This research explores new directions in the representation of signals, based on certain mathematical ideas introduced by Ramanujan many decades ago. In particular the applicability of Ramanujan-sums in the representation of digital signals is studied in detail. Signals acquired and stored in many scientific areas such as speech, music, medical, genomic, and finance often contain hidden periodic patterns bearing important information. These are difficult to identify and localize because of the large volume of data, and extreme localization of information. Conventional representations of signals based on Fourier and wavelet transforms are not efficient for this purpose. This research involves the use of Ramanujan-sums to develop new representations that are efficient and economical. These offer insights into the role of mathematics, especially number theory, in representation of signals in science and technology, with emphasis on particular hidden structures such as periodicities. Such representations are expected 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. This enables one to address some fundamental research problems in signal processing. This includes the minimum information that should be gathered from a discrete-time signal in order to identify multiple periodicities, the design of digital filter banks that serve as projection operators onto periodicity subspaces, and optimal design of dictionaries of atoms to represent sums of periodic signals with unknown integer periods. The research also involves the theory and implementation of gridless methods to achieve super resolution in period-estimation for continuous-time signals, and extension to two- and higher-dimensional signals.
