Analysis of the Band-Pass and Notch Filter with Dynamic Damping of Fractional Order Including Discrete Models

Abstract

The paper presents analysis of the second order band-pass and notch filter with a dynamic damping factor βd of fractional order. Factor βd is given in the form of fractional differentiator of order a, i.e. βd=β/sa, where β and a are adjustable parameters. The aim of the paper is to exploit an extra degree of freedom of presented filters to achieve the desired filter specifications and obtain a desired response in the frequency and time domain. Shaping of the frequency response enables achieving a better phase response compared to the integer-order counterparts which is of great concern in many applications. For the implementation purpose, the paper presents a comparison of four discretization techniques: the Osutaloup’s Recursive Algorithm (ORA+Tustin), Continued Fractional Expansion (CFE+Tustin), Interpolation of Frequency Characteristic (IFC+Tustin) and recently proposed AutoRegressive with eXogenous input (ARX)-based direct discretization method.

Keywords

• Butterworth filter
• Discretization
• Fractional-order filter
• Fractional calculus
• Frequency response