Srinivasa Ramanujan and signal-processing problems
Abstract
The Ramanujan sum c_q(n) has been used by mathematicians to derive many important infinite series expansions for arithmetic-functions in number theory. Interestingly, this sum has many properties which are attractive from the point of view of digital signal processing. One of these is that c_q(n) is periodic with period q, and another is that it is always integer-valued in spite of the presence of complex roots of unity in the definition. Engineers and physicists have in the past used the Ramanujan-sum to extract periodicity information from signals. In recent years, this idea has been developed further by introducing the concept of Ramanujan-subspaces. Based on this, Ramanujan dictionaries and filter banks have been developed, which are very useful to identify integer-valued periods in possibly complex-valued signals. This paper gives an overview of these developments from the view point of signal processing.
Additional Information
© 2019 The Author(s). Published by the Royal Society. Manuscript accepted 10/07/2019; Published online 09/12/2019; Published in print 01/2020. This article is part of a discussion meeting issue 'Srinivasa Ramanujan: in celebration of the centenary of his election as FRS'. The authors wish to thank Prof. Bhaskar Ramamurthi, Director of the Indian Institute of Technology, Chennai, for drawing the first author's attention to [51] many years ago. The authors learned about Ramanujan's sum because it was used in [51], although for a different purpose. Subsequent enthusiastic remarks on this work from Dr Gadiyar and Dr Padma are gratefully acknowledged. Data accessibility: This article has no additional data. Authors' contributions: P.P.V. introduced the idea of Ramanujan subspaces, studied their properties for periodic signals, and introduced the Farey dictionary for signal representation. S.T. further developed the idea of periodicity dictionaries and proved a number of results for them. S.T. also developed Ramanujan filter banks, established the minimum data length results, developed the applications mentioned and produced all computer simulation plots. Both authors read and approved the manuscript. We declare we have no competing interests. This work was supported in parts by the NSF grant no. CCF-1712633 and the ONR grant no. N00014-18-1-2390 of the USA, and an Amazon post doctoral fellowship facilitated through the Information Science and Technology (IST) initiative at the California Institute of Technology.Additional details
- Eprint ID
- 100363
- DOI
- 10.1098/rsta.2018.0446
- Resolver ID
- CaltechAUTHORS:20191218-151131784
- CCF-1712633
- NSF
- N00014-18-1-2390
- Office of Naval Research (ONR)
- Amazon
- Created
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2019-12-18Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field