Uncertainty principle for convolutional tight frames and its applications in signal recovery

Abstract

This paper studies the Donoho–Stark uncertainty principle in the context of convolutional tight frames arising from the analysis phase of filter banks in a finite-dimensional setting. Unlike the classical uncertainty principle, which constrains the sizes of a signal support and support of its Fourier transform, our approach replaces the support of the Fourier transform with the support of frame coefficients, linking it to the problem of signal recovery after erasures. We refine this principle using restriction estimates from classical restriction theory and demonstrate its applications in recovering signals from lost frame coefficients or noisy observations. This study further explores the uncertainty principle and signal recovery conditions for Dirac combs, where their support is replaced with a concentrated set. Finally, we present numerical experiments illustrating signal recovery performance when theoretical conditions are not met, and we further enhance the recovery process using the properties of Ramanujan sums. © 2025 Elsevier B.V., All rights reserved.