Orthogonal Ramanujan Sums, its properties and Applications in Multiresolution Analysis

Abstract

Signal processing community has recently shown interest in Ramanujan sums, which were defined by S. Ramanujan in 1918. In this paper, we propose a family of sequences, termed here as orthogonal Ramanujan sums, $c^{k}_q(n)$ and also derive their various properties. We also prove two new properties of the Ramanujan Sums, namely, shifted multiplication property and interpolation property. These properties are used to define higher order orthogonal Ramanujan sums. These sums are periodic in nature and are orthogonal for different values of $k$ and $q$ . Using orthogonal Ramanujan sums, a new representation of finite length signal is given, termed here as orthogonal Ramanujan periodic transform. The coefficients obtained using this transform can be divided into two parts, first coefficient gives smoothing component and the rest give the detail components of the signal. Further, application of the orthogonal Ramanujan sums in discrete wavelet transform is developed. Construction of Ramanujan wavelet analysis matrix for applying discrete wavelet transform on a signal is described. The application of this matrix on an image of size $N\times N$ , where $N$ is divisible by integer $q$ will result in $q^2-1$ detail components and one average component in transform domain. Some of the applications of this transform are also demonstrated, especially in the field of image processing.