A modified Euclidean algorithm for isolating periodicities from a sparse set of noisy measurements

A modified Euclidean algorithm is presented for determining the period from a sparse set of noisy measurements. The set may arise from measuring the occurrence time of noisy zero-crossings of a sinusoid with very many missing observations. The procedure is computationally simple, stable with respect to noise, and converges quickly. Its use is justified by a theorem that shows that, for a set of randomly chosen positive integers, the probability that they do not all share a common prime factor approaches one quickly as the cardinality of the set increases. Simulations are presented to demonstrate the proposed algorithm.