidya Nath Ray Abstract—In this paper, a new design procedure has been proposed for realization of logarithmic function via three phases: 1) differentiation; 2) division; and 3) integration for any arbitrary analog signal. All the basic building blocks, i.e., differentiator, divider, and integrator, are realized by operational transconduc- tance amplifier, a current mode device. Realization of exponential, power law, and hyperbolic function as the design examples claims that the proposed synthesis procedure has the potential to design a log-based nonlinear system in a systematic and hierarchical manner. The performance of all the proposed circuits has been verified with SPICE simulation. Index Terms— Exponential function, hyperbolic function, integrator, logarithmic function, operational transconductance amplifier (OTA), power law circuit, systematic synthesis. I. I NTRODUCTION L OGARITHMIC and exponential functions, having a wide application in communication and signal processing, are generally implemented using bipolar junction transistors (BJTs) or MOS transistor in weak inversion using their exponential characteristics [5]–[9]. In this paper, a thorough understanding of a mathematical model combined with hardware compatibility has been exploited to realize logarithmic and exponential functions for various analog signals, which leads to an improved design. A low-cost, high-speed architecture for binary logarithm approximation has been proposed by many researchers. Mitchell’s method with a correction stage composed of piecewise linear interpolation and a lookup-table correction is used to compute binary logarithm in [8]. Hardware implementation of log() and antilog() is proposed in [26]. Both the architectures are implemented in an FPGA. Voltage–current relationship of a p-n junction diode and Taylor’s series expansion have been used to realize logarithmic and exponential function [2]–[10]. Significant efforts have been invested in using MOS transistors in saturation region for the implementation of exponential functions [6], [7]. Liu and Liu [5] proposed a compact, low-power, CMOS exponential function generator with a wide dynamic range. A CMOS pseudoexponential Manuscript received September 23, 2014; revised January 9, 2015; accepted February 9, 2015. Date of publication March 19, 2015; date of current version January 19, 2016. This work was supported in part by the Department of Science and Technology—Science and Engineering Research Council, Government of India, under Grant SR/S3/EECE/0065/2010, and in part by the University Grants Commission, Government of India. The authors are with the Department of Electronics and Telecommuni- cation Engineering, Indian Institute of Engineering Science and Technol- ogy at Shibpur, Howrah 711103, India (e-mail: mousumi1988@yahoo.com; bnr@telecom.iiests.ac.in). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVLSI.2015.2406953 function circuit based on Taylor series approximation using MOS transistor operating at saturation region is presented in [6]. Chang and Liu [7] proposed a voltage-mode and a current-mode pseudoexponential function circuit. In [3], a low-voltage CMOS current mode exponential circuit has been described using MOS transistors in weak inversion. Translinear principle is used to implement the approximation to cancel the temperature effect. Maundy et al. [9] introduced a pseudoexponential and pseudologarithmic circuit using operational amplifier (op-amp). Implementation of pseudologarithmic circuit using Taylor’s series expansion has been described in [10]. The above survey reveals that each of the works deals with the design of logarithmic and exponential circuit for a specific input signal, i.e., ramp signal. Design of logarithmic and exponential circuit for any arbitrary input signal remains a serious problem. In that context, this paper presents a sys- tematic design procedure using operational transconductance amplifier (OTA), a current mode device, for logarithmic and exponential function realization that handles any arbitrary input signal. The basic idea of this method is that, if a function q (x ) is expressed as a ratio of two functions, p(x ) and f (x ), where the numerator function, p(x ), is the differentiation of the denominator function f (x ), then integration of q (x) generates ln[ f (x )]. In addition, it is demonstrated that for the parabolic input signal, i.e., q (x ) = 1/x , logarithmic function, ln(x ), is implemented with a single OTA (which consists of eight transistors) and a grounded capacitance. Relevant applications of the logarithmic function generator, such as exponential, power law, and hyperbolic functions, are also described as the design examples. Exponential of any arbitrary time function including ramp has also been proposed. Power law functions are important building blocks of analog instruments. The arbitrary power law circuit proposed in [16] differs for different power, and as a result of it, component overhead increases. OTA-based square-root circuit has been proposed in [17]. In contrast, the proposed power law circuit realizes arbitrary power, including fractional power, using logarithmic circuit as building block. The proposed topology provides programmability. Hyperbolic functions, which are very important tools in neural network analysis, have been realized using BJTs or OTAs [18], [19]. In that context, we have proposed a systematic design procedure of hyperbolic function using logarithmic-based exponential function circuit. Different sections of this paper are laid out as follows. The synthesis methodology and OTA implementation of the logarithmic function have been discussed in Section II. Some relevant applications of the logarithmic circuits have 1063-8210 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.