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

The problems addressed by this paper is the following: Given a set of measurements over the range of normalized frequencies (thetas₁ ,thetas₂) on the magnitude and/or phase of a real FIR but otherwise unknown filter, to estimate the order of the FIR filter. The range (thetas₁,thetas₂) 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