Stability in unlimited sampling

TL;DR

This paper investigates the stability of reconstructing bandlimited signals from folded samples in analog-to-digital conversion, demonstrating that energy constraints are essential for stable recovery across various sampling schemes.

Contribution

It proves that imposing an a priori energy bound restores stability in folded sampling, extending the analysis to non-uniform sampling and connecting it to lattice and harmonic analysis.

Findings

Stability is inherently lost in equispaced folded sampling without energy constraints.

Energy bounds restore stability in both uniform and non-uniform sampling.

The stability problem is linked to lattice shortest-vector problems and solved using harmonic analysis and polynomial bounds.

Abstract

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.