Difference-Based Recovery for Modulo Sampling: Tightened Bounds and Robustness Guarantees

Conventional analogue-to-digital converters (ADCs) suffer from clipping when inputs exceed their dynamic range. Modulo (unlimited) sampling mitigates this by folding the signal prior to digitisation. However, existing recovery methods are either computationally intensive or require high sampling rates due to loose oversampling factors. Moreover, prior analyses do not account for sampling jitter, which is unavoidable in practice. This paper revisits the difference-based recovery and establishes refined theoretical and practical guarantees. In particular, we first establish an improved Bernstein-type inequality for $N$-th order divided differences of bandlimited signals. We then revise the baseline difference-based recovery algorithm through an $N$-dependent window length. Based on this refinement, we derive recovery conditions for both noiseless and noisy settings. In the noiseless case, the sufficient oversampling factor is reduced from $2\pi e$ to 3, while in the noisy case, explicit robustness guarantees are obtained. We further extend the analysis to non-uniform sampling and establish recovery conditions under bounded sampling jitter. Finally, we identify the second-order setting ($N = 2$) as a practically favourable design choice, offering an effective trade-off between sampling rate and robustness. An FPGA-based prototype demonstrates reliable reconstruction with amplitude expansion up to $\rho = 108$, validating the framework and highlighting the potential of low-complexity, high-performance unlimited sensing.