Stability of second order time-varying constrained notch filters

The stability of second order time-varying constrained notch filters (TVCNF) which are capable of rejecting a sine signal with time-varying frequency is analysed. Two cases are considered: TVCNF₁, where one of the two coefficients of the filter is allowed to vary with time while the value of the second coefficient is kept constant; TVCNF₂, where both the coefficients can vary with time. It is shown that TVCNF₂ is always asymptotically stable while the TVCNF₁ possesses unstable regions. Thus for tracking a sine signal with time-varying frequency, TVCNF₂ is preferable to TVCNF₁.