COMPLEMENTARY POLE-PAIR FILTERS AND POLE-PARAMETER TRANSFORMATIONS

Abstract

It is well known that the poles of a Butterworth filter (BF) are uniformly placed on the unit circle in the s-plane. If the order of the filter is a binary power (i.e., n=2k, k being an integer), then the pole-locations of such Butterworth filters exhibit very interesting symmetry properties in the s-plane. In this paper, first we point out that the pole-vectors of the Butterworth filters (of order n=2k) can be subjected to “prescribed symmetrical swinging”, such that certain symmetry properties present in the original pole-pattern can be maintained invariant. Then, we introduce a new family of filters called “Complementary Pole-Pair Filters (CPPFs)”, generated by judiciously exploiting the symmetry-invariant property referred to above. The Q-constraint can be easily incorporated into the design considerations of the new family of filters; accordingly, the CPPFs are classified as Low-Q filters (LQFs) and High-Q Filters (HQFs). Performances of LQFs and HQFs are analyzed and compared with that of the generic Butterworth filter. Also, the possibility of generating additional transitional filters is indicated. Design constraints are derived. Numerical examples are worked out for illustrations. A new transformation referred to as the “Pole-Parameter Transformation (PPT)”, which modifies the pole-parameters of a given reference filter to obtain a low-Qp filter, but with increased order of complexity, is formulated. This enables us to generate new families of Transitional Filters. The transitional nature involved between Butterworth and Chebyshev filters is brought out. The performance of new filters obtained are compared with those available in the literature. The CPP Filters coupled with the PPT provide a basis for generating a large family of transitional filters, and serves to commendably place the known classical filters as particular members of a large general family.