CIF:Small: Ramanujan-sums in signal representation

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This research explores new directions in the representation of signals, based on certainmathematical ideas introduced by Ramanujan many decades ago. In particular the applicability ofRamanujan-sums in the representation of digital signals is studied in detail. Signals acquired andstored in many scientific areas such as speech, music, medical, genomic, and finance oftencontain hidden periodic patterns bearing important information. These are difficult to identify andlocalize because of the large volume of data, and extreme localization of information.Conventional representations of signals based on Fourier and wavelet transforms are not efficientfor this purpose. This research involves the use of Ramanujan-sums to develop newrepresentations that are efficient and economical. These offer insights into the role ofmathematics, especially number theory, in representation of signals in science and technology,with emphasis on particular hidden structures such as periodicities. Such representations areexpected to have significant impact in scientific applications involving large data bases.More specifically, the research goes deep into Ramanujan subspaces, nested periodic subspaces,and Ramanujan filter banks, which are crucial to the representation of periodic sequences. Thisenables one to address some fundamental research problems in signal processing. This includesthe minimum information that should be gathered from a discrete-time signal in order to identifymultiple periodicities, the design of digital filter banks that serve as projection operators ontoperiodicity subspaces, and optimal design of dictionaries of atoms to represent sums of periodicsignals with unknown integer periods. The research also involves the theory and implementationof gridless methods to achieve super resolution in period-estimation for continuous-time signals,and extension to two- and higher-dimensional signals.