Adaptive Frequency Estimation and New Convergence Properties for the Least Mean Square Algorithm

Book excerpt: The convergence properties of the least mean square (LMS) algorithm are interpreted in terms of a vector space associated with the coefficients of the adaptive linear prediction filter (ALPF). Signal planes defined in this weight vector space are used to describe the frequency tracking by characteristics of spectral estimators based on the ALPF and are used to explain the effects of both the filter parameters and the algorithm on tracking speed. The performances of three different adaptive frequency estimators derived from the ALPF are compared. Two of these employ Fourier transforms of the coefficients and the third is based on a transform of the ALPF output. Comparisons with the conventional periodogram spectrum estimator are presented in terms of a signal-to-noise ratio (SNR) defined in frequency domain parameters. Specific calculations for one ALPF frequency estimator (The maximum entropy estimator) are used to demonstrate a bias in this estimator.