An Algebraic Approach to the Estimation of the Order of FIR Filters From Complete and Partial Magnitude and Phase Specifications

Abstract

Abstract-The problems addressed by this paper is the following: Given a set of measurements over the range of normalized frequencies ( 1 2 ) on the magnitude and/or phase of a real FIR but otherwise unknown filter, to estimate the order of the FIR filter. The range ( 1 2 ) may be partial or it may cover the entire range of frequencies. The purpose of the paper is to propose a new algebraic approach to solve the above collection of problems. Specific new results include FIR order estimation from partial or complete noiseless measurements for a real system from: a) phase alone, from b) magnitude alone (not necessarily piece-wise constant), and from c) joint magnitude and phase. The proposed approach is not only capable of dealing with specifications that go beyond the conventional formulas for the standard piece-wise-constant magnitude FIR filter order estimation, but it also furnishes a nexus for order estimation from phase (or group delay) specifications, areas which have remained hitherto unaddressed. The approach is based on the use of Root Moments. In this context, the novel concept of Fractional Root Moments is used in a key fashion to provide an estimate on the number of zeros inside the unit circle. Open problems and new directions of exploration and research are mentioned in the body of the paper.

Digital filter design using root moments for sum-of-all-pass structures from complete and partial specifications

“…Clearly, the procedures based on magnitude response are inapplicable [45], [47], [52], [65], [66]. The unwrapped phase response in the fundamental relationships of (54), evaluated at , yields immediately the all-pass order as (58) This is also a consequence of the argument principle in complex variable theory, and it essentially corresponds to the winding number of the FIR transfer function [67], [68].…”

“…The phase response of an all-pass transfer function also furnishes appropriate bounds [68] as indicated below.…”

“…The root with the largest modulus can furnish a bound on the order as follows [68]. We assume that we are examining a minimum-phase polynomial transfer function for which have its root moments.…”

“…There is an alternative, and more general route, to estimating the order, which relies on the estimation of the order of a polynomial from its root moments [68]. The root moments of the denominator of an all-pass filter are directly derived from (54), and the problem can now be posed as indicated below.…”

“…The various ramifications and solutions to it are dealt with in a forthcoming paper [68]. The pivotal observation is made that if the order is , then the Newton Identities alone can yield the root moment , obtainable from all the previous root moments and the estimated polynomial coefficients .…”