A unified approach to the design of IIR and FIR notch filters

This paper presents a unified, optimization-driven solution for designing IIR and FIR notch filters with prescribed, possibly varying notch levels in the given stop-bands, and near unit magnitude frequency response at the pass-bands. Although the original IIR notch filter optimization problem is non-convex, we show that it can be well approximated by a convex problem, by replacing a non-positive semi-definite 2 × 2 Hermitian matrix with its nearest positive semi-definite counterpart. With this approach, the IIR filter design can be efficiently solved via Newton iteration. The same approach can be directly applied to the FIR filter design since it is a degenerated case of the IIR filter. Moreover, we show that the FIR design problem is convex and therefore can be solved optimally. Numerical examples are presented to verify the effectiveness of the proposed design.