Modified Pole Re-position Technique for Optimal IIR Multiple Notch Filter Design

Abstract

This paper presents a novel and useful technique based on pole optimal re-position techniques and Groebner basis theory for solving the gain non-uniformity problem in the frequency band between notch frequencies, normally emerged when cascading several single notch filters to form a multiple notch filter. The previously proposed pole-reposition technique can apply only to the two notch case and does not consider the minimization of the passband gain deviation. The optimality considered in this paper is based on the minimization of resulting filter magnitude response from the ideal notch magnitude response, measured in term of numerical sum of error square. Furthermore, by using GrÄobner basis, the symbolic solution and design method can be generalized to the higher number of notch frequencies case. The design algorithm can be implemented directly on Matlab platform and optimal ¯lter coe±cients can be found via numerical optimization/search. The proposed method can 1) limit the maximum passband gain 2) maximize the passband gain flatness and 3) generalize to the higher order of multi-notch filter. The searching space of the possible filters are maximized by increasing the number of free-variables, namely the gains at critical frequencies. In addition, the proposed technique allows the inclusion of stability margin of the IIR (in¯nite impulse response) notch filter as one of optimization objectives, resulting in a robustly stable filter.