### Browsing by Author "Casey, Stephen D."

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Item Modifications of the Euclidean Algorithm for Isolating Periodicities from a Sparse Set of Noisy Measurements(1995) Casey, Stephen D.; Sadler, Brian M.; ISRModifications of the Euclidean algorithm are presented for determining 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. This problem arises in radar pulse repetition interval (PRI) analysis, in bit synchronization in communications, and other scenarios. The proposed algorithms are computationally straightforward and converge quickly. A robust version is developed that is stable despite the presence of arbitrary outliers. The Euclidean algorithm 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 one quickly as the cardinality of the set increases. In the noise-free case this implies convergence with only 10 data samples, independent of the percentage of missing measurements. In the case of noisy data simulation results show, for example, good estimation of the period from 100 data samples with 50 percent of the measurements missing and 25 percent of the data samples being arbitrary outliers.Item Modulation and Sampling Techniques for Multichannel Deconvolution(1996) Casey, Stephen D.; ISRThe problem recovering information from single component linear translation invariant systems is inherently ill-posedness may be circumvented in a multichannel system if the components of the system satisfy the strongly coprime condition. The signal may be completely recovered from a strongly coprime system by filtering the output of each channel with a deconvolver, and adding. This approach has been labeled multichannel deconvolution.We explore the state of the art for the multichannel theory. We then use sampling theory to develop an alternative method for creating these systems. Modulation techniques are then used to create a strongly coprime system, for which the corresponding deconvolvers are developed. We close by discussing several applications of the theory.

Item Number Theoretic Methods in Parameter Estimation(1996) Casey, Stephen D.; Sadler, Brian M.; ISRNumber 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.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.

Item On Periodic Pulse Interval Analysis with Outliers and Missing Observations(1996) Sadler, Brian M.; Casey, Stephen D.; ISRAnalysis of periodic pulse trains based on time of arrival is considered, with perhaps very many missing observations and contaminated data. A period estimator is developed based on a modified Euclidean algorithm. This algorithm is a computationally simple, robust method for estimating the greatest common divisor of a noisy contaminated data set. The resulting 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. An extension using multiple independent data records is also developed that overcomes high levels of contamination.Item Using Dimension Theory to Analyze and Classify the Generation of Fractal Sets(1996) Casey, Stephen D.; ISRThis article discusses the interplay in fractal geometry occurring between computer programs for developing (approximations of) fractal sets and the underlying dimension theory. The computer is ideally suited to implement the recursive algorithms needed to create these sets, thus giving us a laboratory for studying fractals and their corresponding dimensions. Moreover, this interaction between theory and procedure goes both ways. Dimension theory can be used to classify and understand fractal sets. This allows us, given a fixed generating pattern, to describe the resultant images produced by various programs. Thus, dimension theory can be used as a tool which enables to predict and classify behavior of certain fractal generating algorithms.We also tie these two perspectives in with some of the history of the subject. Four examples of fractal sets developed around the turn of the century are introduced and studied from both classical and modern viewpoints. Then, definitions and sample calculations of fractal and Hausdorff-Besicovitch dimension are given. We discuss various methods for extracting dimension from given fractal sets. Finally, dimension theory is used to classify images