On Unlimited Sampling and Reconstruction

Shannon's sampling theorem, at the heart of digital signal processing, is well understood and explored. However, its practical realization still suffers from a fundamental bottleneck due to dynamic range limitations of the underlying analog-to-digital converters (ADCs). This results in clipping or saturation for signal amplitudes exceeding their maximum recordable voltage thus leading to a significant information loss. In this paper, we develop an alternative paradigm for sensing and recovery, called the Unlimited Sampling Framework. The key observation is that applying a modulo operation to the signal before the ADC prevents saturation; instead, one encounters a different type of information loss. Such a setup can be implemented, for example, via so-called folding or self-reset ADCs, as proposed in various contexts in the circuit design literature. The key challenge for this new type of information loss is to recover a bandlimited signal from its modulo samples. We derive conditions when perfect recovery is possible and complement them with a stable recovery algorithm. The required sampling density is independent of the maximum recordable ADC voltage and depends on the signal bandwidth only. Our guarantees extend to measurements affected by bounded noise, which includes round-off quantization. Numerical experiments validate our approach. For example, it is possible to recover functions with amplitudes orders of magnitude higher than the ADC's threshold from quantized modulo samples up to the unavoidable quantization error. Applications of the unlimited sampling paradigm can be found in a number of fields such as signal processing, communication and imaging.