Ramanujan Sums as Derivatives and Applications

Abstract

In 1918 S. Ramanujan defined a family of trigonometric sum now known as Ramanujan sums. In the last few years, Ramanujan sums have inspired the signal processing community. In this paper, we have defined an operator termed here as Ramanujan operator. In this paper it has been proved that these operator possesses properties of first derivative and second derivative with a particular shift. Generalised multiplicative property and new method of computing Ramanujan sums are also derived in terms of interpolation.

Orthogonal Ramanujan Sums, its properties and Applications in Multiresolution Analysis

“…By construction each column of R N are pairwise orthogonal.Hence R N is invertible and β N = R −1 N x N . In an earlier work authors have shown that Ramanujan sequences are basically first order derivative [1]. Hence β 1 is smoothing coefficient and β N (N > 1) can be interpreted as finer details of a signal.…”

“…The great Indian mathematician S.Ramanujan defined a trigonometric sum [9] as c q (n) = q k=1 (k,q)=1 exp j2πkn q (1) where (k, q) = 1 implies that k and q are relatively prime. Various standard arithmetic functions like Mobius function µ(n), Euler's toient function φ(n),von Mangoldt function , Riemann-zeta function ζ(s) were represented using these Ramanujan sums.…”

“…Recently authors have shown application of Ramanujan sums in image processing in particular for finding edges and noise level estimation [1]. In this paper a new property of Ramanujan sums has been derived.…”

“…Planat et al showed that these sums can be used for analysis of long-period sequences and 1 f noise [4] [5]. M.Planat et al in [5] have shown that the result obtained in [4] was quite ambiguous.Authors [1] have recently shown that Ramanujan Sums behave as first derivative which gives more insight to Ramanujan Sums. P.P.…”

On the Relation Between Fourier Frequency and Period for Discrete Signals, and Series of Discrete Periodic Complex Exponentials

“…Ramanujan sums have been considered for application in Signal Processing, e.g. in [11] and [22]. In particular, they have been used to study the periodicity structure of streaming data [10], and also how the periodicity of a sampled continuous signal does not imply the periodicity of the continuous signal [9].…”