On Identifiability in Unlimited Sampling

In recent work [1], the authors introduced the Unlimited Sampling framework which establishes that a bandlimited function can be perfectly recovered from a constant-factor oversampling of its modulo samples, hence complementing recent developments in sensor design. This new sensing framework allows to overcome the clipping or saturation problem that is a fundamental limitation common to all formats of conventional digital sensing that rely on Shannon's sampling theorem. In contrast to critical sampling rate of one sample per second, the sampling density criterion prescribed by the Unlimited Sampling Theorem requires a factor of 2πe oversampling. In this paper, we prove identifiability conditions linked with the unlimited sensing setup. Our main result establishes that any sampling rate that is faster than critical sampling allows for one-to-one mapping between a finite energy bandlimited function and its modulo samples. This result is corroborated by experiments and opens further interesting questions around the topic as it relaxes the previously held oversampling criterion.