The least-square design of minimum-order allpass-based infinite impulse response multi-notch filters

Abstract

The paper investigates the least-square design of minimum-order infinite impulse response multi-notch filters. Monotonically decreasing nature of the stable allpass filter's phase response, characterizing the allpass-based multi-notch filter, along with the simple relation between coefficients and the phase response of the allpass filter, allows the formulation of the multi-notch filter's magnitude response specifications as the linear equality constraints regarding the positions of notch, left- and right-hand cutoff frequencies. Since the number of such constraints equals triple the number of notch frequencies, while the number of unknown filter coefficients equals double the number of notch frequencies, variable elimination is first performed to ensure that specifications regarding the notch frequencies positions are strictly satisfied. The remaining overdetermined system of linear equations is then solved in the least-square sense, leading to the approximate satisfaction of left- and right-hand cutoff frequencies positions. Proposed design method is also compared with some of the existing infinite impulse response multi-notch filter design methods.