Stability in unlimited sampling
This addresses signal reconstruction issues in analog-to-digital conversion, providing theoretical support for folded sampling under practical constraints, though it is incremental as it builds on prior work on injectivity and encoding.
The paper tackled the instability of reconstructing bandlimited functions from folded samples in analog-to-digital converters, showing that equispaced sampling is inherently unstable, but stability is restored by imposing an energy bound, which extends to non-uniform sampling.
Folded sampling replaces clipping in analog-to-digital converters by reducing samples modulo a threshold, thereby avoiding saturation artifacts. We study the reconstruction of bandlimited functions from folded samples and show that, for equispaced sampling patterns, the recovery problem is inherently unstable. We then prove that imposing any a priori energy bound restores stability, and that this regularization effect extends to non-uniform sampling geometries. Our analysis recasts folded-sampling stability as an infinite-dimensional lattice shortest-vector problem, which we resolve via harmonic-analytic tools (the spectral profile of Fourier concentration matrices) and, alternatively, via bounds for integer Tschebyschev polynomials. Our work brings context to recent results on injectivity and encoding guarantees for folded sampling and further supports the empirical success of folded sampling under natural energy constraints.