Sinusoidal interference rejection analysis of an LMS adaptive feedforward controller with a noisy periodic reference

This paper applies a new vector subspace model to determine the non-Wiener solutions of the LMS algorithm when the reference input is an arbitrary noisy periodic signal. The LMS weights are modeled as a deterministic time-varying mean plus a zero-mean fluctuating part. For the mean weight, each harmonic component of the periodic reference is shown to excite only a two-dimensional (2-D) subspace of the N-dimensional tap weight space. The fluctuating part of the weight is due to the reference noise that excites the full N-dimensional space. The power spectrum of the error is computed using the correlation function of the weight fluctuations. The error is shown to be primarily the sum of a white noise and the canceled periodic residual for an independent noise vector model. When the effects of the tapped delay line are incorporated in the model, the noise becomes colored with a spectrum centered about the canceled sinusoidal frequency. This weight model significantly simplifies the understanding of the non-Wiener behavior and can be applied to active noise cancellation problems when filters appear in the cancellation loop.