Sampling and reconstruction of operators

TL;DR

This paper develops explicit formulas and conditions for sampling and reconstructing operators with bandlimited symbols, extending classical results to more general and unknown bandlimiting sets, with applications in communication channels.

Contribution

It generalizes operator sampling formulas to arbitrary bandlimiting sets and establishes necessary conditions for sampling rates based on set geometry.

Findings

Reconstruction formulas for bandlimited operators are derived.

Sampling is possible for operators with unknown bandlimiting sets of area less than one-half.

Necessary and sufficient conditions depend on the size and shape of the bandlimiting set.

Abstract

We study the recovery of operators with bandlimited Kohn-Nirenberg symbol from the action of such operators on a weighted impulse train, a procedure we refer to as operator sampling. Kailath, and later Kozek and the authors have shown that operator sampling is possible if the symbol of the operator is bandlimited to a set with area less than one. In this paper we develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions. We give necessary and sufficient conditions on the sampling rate that depend on size and geometry of the bandlimiting set. Moreover, we show that under mild geometric conditions, classes of operators bandlimited to an unknown set of area less than one-half permit sampling and reconstruction. A similar result considering unknown sets of area less than one was independently achieved by Heckel and Boelcskei. Operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly-time-varying impulse response. The results in this paper are rooted in work by Bello and Kailath in the 1960s.