State-space analysis of the adaptive notch filter

The adaptive notch filter has a variety of applications in the field of communications including 60-Hz noise cancellation, phase-locked loops, frequency-hop filtering, and channel equalization. We present a simple derivation of the time-invariant state equations for the adaptive notch filter. Unlike previous work, this analysis provides a closed-form solution for the LMS weight vector as a rotated and filtered product of the desired response. This analysis also lends a simple explanation of non-Wiener type adaptive behavior.