A study of techniques for finding the zeros of linear phase FIR digital filters

Since the majority of the zeros of standard finite-duration impulse-response (FIR) linear phase digital filters (e.g., low pass, band-pass, etc.) are located on the unit circle in the z-plane, it is possible to exploit this information in devising an efficient method for accurately solving for the locations of these zeros. To perform this task, three standard algorithms for finding the roots of a polynomial were evaluated. These methods were the bisection method, the modified false position method, and the Newton-Raphson method. Using a convergence criterion based on the value of the function (rather than the uncertainty in the position of the root), it was experimentally found that the Newton-Raphson method was the most accurate in determining the location of the roots, as well as being the most computationally efficient of the three methods. A second study was made to compare methods for determining all the zeros of the filter (i.e., the zeros off the unit circle, as well as those on the unit circle). Two sophisticated algorithms (the Jenkins-Traub Three-Stage Algorithm, and the Madsen-Reid Algorithm based on Newton´s method) were used in this study as well as a deflation method in which the unit circle roots were first located, and then used to form a deflated polynomial which was used to find the roots which occurred off the unit circle. It was found that for polynomials of degree greater than about 100, both the Jenkins-Traub and Madsen-Reid methods were far superior (in terms of accuracy in locating the roots off the unit circle) to the deflation method because of inaccuracies incurred in deflating high-order polynomials. It was also found that the accuracies of both the Jenkins-Traub and Madsen-Reid methods were comparable; however, the Madsen-Reid method was between two and four times faster than the Jenkins-Traub method for the examples tested.