Matrix-Based Ramanujan-Sums Transforms

In this letter, we study the Ramanujan Sums (RS) transform by means of matrix multiplication. The RS are orthogonal 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 literature for non-even function
$ ({\rm mod}~ {\rm M}) $
. We solved this problem by using matrix multiplication in this letter.