On Unlimited Sampling and Reconstruction

Abstract

Shannon's sampling theorem is one of the cornerstone topics that is well understood and explored, both mathematically and algorithmically. That said, practical realization of this theorem still suffers from a severe bottleneck due to the fundamental assumption that the samples can span an arbitrary range of amplitudes. In practice, the theorem is realized using so-called analogto-digital converters (ADCs) which clip or saturate whenever the signal amplitude exceeds the maximum recordable ADC 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. It is based on the observation that when a signal is mapped to an appropriate bounded interval via a modulo operation before entering the ADC, the saturation problem no longer exists, but one rather encounters a different type of information loss due to the modulo operation. Such an alternative setup can be implemented, for example, via so-called folding or self-reset ADCs, as they have been proposed in various contexts in the circuit design literature.The key task that we need to accomplish in order to cope with this new type of information loss is to recover a bandlimited signal from its modulo samples. In this paper we derive conditions when this is possible and present an empirically stable recovery algorithm with guaranteed perfect recovery. The sampling density required for recovery is independent of the maximum recordable ADC voltage and depends on the signal bandwidth only.Numerical experiments validate our approach and indeed show that it is possible to perfectly recover functions that take values that are orders of magnitude higher than the ADC's threshold.Applications of the unlimited sampling paradigm can be found in a number of fields such as signal processing, communication and imaging.

Unlimited Sampling from Theory to Practice: Fourier-Prony Recovery and Prototype ADC

“…Note that the sampling criterion is independent of λ and only depends on the bandwidth of the signal. This is somewhat surprising and indeed the reconstruction becomes less stable with respect to noise for large amplitudes [3]. Also, the sampling rate required for local reconstruction from a finite set of modulo samples grows in proportion to the dynamic range of the signal, relative to λ, see for example the local reconstruction theorem discussed in [4], [5].…”

“…1. Hardware demonstration for unlimited sampling [1]- [3]. We show an oscilloscope screen shot plotting a continuous-time function (ground truth, black) and its folded version (pink).…”

“…How does the Unlimited Sampling algorithm [3] work? Let us formally define the centered modulo operation using the mapping…”

“…I N the recent line of work [1]- [3], the authors introduced the Unlimited Sensing Framework (USF). The USF allows for the acquisition of signals that are orders of magnitude larger than the dynamic range of the analog-to-digital converter (ADC) used in the sampling process.…”

“…• On the algorithmic side, one needs to solve the ill-posed inverse problem of recovering a signal from folded measurements. The solution approach of [1]- [3], which capitalizes on certain commutativity properties of the modulo non-linearity, and the associated reconstruction guarantees are reviewed in the next subsection.…”