The Analysis of Periodic Point Processes (Pi, the Primes, Periodicities, and Probability)

Our talk addresses the problems of extracting information from periodic point processes. These problems arise in numerous situations, from radar pulse repetition interval analysis to bit synchronization in communication systems. We divide our analysis into two cases - periodic processes created by a single source, and those processes created by several sources. We wish to extract the fundamental period of the generators, and, in the second case, to deinterleave the processes.

We first present very efficient algorithm for extracting the fundamental period from a set of sparse and noisy observations of a single source periodic process. The procedure is computationally straightforward, stable with respect to noise and converges quickly. Its use 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. The proof of this theorem rests on a probabilistic interpretation of the Riemann zeta function. We then build upon this procedure to deinterleave and then analyze data from multiple periodic processes. This relies on the probabilistic interpretation of the Riemann zeta function, the equidistribution theorem of Weyl, and techniques from spectrum analysis. We close by demonstrating simulations of the procedures, which were developed jointly by the speaker and Kevin Duke of American University.

Speaker Bio: Professor Stephen D. Casey is founding member of the Editorial Board for Sampling Theory in Signal and Image Processing. His research is in complex analysis, harmonic analysis, and number theory with applications to signal and image processing. He has been a Visiting Research Professor at the Institute for Systems Research (ISR) at the University of Maryland, and was a Visiting Research Professor at the Norbert Wiener Center for Harmonic Analysis (NWC) at the University of Maryland. He was recently invited to be an Associate Editor of The Journal of Signal and Image Processing. He has published nearly fifty articles, and has given over one hundred talks. His research has been funded by nineteen research grants and four contracts, including grants from the National Science Foundation, the Office of Naval Research, the Army Research Office and the Air Force Office of Scientific Research. He was awarded two provisional patents for his work in adaptive signal processing. A full patent is under review.

Presentation Slides: PDF

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