FRACTIONAL-ORDER SIGNAL PROCESSING AND SIGNAL GENERATING CIRCUITS

Abstract

Fractional calculus is the branch of mathematics concerning differentiations and integrations of non-integer orders. The inherent ability of fractional calculus to provide more accurate descriptions of real-world phenomena compared to classical integer methods is making it a preferred choice among system designers. Fractional-order systems are found in diverse scientific disciplines including thermodynamics, electrical engineering, control systems, biomedical science, mechanics, electronics, communication, and image encryption. As a result, there is significant potential for multidisciplinary research and exploration in the field of fractional-order systems. Electronic filters are essential components of modern electronic systems. The order of a filter is characterized by the slope it offers in transition band. Fractional-order filters (FOF) are more general case of classical integer-order filters, with order of the filter (n + α), where n is the integer and α is the fractional part (0 < α < 1). FOF offers more precise control over the transitional slope between pass band and stop band, as the slope is given by −20(n+α) dB/decade, while for classical filters it is 20 dB/decade. Fractional order provides an extra degree of freedom, which increase flexibility of the design. In many applications, such as telecommunication, biomedical signal processing etc., the cascading of voltage-mode and current-mode filters necessitates the use of a voltage-to-current (V-I) converter. The total efficacy of the filter can be increased if signal processing can be combined with V-I converter interfacing. A transadmittance-mode filter is suitable for such applications. As OTA exhibits high impedance at both input and output terminals, it is an ideal choice for implementing trans-admittance mode (TAM) signal processing applications. Thus, a resistor-less, α-order TAM FOF utilizing two OTAs and one fractional-order capacitor is presented to improve the cascadibility between voltage-mode and current-mode filters. The electronic tuning of the proposed TAM FOF’s parameters is achieved through transconductance gain of OTA. There are two main methods to achieve the electronic tuning of filter’s parameter. xi xii Abstract The first method involves adjusting the transconductance/current/voltage gain of the constituent active elements. The second method utilizes controlling the gain of the external amplifier introduced in the feedback loop of the core filter. The later approach is known as the shadow concept, and filters employing this method are referred to as shadow filters. The theory of shadow filters, originally developed for integer-order filters, is generalized to fractional domain. Mathematical formulas are drafted for pole frequency and pole quality factor of the fractional-order shadow filters. The proposed theory is validated through OTA-based two active filters. Sinusoidal oscillators are extensively used in the field of communication, control systems, testing and measurements. Fractional-order oscillators (FOOs) offer the advantage of achieving higher oscillation frequencies compared to their integer-order counterparts, while still maintaining the same values of passive components. FOOs also offer arbitrary phase shifts between their output signals, providing added flexibility. OTA based three new sinusoidal FOOs and one fractional-order multivibrator are presented. The first two circuits of the sinusoidal FOO are designed using the trans-admittance mode FAPF with a trans-impedance mode integrator/differentiator topology. The third circuit of the sinusoidal FOO features a unique design that enables independent control of the phase difference between its two output voltages. Further, an electronically tunable fractional-order multivibrator based on OTA has been generalized to fractional domain. The mathematical formula for the time period has been derived using Reimann-Liouville fractional integral. As all the circuits of FOFs and FOOs, proposed in this thesis employs fractional-order elements (FOEs), specifically fractional-order capacitor (FOC), thus a compact design to approximate the behaviour of FOC based on active inductor is proposed. The circuit is modular in nature and allows for the higher order approximations through parallel connection or impedance multiplication to realize FOC. Furthermore, a circuit is presented that implements a floating version of the higher order FOE (1 < α < 2) using OTA based IIMC. The functionality of all the proposed structures has been verified through SPICE simulations with 180 nm CMOS technology parameters. Mathematical formulation for sensitivity of the proposed FOFs and FOOs is included. The robustness of the proposed FOFs and FOOs is also investigated through corner and Monte-Carlo analysis.