Fractional-order band-pass filter design using fractional-characteristic specimen functions

Abstract

Two (α + β)-order transfer functions realizing fractional-order band-pass filter responses are presented and analysed, where 0 < α ≤ 1 and 0 < β ≤ 1. Generalizing the transfer functions from integer to fractional order offers continuous and independent control of the magnitude characteristic slopes both below and above the filter center frequency. The transfer function coefficients that minimize the magnitude error between the discussed fractional-order transfer functions and proposed fractional-characteristic specimen functions in a specified frequency band are determined using a numerical least squares optimization search routine. The relations for computing the transfer function coefficients as functions of α and β are given. The simulations show that the magnitude frequency characteristics of the transfer functions with the new coefficients are very well matched with the specimen functions. Further verification of these results was carried out using the Tow-Thomas filter topology with RC approximations of fractional-order capacitors. The computation of filter element parameters is demonstrated for selected values of α and β and the simulated and measured frequency characteristics of the filter are included to validate the fractional-order responses.

Introduction

Recently, research targeting fractional-order (FO) systems (i.e. systems described by differential equations containing derivatives of non-integer order) has gained interest in many branches of science and engineering. These branches include electronics and signal processing, thermodynamics, biology, electrochemistry, medicine, mechanics, control theory, nanotechnologies, and finances [1,2] to name a few. This interest is due to the fact that FO systems offer greater flexibility in terms of realizable system characteristics, compared to their integer-order counterparts. This greater flexibility often results in FO systems with lower order than their integer-order counterparts that are able to accurately model various phenomena of the world around us.

While analog frequency filters are traditionally of integer order, they can be also designed with a fractional order [[3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]]. This provides more arbitrary filter characteristics, including particularly the control of the slope of the magnitude frequency response [3]. For example, in the case of FO low-pass (LP) and high-pass (HP) filters the stopband slopes are −20 (n + α) dB/dec and +20 (n + α) dB/dec, respectively, where n is the integer order and 0 < α < 1 is the fractional component. This increases the range of attenuation values in the filter magnitude response that are easily realizable using FO filters over their integer-order counterparts.

During the design of any analog filter it is important to determine the transfer function coefficients that yield the desired filter characteristics, i.e. magnitude, phase, or delay. Systematic design procedures are widely available for integer-order filters to determine these coefficients, but similar procedures to design FO filters are not common yet. Though several methods have been investigated and published to deal with this issue. These design procedures are usually based on approximating a classic integer-order filter magnitude response (Butterworth [3], Chebyshev [4], Inverse-Chebyshev [5], Elliptic [6], arbitrary quality factor [7]) by the FO filter transfer function; searching for the coefficients that minimize the error between the FO filter and the integer-order response over a selected frequency band. While this approach has been successful, these works have focused largely on LP transfer functions, with little attention on the investigation of other kinds of FO filter transfer functions, such as HP or band-pass (BP).

In Ref. [8] the coefficients solely for FO BP transfer functions are obtained using both analytical and numerical methods. However, the analytical method is applied to a particular FO transfer function that cannot be implemented by a circuit that stems from a second-order active BP filter with both traditional capacitors replaced by FO elements with capacitive impedance, as is the case of this paper. Moreover, the authors of [8] claim that the analytic approach does not always result in successful coefficient finding. The numerical approach can be used for setting of parameters of FO BP filter based on parallel connection of two FO LP filters with opposite gains, however the resulting coefficient values are provided in Ref. [8] only for one specific FO characteristic.

An additional approach to the design of a FO BP filter was presented in Ref. [9], utilizing a fractional-order parallel resonator circuit. This topology is characterized by a narrow pass-band and high quality factor (Q > 220 in the hardware realizations), with both resonant frequency and quality factor that can be independently tuned. These filters also possess asymmetric attenuation characteristics that can be tuned using the fractional-order of the resonator topology components. This highlights the potential of fractional-order filters to be practically implemented and with tunability. Additionally, there is still a need to expand the processes to design and realize fractional-order filters to achieve characteristics with both high and low quality factors which requires further exploration of transfer function characteristics and circuit topologies.

In the works that have investigated FO HP and BP filters [10,11], the FO transfer functions have been obtained after applying transformations and using the coefficients determined for the LP function. While this does realize FO HP and BP filters, in our recent analysis discussed in Ref. [12] we show that the properties of the FO HP and BP transfer function (e.g. pass-band ripple of magnitude response, characteristic frequency, pass-band gain, etc.) obtained in this way may not exactly correspond to the original FO LP function. As shown in Ref. [12], in the case of FO HP filter the only correct FO LP to HP transformation that maintains the FO LP properties is the standard substitution of s by 1/s in the prototype transfer function. The HP characteristic is then exactly symmetrical to the LP response with regard to the frequency 1 rad/s. In the case of FO BP filter, the traditional integer-order transformation can also be used; replacing s in FO LP function with (s²+1)/(s·B), where B is the desired bandwidth (in rad/s). This transformation maintains the original LP filter properties, however it leads to doubling the filter order (e.g. the order between one and two increases to the order between two and four), increased circuit complexity and symmetrical slopes of magnitude frequency characteristics below and above center frequency, although these slopes can be set continuously.

For completeness' sake it should be noted that in many works that deal with FO BP filters (for example [[13], [14], [15], [16]]), the coefficients of the transfer functions are determined without prior definition of the desired filter specifications.

From the above it is obvious that there is currently an absence of practical methods for the systematic design of FO BP transfer functions based on predetermined properties, which is the motivation for this paper. In this work, FO BP filters of order (α + β) composed of two fractional components, with 0 < α ≤ 1 and 0 < β ≤ 1, are investigated. These transfer functions allow continuous and independent adjustment of the magnitude characteristic slopes, from 0 to +20 dB/dec below and 0 to −20 dB/dec above the center frequency. Here, we present the transfer function coefficients to achieve desired stopband characteristics with a flat passband for two general FO transfer functions.

The paper is organized as follows: Section 2 introduces two FO BP transfer functions that are utilized to achieve the filter responses with asymmetric stopbands characteristics. Section 3 describes the optimization procedure applied to determine the coefficients necessary for the FO BP transfer functions to reproduce the fractional-characteristic specimen function and gives the relations for computing them. The stability analysis is also included here to verify that the transfer functions with the determined coefficients are always stable. The simulations that numerically evaluate the conformity of the transfer functions with the new coefficients and the specimen function are presented in Section 4 and in the Appendix. The discussion of these results is in Section 5. Section 6 contains the design, simulation and experimental verification of FO Tow-Thomas filter to validate the previous assumptions and Section 7 concludes the paper.