Design of nearly linear-phase double notch digital filters with close notch frequencies

In this study, the authors present a new procedure for the synthesis of the nearly linear-phase infinite impulse response (IIR) filters with two close notch frequencies based on all-pass filter application. The filter is obtained using a parallel connection of the delay line and IIR all-pass filter with a piecewise approximately linear phase. The desired magnitude characteristic is achieved by equiripple approximation of the all-pass filter phase. The magnitude characteristic is equiripple in all passbands. The proposed method is also suitable for the synthesis of complementary filters for extraction of the signal components at prescribed frequencies. The initial solution for the arbitrary order of an all-pass filter is proposed, which attains a sufficient number of phase error extrema. All phase error minima and maxima corresponding to the initial solution have almost identical values, excluding the ones near notch frequencies. As a result, the method exhibits fast and strong convergence. The described method can also be applied for the design of multiple notch filters with two very close notch frequencies.