  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
analog-to-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 perfect recovery is possible and complement them with
a stable recovery algorithm. The sampling density required to guarantee
recovery is independent of the maximum recordable ADC voltage and depends on
the signal bandwidth only. Our recovery guarantees extend to measurements
affected by bounded noise, which includes the case of round-off quantization.
Numerical experiments validate our approach. Applications of the unlimited
sampling paradigm can be found in a number of fields such as signal processing,
communication and imaging.
