Pole-zero placement algorithm for the design of digital filters with fractional-order rolloff

Highlights

• Lowpass filter coefficients are computed directly in the discrete domain.

• Filters are not subject to the error and stability issues possible in discretized analog designs.

• Filter cutoff frequency is accurate to an arbitrary level of precision.

• The proposed technique has lower mean-squared and minimum absolute error than existing approaches.

• Parameter estimation curves can be precomputed to ease the burden of the full iterative approach.

Abstract

Digital and analog filters are often used in the modeling of real-world systems. Many practical systems are better modeled by filters with fractional order rolloff. Finite integer-order transfer functions cannot represent fractional-order systems exactly. They can, however, approximate fractional-order systems, with the approximation quality depending on the order of the transfer functions and the methods used to design them. Several analog design techniques have been developed to realize and improve such approximations. To date, digital design techniques have been largely restricted to discretizations of existing analog solutions. This paper presents a novel approach to designing digital lowpass filters with fractional-order rolloff directly in the discrete domain through pole-zero placement. Filters designed using the proposed iterative technique are stable, have precisely definable cutoff frequencies, and do not suffer from the variations that can arise from transforming an existing analog design. The proposed technique is shown to outperform certain existing analog and digital design methods, both subjectively and by objective measures. To complement the proposed approach, a parameter-estimation routine is also introduced to alleviate some of the associated computational burden by reducing its reliance on iterative methods.

Introduction

Modeling real-world systems as analog or digital filters with specific transfer function characteristics has been a popular subject of research for years. Often complex systems can be reduced to or modeled by some combination of much more simple filter stages. Historically, these filters have been defined by finite, integer-order transfer functions. Such filters have an ideal transition band slope (rolloff) of ±6n decibels per octave (dB/oct) or ±20n dB per decade (dB/dec), with integer n. Yet the real world is rarely ideal, and many systems would be more accurately modeled with fractional-order rolloff filters. Fractional-order rolloff filters are required in such applications as underwater acoustics (to model ambient sea noise [1]) and speech intelligibility (to model the long-term spectrum of human speech [2]). Fractional integrators and differentiators have also been studied extensively.

The concept of extending integer-order mathematics to their fractional-order generalizations is not new. Fractional calculus, the study of generalizing integration and differentiation to fractional orders, dates back more than three centuries [3]. However, early research into fractional calculus was impossible due to the computational complexity involved. By the latter half of the 20th century, computational capability had advanced such that practical applications of fractional calculus could be investigated. Since exact fractional-order representations of systems would generally require systems of infinite order, much early research was devoted to finding finite-order approximations. An excellent review of these methods was provided by Vinagre et al. [4], including some of the more significant methods developed by Carlson [5], Roy [6], Oustaloup [7], Charef [8], and Matsuda [9].

The earliest practical applications of fractional-order systems were in analog circuitry. Several early papers, and even a few more recent ones, were devoted to this topic [6], [8], [10], [11], [12]. Much research has also been conducted into using fractional calculus in control theory to describe plants and controllers of fractional order [7], [13], [14], [15], [16], [17], [18]. Since highpass and lowpass filters can be viewed as band-limited differentiators and integrators, some researchers have investigated using fractional calculus concepts in signal processing and filter design [19], [20], [21], [22], [23], [24], [25], [26].

Since early work with fractional-order systems was focused on analog theory and implementations, all the finite-order approximations cited above were developed in the analog domain. With the advent of the digital age, the most straightforward digital implementations were those that “discretized” previously existing analog solutions. Several early papers proposed discretization routines [27], [28], [29], and this technique continues to maintain popularity today [30], [31], [32], [33], [34], [35]. Building on the success of discretization techniques, parameter optimization is now emerging as a very active area of research [36], [37]. While other methods have been introduced (such as the use of the so-called fractional delay operator [38]), this paper focuses its comparisons on discretization and optimization.

Fractional-order digital filters have already been designed, and continue to be designed. However, the literature currently neglects the problem of generating the digital filter coefficients directly in the discrete domain. Direct digital design is desirable, as it is often not possible to simultaneously preserve both the stability and frequency response of analog filters when they are discretized [39]. This paper proposes a method to design fractional-order digital filters directly in the discrete domain through pole-zero placement and subsequent parameter optimization. The poles and zeros are placed logarithmically on the real axis inside the unit circle to maximize performance and guarantee stability.

The remainder of this paper has the following organization. Section 2 provides technical background information, including an overview of selected existing design approaches. Section 3 presents the proposed pole-zero placement algorithm, along with guidelines for parameter selection and optimization. Section 4 examines the results of some example cases, and compares the proposed method with other approaches. Section 5 concludes the paper.