IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 10, OCTOBER 2013 941 Matrix-Based Ramanujan-Sums Transforms Guangyi Chen, Sridhar Krishnan, and Tien D. Bui Abstract—In this letter, we study the Ramanujan Sums (RS) transform by means of matrix multiplication. The RS are orthog- onal in nature and therefore offer excellent energy conservation capability. The 1-D and 2-D forward RS transforms are easy to calculate, but their inverse transforms are not defined in the lit- erature for non-even function . We solved this problem by using matrix multiplication in this letter. Index Terms—Fourier transform (FT), Gaussian white noise, Ramanujan Sums (RS). I. INTRODUCTION T HE Ramanujan Sums (RS) were proposed by S. Ramanujan in 1918 [1], and were applied to time-fre- quency analysis, signal processing, moment invariants, and shape recognition recently ([2]–[9]). The RS are orthogonal in nature and therefore offer excellent energy conservation, similar to the Fourier transform (FT). The RS are operated on integers and hence can obtain a reduced quantization error implementation. Even though the RS transform has so many important properties, it does not have the inverse RS transform for non-even function signals. In this letter, we analyse the RS transform by means of ma- trix multiplication, which can invert the RS transform easily. We derive both the forward and inverse RS transforms for 1-D signals and 2-D images. A few examples are also tested and our method can recover the 1-D signals and 2-D images perfectly without any errors. The organization of this letter is as follows. Section II presents a short review of the RS transform and proposes the matrix-based RS transforms for 1-D signals and 2-D images. The inverse RS transforms can recover the signals and images perfectly without any errors. Finally, Section III concludes the letter and proposes future research directions about the RS transform. Manuscript received May 31, 2013; revised July 10, 2013; accepted July 16, 2013. Date of publication July 18, 2013; date of current version July 25, 2013. This work was supported by research grants from the Natural Sciences and En- gineering Research Council of Canada (NSERC). The associate editor coordi- nating the review of this manuscript and approving it for publication was Dr. Zhu Liu. G. Chen is with the Department of Electrical and Computer Engineering, Ry- erson University, Toronto, ON Canada M5B 2K3, and also with the Department of Computer Science and Software Engineering, Concordia University, Mon- treal, QC Canada H3G 1M8 (e-mail: guang_c@cse.concordia.ca). S. Krishnan is with the Department of Electrical and Computer En- gineering, Ryerson University, Toronto, ON Canada M5B 2K3 (e-mail: krishnan@ee.ryerson.ca). T. D. Bui is with the Department of Computer Science and Software Engineering, Concordia University, Montreal, QC Canada H3G 1M8 (e-mail: bui@cse.concordia). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2013.2273973 II. MATRIX-BASED RS TRANSFORM The RS transform has been used as means of representing arithmetical functions by an infinite series expansion. The basis of this transform is the building block of number-theoretic func- tions. The RS are sums of the powers of primitive roots of unity, defined as (1) where means that the greatest common divisor (GCD) is unity, i.e., and are co-prime. An alternate compu- tation of RS can be given as (2) Let ( prime). Then, we have . The Möbius function is equal to 0 if contains a square number; 1 if ; and if is a product of distinct prime numbers. We tabulate with in Table I in this letter. The RS have the following multiplicative property: (3) and the orthogonal property: (4a) (4b) We can also derive by using Euler’s formula and basic trigonometric identities. 1070-9908/$31.00 © 2013 IEEE Authorized licensed use limited to: Ryerson University Library. Downloaded on October 28,2022 at 13:25:37 UTC from IEEE Xplore. Restrictions apply.