  Shannon's sampling theorem provides a link between the continuous and the
discrete realms stating that bandlimited signals are uniquely determined by its
values on a discrete set. This theorem is realized in practice using so called
analog--to--digital converters (ADCs). Unlike Shannon's sampling theorem, the
ADCs are limited in dynamic range. Whenever a signal exceeds some preset
threshold, the ADC saturates, resulting in aliasing due to clipping. The goal
of this paper is to analyze an alternative approach that does not suffer from
these problems. Our work is based on recent developments in ADC design, which
allow for ADCs that reset rather than to saturate, thus producing modulo
samples. An open problem that remains is: Given such modulo samples of a
bandlimited function as well as the dynamic range of the ADC, how can the
original signal be recovered and what are the sufficient conditions that
guarantee perfect recovery? In this paper, we prove such sufficiency conditions
and complement them with a stable recovery algorithm. Our results are not
limited to certain amplitude ranges, in fact even the same circuit architecture
allows for the recovery of arbitrary large amplitudes as long as some estimate
of the signal norm is available when recovering. Numerical experiments that
corroborate our theory indeed show that it is possible to perfectly recover
function that takes values that are orders of magnitude higher than the ADC's
threshold.
