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Published August 15, 2014 | public
Journal Article

Ramanujan Sums in the Context of Signal Processing—Part II: FIR Representations and Applications

Abstract

The mathematician Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). In a companion paper (Part I), properties of Ramanujan sums were reviewed, and Ramanujan subspaces S_q introduced, of which the Ramanujan sum is a member. In this paper, the problem of representing finite duration (FIR) signals based on Ramanujan sums and spaces is considered. First, it is shown that the traditional way to solve for the expansion coefficients in the Ramanujan-sum expansion does not work in the FIR case. Two solutions are then developed. The first one is based on a linear combination of the first N Ramanujan-sums (with N being the length of the signal). The second solution is based on Ramanujan subspaces. With q_1, q_2,..., q_K denoting the divisors of N; it is shown that x(n) can be written as a sum of K signals x_(qi) (n) ∈ S_(qi). Furthermore, the i_(th) signal x_(qi) (n) has period q_i, and any pair of these periodic components is orthogonal. The components x_(qi) (n) can be calculated as orthogonal projections of x(n) onto Ramanujan spaces S_(qi). Then, the Ramanujan Periodic Transform (RPT) is defined based on this, and is useful to identify hidden periodicities. It is shown that the projection matrices (which compute x_(qi) (n) from x(n)) are integer matrices except for an overall scale factor. The calculation of projections is therefore rendered easy. To estimate internal periods N_∞ <; N of x(n), one only needs to know which projection energies are nonzero.

Additional Information

© 2014 IEEE. Manuscript received January 10, 2014; revised April 20, 2014 and June 13, 2014; accepted June 13, 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 Caltech.

Additional details

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