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.

翻译：折叠采样通过在模阈值下对样本进行约减，替代了模数转换器中的削波处理，从而避免饱和伪影。本文研究了带限函数从折叠样本中重建的问题，并证明对于等间距采样模式，重建问题本质上是非稳定的。随后我们表明，施加任意先验能量约束可恢复稳定性，且该正则化效应可推广至非均匀采样几何结构。我们的分析将折叠采样稳定性转化为无限维格最短向量问题，通过调和分析工具（傅里叶浓度矩阵的谱剖面）以及整数切比雪夫多项式的界值两种途径加以解决。本研究为近期关于折叠采样可逆性与编码保证的成果提供了理论背景，并进一步支持了折叠采样在自然能量约束下取得的实证成功。