Application of approximation theory methods to recursive digital filter design

In this paper, the ε-Herglotz tansform (or ε-outer function) is utilized to yield an analytic function\hat{H}_{\epsilon}(z)on the unit disk from the given amplitude characteristic of a digital filter. This is motivated by a two-fold objective: 1) the analytic function can be written in a power series expansion so that Padé approximants can be obtained and 2)\hat{H}_{\epsilon}(z)is a zero-free function in the Hardy space H²so that its polynomial least-squares inverse exists and does not vanish on the unit disk. Hence, we can immediately obtain recursive filters by Padé approximants and all-pole filters by means of least-squares inverses. There are two advantages in these methods: 1) we only need to solve linear systems of algebraic equations to obtain the filter coefficients; and 2) the phases of these filters exhibit almost linear in the passband, while the amplitude characteristics of these filters closely approximate the given one. If some adjustments are needed for the all-pole filters, we can use a standard least-squares technique to change an all-pole filter into a pole-zero filter. The detailed procedures are presented and the stability and approximation of these methods are discussed. Several examples are used for illustration purposes in this paper.