Multiple Frequency Estimation in Mixed-Spectrum Time Series by Parametric Filtering
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A general parametric filtering procedure (the PF method) is proposed for the problem of multiple frequency estimation in mixed-spectrum times series (i.e., superimposed sinusoids in additive noise). The method is based on the fact that a sum of sinusoids satisfies an homogeneous autoregressive (AR) equation. The gist of the method is to parametrize a linear filter so that it possesses a certain parametrization property as suggested by the particular form of the bias encountered by Prony's (least squares) estimator. For any parametric filter with this property, in addition to some mild regularity conditions, the least squares estimator from the filtered data, as a function of the filter parameter, constitutes a contractive mapping - whose multivariate fixed-point serves as a consistent AR estimator. The chronic bias of Prony's estimator is thus eliminated. Coupled with the all- pole (AR) filter endowed with an extra bandwidth parameter, the PF method can achieve the accuracy of nonlinear least squares by a simple iterative procedure consisting of linear least squares estimation followed by linear recursive filtering. Crude initial guesses such as those from Prony's estimator are sufficient to initiate the iteration. The method is also capable of resolving closely-spaced frequencies which are unresolvable by periodogram analysis or DFT.<P>To analyze the statistical properties of the PF method, some classical asymptotic results concerning the sample autocovariances are extended to accommodate mixed-spectrum time series and parametric filtering. In particular, under regularity conditions, uniform strong consistency and asymptotic normality are proved for the sample autocovariances of a mixed- spectrum time series after parametric filtering. Equipped with these results, some statistical properties of the PF method itself are investigated. These include the existence of the PF estimator as a fixed-point of the parametric least squares mapping, the convergence of an iterative algorithm that calculates the PF estimator, as well as the strong consistency and asymptotic normality of the PF estimator.<P>Computer simulations are also presented to demonstrate the effectiveness of the PF method. Direction for future research are briefly discussed at the end of the dissertation.