Strong Consistency of the Contraction Mapping Method for Frequency Estimation.

摘要

Consider the stationary process y sub t =beta cos(omega sub 0 t + phi) + epsilon sub t, and a parametric filter L sub alpha, and let rho(alpha) be the first-order antocorrdation of the filtered process (L sub alpha (y) sub t). Under a certain assnmption on the filtered noise spectrum, rho(alpha) is contractive at cos omega sub 0. It is shown that the sample estimate of rho(alpha), denoted by rho-dot(alpha) and obtained from a finite sampte of length n, has with probability one a fixed point alpha-dot sub n in a neighborhood of cos omegas sub 0, and that the sequence of fixed points (&n) converges with probability one to cosw0. The proof is based on a general result regarding the uniform consistency of the sample autocorrelation. The developed theory is illustrated by two nnmerical examples pertaining to two different parametric time invariant filters.