All-pass based IIR multiple notch filter design using Gröbner Basis

In this paper, the digital IIR multiple notch filter based on an all-pass filter design of order 2N is considered. The most important step is about the calculation of the exact notch frequencies. The previous method requires solving of non-linear polynomial equation system which can be difficult to tackle especially for the case with higher number of notch frequencies. The analytical result was derived only for the case of N = 2 not for N = 3 and higher, then the main idea proposed in this paper is focused on solving of the non-linear systems for the cases of higher number of notch frequencies, e.g. N ≥ 3, by employing “Gröbner Basis” theory which has ability to solve multi-variance polynomial equation systems by using the benefit of “lexicographical orderings” in the polynomial rings to make those systems to be “triangular systems” which can be solved by backward substitution. Although the proposed filter design is a one dimensional but the proposed technique involved multi-variance polynomials. Another advantage is that the Gröbner Basis provides symbolic solutions which allow further optimization to satisfy additional constraints such as having minimum group delay.