Number Theoretic Methods in Parameter Estimation
Casey, Stephen D.
Sadler, Brian M.
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Number theory has proven to be an effective tool in harmonic analysis, used to extend existing theories (e.g., sampling theory, fast transform computations) and develop new approaches to problems (e.g., interpolation). Number theoretic methods have also been successfully applied to the analysis of periodic point processes, leading to computationally straightforward algorithms for several parameter estimation problems.<P>We first present modifications of the Euclidean algorithm which determine the period from a sparse set of noisy measurements. The elements of the set are the noisy occurrence times of a periodic event with (perhaps very many) missing measurements. The approach is justified by a theorem which shows that, for a set of randomly chosen positive integers, the probability that they do not all share a common prime factor approaches on quickly as the cardinality of the set increases. A robust version is developed that is stable despite the presence of arbitrary outliers. We then use these algorithms in the analysis of periodic pulse trains, getting an estimate of the underlying period. This estimate, while not maximum likelihood, is used as initialization in a three-step algorithm that achieves the Cramer-Rao bound for moderate noise levels, as shown by comparing Monte Carlo results with the Cramer-Rao bounds. We close by discussing our work on deinterleaving. Here we discuss a variation on Weyl's Equidistribution Theorem, which works for noisy measurements. we then use periodogram-like operators in a multistep procedure to isolate fundamental periods.