A nonlinear error adaptive notch filter for separating two sinusoidal signals

For a primary input consisting of two sinusoids, the adaptive notch filter coefficients have sinusoidally varying components in steady-state which reduce the rejection at the notch frequency due to weight misadjustment. The authors present a simple modification of the adaptive notch filter that removes these sinusoidal variations, improves rejection, and enhances tracking performance. Using a general theory of non-mean-square error stochastic gradient adaptation, they show that the optimum nonlinear error algorithm adapts the system only when the instantaneous error magnitude is greater than the amplitude of the interfering sinusoid. This amplitude can be easily estimated at the output of the notch filter when the system nears convergence. To follow any slow changes in the interfering sinusoid amplitude, the authors introduce a time-varying error criterion to keep the algorithm optimal. Simulations show a 6-dB reduction of the excess mean-square error, a 10-13-dB improvement in rejection at the filter outputs, and an increase in adaptation speed near the solution over the least-mean-square adaptive notch filter when using this modified structure with a time-varying nonlinearity