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In this paper, a novel and simple method based on an all-pass filter is presented by using the benefits of a pole re-position technique. The gains at frequencies of π of each single notch filter are adjusted by adding N − 1 tuning variables. The proposed method ensures that the passband gains are uniformly flat, the notch frequencies exactly meet the specifications, and the realized 3 dB bandwidths are approximately the same as those specified. This technique is very useful for designers because there are only N − 1 tuning variables required to adjust, and it does not need a complicated mathematical calculation. Although this technique is simple and easy to implement, it requires many iterations to find a suitable parameter for making the passband gain between two notch frequencies uniformly flat. Hence in this paper, three searching algorithms have been applied to reduce the number of iterations. We obtain a faster search and a closer frequency response to the ideal one.

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.