Stability in unlimited sampling

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.

Figures (4)

• Figure 1: (a) Ideal sampling (expensive hardware); (b) clipped samples due to ADC saturation; (c) folding architecture - for dense sampling patterns, one can guess how to piece back the original functions, but less redundant sampling presents a challenge.
• Figure 2: The functions $\frac{f}{\gamma}$ (black) and $\gamma f$ (gray) from Lemma \ref{['lem_disc']} sampled around $x_0$.
• Figure 3: Bandlimited functions whose samples are given by the sequences in the proof of Theorem \ref{['thm:two_thirds_binom']}.
• Figure 4: Left: the functions $f_N$, sampled at $0.7{\mathbb Z}$ (black dots) and the measurements with $|\mathcal{M}_{\lambda}f(0.7 k)|<0.25$, $\lambda=\frac{1}{2}$ (crosses). Right: log-plot of the norms of $P_{\alpha{\mathbb Z}} m_N$ (black circles) and the norms of $n-P_{\alpha{\mathbb Z}} m_N$ (gray squares) $1\leq N\leq 30$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 2.1: Instability of folded sampling with unbounded inputs
• Theorem 2.2
• Corollary 2.3
• Theorem 3.1: Qualitative version of the sampling theorem
• Lemma 4.1: Discontinuity of the folded sampling operator with the Euclidean metric
• proof
• Lemma 4.2
• proof
• Lemma 4.3