Unlimited Sampling via Generalized Thresholding

The Unlimited Sensing Framework (USF) provides an alternative protocol for high dynamic range (HDR) acquisition of real world signals. By incorporating a modulo non-linearity prior to sampling, an HDR signal, which is prone to clipping due to sensor saturation, is folded back into the dynamic range of the analog-to-digital converter (ADC). Thereafter, the modulo samples are algorithmically unfolded to their native dynamic range, thus allowing for recovery of HDR input signals. For bandlimited functions, a sampling density criterion akin to the Shannon-Nyquist theorem guarantees recovery from the modulo samples. Recently, a hardware implementation of the modulo ADC has motivated a generalized acquisition model called modulo-hysteresis that can handle non-idealities observed in practice. The recovery guarantees for this model are currently based on finite difference filters. Such filters can not handle practical scenarios where noise and perturbation play a role. The main goal of this work is to introduce a principled approach that explains the role of general filters for signal recovery. Based on the modulo-hysteresis acquisition model, we formulate input recovery guarantees for thresholding with general filters and give numerical simulations to show cases where the finite difference filter is not an optimal choice for reconstruction.