Identification of Operators on Elementary Locally Compact Abelian Groups
Benedetto, John J
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Measurement of time-variant linear channels is an important problem in communications theory with applications in mobile communications and radar detection. Kailath addressed this problem about half a century ago and developed a spreading criterion for the identifiability of time-variant channels analogous to the band limitation criterion in the classical sampling theory of signals. Roughly speaking, underspread channels are identifiable and overspread channels are not identifiable, where the critical spreading area equals one. Kailath's analysis was later generalized by Bello from rectangular to arbitrary spreading supports. Modern developments in time-frequency analysis provide a natural and powerful framework in which to study the channel measurement problem from a rigorous mathematical standpoint. Pfander and Walnut, building on earlier work by Kozek and Pfander, have developed a sophisticated theory of "operator sampling" or "operator identification" which not only places the work of Kailath and Bello on rigorous footing, but also takes the subject in new directions, revealing connections with other important problems in time-frequency analysis. We expand upon the existing work on operator identification, which is restricted to the real line, and investigate the subject on elementary locally compact abelian groups, which are groups built from the real line, the circle, the integers, and finite abelian groups. Our approach is to axiomatize, as it were, the main ideas which have been developed over the real line, working with lattice subgroups. We are thus able to prove the various identifiability results for operators involving both underspread and overspread conditions in both general and specific cases. For example, we provide a finite dimensional example illustrating a necessary and sufficient condition for identifiability of operators, owing to the insight gleaned from the general theory. In working up to our main results, we set up the quite considerable technical background, bringing some new perspectives to existing ideas and generally filling what we consider to be gaps in the literature.