The classical sampling theorem, attributed to Whittaker, Shannon, Nyquist, and Kotelnikov, states that a bandlimited function can be recovered from its samples, as long as we use a sufficiently dense sampling grid. Here, we review the recent development of an operator sampling theory which allows for a "widening" of the classical sampling theorem. In this realm, bandlimited functions are replaced by "bandlimited operators". that is, by pseudodifferential operators which have bandlimited Kohn-Nirenberg symbols. Similar to the Nyquist sampling density condition alluded to above, we discuss sufficient and necessary conditions on the bandlimitation of pseudodifferential operators to ensure that they can be recovered by their action on a single distribution. In fact, we show that an operator with Kohn-Nirenberg symbol bandlimited to a Jordan domain of measure less than one can be recovered through its action on a distribution defined on a appropriately chosen sampling grid. Further, an operator with bandlimitation to a Jordan domain of measure larger than one cannot be recovered through its action on any tempered distribution whatsoever, pointing towards a fundamental difference to the classical sampling theorem where a large bandwidth could always be compensated through a sufficiently fine sampling grid. The dichotomy depending on the size of the bandlimitation is related to Heisenberg's uncertainty principle. Further, we discuss an application of this theory to the channel measurement problem for Multiple-Input Multiple-Output (MIMO) channels.

Date Published: 27 September 2007
PDF: 14 pages
Proc. SPIE 6701, Wavelets XII, 67010T (27 September 2007); doi: 10.1117/12.734755

Published in SPIE Proceedings Vol. 6701:
Wavelets XII
Dimitri Van De Ville; Vivek K. Goyal; Manos Papadakis, Editor(s)