Ramanujan sums in signal recovery and uncertainty principle inequalities
This paper explores the perfect reconstruction property of filter banks based on Ramanujan sums and their applications in signal recovery. Originally introduced by Srinivasa Ramanujan, Ramanujan sums serve as powerful tools for extracting periodic components from signals and form the foundation of Ramanujan filter banks. We investigate the perfect reconstruction property of these filter banks and analyze their robustness against erasures for discrete-time signals in a finite-dimensional space $\mathbb C^N$ . The study is further extended to non-uniform Ramanujan filter banks, showcasing their ability to address the limitations of uniform ones. Employing the reconstruction properties of uniform Ramanujan filter banks, we present an uncertainty principle associated with a tight frame of shifts of Ramanujan sums. This principle establishes representation inequalities in terms of Euler's totient function that provide sufficient conditions for the perfect recovery of signals in scenarios where signal information is lost during transmission or corrupted by noise. Finally, we illustrate that utilizing the signal's periodicity information through Ramanujan filter banks significantly improves the efficiency of signal recovery optimization algorithms, resulting in enhanced signal-to-noise ratio (SNR) gains and more precise reconstruction.