Shannons sampling theorem provides a link between the continuous and thediscrete realms stating that bandlimited signals are uniquely determined by itsvalues on a discrete set. This theorem is realized in practice using so called analog to digital converters (ADCs). Unlike Shannons 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. In this talk,we 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 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 whenrecovering.
This is joint work with Ayush Bhandari (Imperial College London) and Ramesh Raskar (MIT).
Felix Krahmer received his PhD in Mathematics in 2009 from New York University under the supervision of Percy Deift and Sinan Güntürk. He was a Hausdorff postdoc in the group of Holger Rauhut at the University of Bonn, Germany from 2009-2012. In 2012 he joined the University of Göttingen as a an assistant professor for mathematical data analysis, where he has been awarded an Emmy Noether Junior Research Group. Since 2015 he has been tenure track assistant professor for optimization and data analysis in the department of mathematics at the Technical University of Munich.