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Published August 15, 2014 | public
Journal Article

Ramanujan Sums in the Context of Signal Processing—Part I: Fundamentals

Abstract

The famous mathematician S. Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). For any fixed integer q, this is a sequence in n with periodicity q. Ramanujan showed that many standard arithmetic functions in the theory of numbers, such as Euler's totient function φ(n) and the Möbius function μ(n), can be expressed as linear combinations of c_q(n), 1 ≤ q ≤ ∞. In the last ten years, Ramanujan sums have aroused some interest in signal processing. There is evidence that these sums can be used to extract periodic components in discrete-time signals. The purpose of this paper and the companion paper (Part II) is to develop this theory in detail. After a brief review of the properties of Ramanujan sums, the paper introduces a subspace called the Ramanujan subspace S_q and studies its properties in detail. For fixed q, the subspace S_q includes an entire family of signals with properties similar to c_q(n). These subspaces have a simple integer basis defined in terms of the Ramanujan sum c_q(n) and its circular shifts. The projection of arbitrary signals onto these subspaces can be calculated using only integer operations. Linear combinations of signals belonging to two or more such subspaces follows certain specific periodicity patterns, which makes it easy to identify periods. In the companion paper (Part II), it is shown that arbitrary finite duration signals can be decomposed into a finite sum of orthogonal projections onto Ramanujan subspaces.

Additional Information

© 2014 IEEE. Manuscript received January 10, 2014; revised April 20, 2014; accepted April 21, 2014. Date of publication June 18, 2014; date of current version July 18, 2014. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Hing Cheung So. This work was supported in parts by the ONR grant N00014-11-1-0676, and the Information Science and Technology (IST) initiative of the California Institute of Technology. The author wishes to thank Prof. B. Ramamurthi, Director of the Indian Institute of Technology, Chennai, for drawing my attention to [4] (and to [11], [12] via Dr. Gadiyar of the Vellore Institute of Technology). Subsequent enthusiastic remarks on this work from Dr. Gadiyar and Dr. Padma (VIT) are gratefully acknowledged.

Additional details

Created:
August 20, 2023
Modified:
October 17, 2023