An Adaptive Estimation of Periodic Signals Using a Fourier Linear Combiner

Here, we present an adaptive algorithm for estimating from noisy observations, periodic signals of known period subject to transient disturbances. The estimator is based on the LMS algorithm and works by tracking the Fourier coefficients of the data. The estimator is analyzed for convergence, noise misadjustment and lag misadjustment for signals with both time invariant and time variant parameters. The analysis is greatly facilitated by a change of variable that results in a time invariant difference equation. At sufficiently small values of the LMS step size, the system is shown to exhibit decoupling with each Fourier component converging independently and uniformly. Detection of rapid transients in data with low signal to noise ratio can be improved by using larger step sizes for more prominent components of the estimated signal. An application of the Fourier estimator to estimation of brain evoked responses is included.