Signal analysis using Ramanujan and affine Ramanujan sums

Abstract

This thesis studies the mathematical foundations and signal processing applications of Ramanujan sums, Ramanujan subspaces, and Ramanujan filter banks, together with their affine generalizations for non-stationary signal analysis. Ramanujan sums provide an efficient framework for detecting and representing periodic structures in discrete-time signals due to their orthogonality and arithmetic properties. Using these properties, finite-dimensional Ramanujan subspaces and finite Ramanujan representations are developed for signal decomposition and periodicity analysis. The thesis further investigates Ramanujan filter banks constructed from Ramanujan sums and analyzes their frame-theoretic properties. In particular, sufficient conditions for frame and tight frame structures are established, ensuring stable signal representation and perfect reconstruction. These results demonstrate the usefulness of Ramanujan systems in signal denoising, reconstruction, and periodic component extraction. To address the limitations of classical Ramanujan systems for non-stationary signals, affine Ramanujan sums and affine Ramanujan Fourier transforms are introduced. Their orthogonality properties, recursive structures, and finite-dimensional representations are studied in detail. Affine Ramanujan subspaces and finite affine Ramanujan transforms are constructed for chirp and time-varying signal analysis. The thesis also discusses the potential development of affine Ramanujan filter banks for improved non-stationary signal processing applications.