A statistically and computationally efficient method for frequency estimation

Traditional methods of estimating frequencies of sinusoids from noisy data include periodogram maximization and nonlinear least squares, which lead to efficient estimates with the rate O(n−3/2). To actually compute these estimates, some iterative search procedures have to be employed because of the high nonlinearity in the frequency parameters. The presence of many local extrema requires the search be started with a very good initial guess – the required precision is typically o(n−1), which is not readily available even from the fast Fourier transform of the data. To overcome these problems, we consider an alternative approach, the contraction-mapping (CM) method. Contributions of this paper include: (a) the establishment, for the first time, of the crucial connection between the accuracy of the initial guess required for convergence in the fixed-point iteration and the precision of the CM estimator as the fixed point of the iteration; (b) the quantification of the asymptotic relationship between the initial guess and the final CM estimator, together with limiting distributions and almost sure convergence of the fixed point; and (c) the construction of a single algorithm adaptable to possibly poor initial values without requiring separate procedures to provide initial guesses. It is shown that the CM algorithm, endowed with an adaptive regularization parameter, can accommodate possibly poor initial values of precision O(1) and converge to a final estimator whose precision is arbitrarily close to the optimal O(n−3/2).